Chapter 13 Forward Rate Modeling This chapter is concerned with interest rate modeling, in which the mean reversion property plays an important role. We consider the main short rate models (Vasicek, CIR, CEV, affine models) and the computation of fixed income products, such as bond prices, in such models. Next we consider the modeling of forward rates in the HJM and BGM models, as well as in two- factor models. 13.1 Short Term Models and Mean Reversion Vasicek Model The first model to capture the mean reversion property of interest rates, a property not possessed by geometric Brownian motion, is the Vasicek [Vaš77] model, which is based on the Ornstein-Uhlenbeck process. Here, the short term interest rate process (r t ) t∈R+ solves the equation dr t =(a - br t )dt + σdB t , (13.1) where a, σ ∈ R, b> 0, and (B t ) t∈R+ is a standard Brownian motion, with solution r t = r 0 e -bt + a b (1 - e -bt )+ σ w t 0 e -b(t-s) dB s , t ∈ R + . (13.2) The probability distribution of r t is Gaussian at all times t, with mean IE[r t ]= r 0 e -bt + a b (1 - e -bt ), and variance Var[r t ]= σ 2 w t 0 (e -b(t-s) ) 2 ds = σ 2 w t 0 e -2bs ds = σ 2 2b (1 - e -2bt ), t ∈ R + , 411
Transcript
Chapter 13Forward Rate Modeling
This chapter is concerned with interest rate modeling in which the meanreversion property plays an important role We consider the main short ratemodels (Vasicek CIR CEV affine models) and the computation of fixedincome products such as bond prices in such models Next we consider themodeling of forward rates in the HJM and BGM models as well as in two-factor models
131 Short Term Models and Mean Reversion
Vasicek Model
The first model to capture the mean reversion property of interest rates aproperty not possessed by geometric Brownian motion is the Vasicek [Vaš77]model which is based on the Ornstein-Uhlenbeck process Here the shortterm interest rate process (rt)tisinR+ solves the equation
drt = (aminus brt)dt+ σdBt (131)
where a σ isin R b gt 0 and (Bt)tisinR+ is a standard Brownian motion withsolution
rt = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs t isin R+ (132)
The probability distribution of rt is Gaussian at all times t with mean
IE[rt] = r0 eminusbt + a
b(1minus eminusbt)
and variance
Var[rt] = σ2w t
0( eminusb(tminuss))2ds = σ2
w t
0eminus2bsds = σ2
2b (1minus eminus2bt) t isin R+
411
N Privault
iert N
(r0 eminusbt + a
b(1minus eminusbt) σ
2
2b (1minus eminus2bt)) t gt 0
In large time t with b gt 0 we have
limtrarrinfin
IE[rt] = a
band lim
trarrinfinVar[rt] = σ2
2b
and this distribution converges to the Gaussian N (ab σ2(2b)) distributionwhich is also the invariant (or stationary) distribution of (rt)tisinR+ and theprocess tends to revert to its long term mean ab = limtrarrinfin IE[rt]
Figure 131 presents a random simulation of t 7minusrarr rt in the Vasicek modelwith r0 = 3 and shows the mean reverting property of the process withrespect to ab = 25
-2
-1
0
1
2
3
4
5
6
7
8
0 01 02 03 04 05 06 07 08 09 1
ab
rt
()
t
Fig 131 Graph of the Vasicek short rate t 7rarr rt with a = 25 b = 1 and σ = 01
As can be checked from the simulation of Figure 131 the value of rt in theVasicek model may become negative due to its Gaussian distribution Al-though real interest rates can sometimes fall below zero this can be regardedas a potential drawback of the Vasicek model
Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) [CIR85] model brings a solution to the posi-tivity problem encountered with the Vasicek model by the use the nonlinearstochastic differential equation
drt = β(αminus rt)dt+ σradicrtdBt α gt 0 β gt 0
The probability distribution of rt at time t gt 0 admits the noncentral Chisquare probability density function given by
412
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
ft(x) (133)
= 2βσ2(1minus eminusβt) exp
(minus2β(x+ r0 eminusβt)
σ2(1minus eminusβt)
)(x
r0 eminusβt
)αβσ2minus12I2αβσ2minus1
(4βradicr0x eminusβt
σ2(1minus eminusβt)
)
x gt 0 where
Iλ(z) =(z
2
)λ infinsumk=0
(z24)kkΓ (λ+ k + 1) z isin R
is the modified Bessel function of the first kind cf Corollary 24 in [AL05]Note that ft(x) is not defined at x = 0 if αβσ2 minus 12 lt 0 ie σ2 gt 2αβ inwhich case the probability distribution of rt admits a point mass at x = 0On the other hand rt remains almost surely strictly positive under the Fellercondition 2αβ gt σ2 cf the study of the associated probability density inLemma 4 of [Fel51]
Figure 132 presents a random simulation of t 7minusrarr rt in the CIR model inthe case σ2 gt 2αβ in which the process is mean reverting with respect toα = 25 and has a nonzero probability of hitting 0
0
1
2
3
4
5
6
7
8
0 01 02 03 04 05 06 07 08 09 1
α=25
r
t (
)
t
Fig 132 Graph of the CIR short rate t 7rarr rt with α = 25 β = 1 and σ = 13
In large time t using the asymptotics
Iλ(z) zrarr01
Γ (λ+ 1)
(z2
)λ
the density (133) becomes the Gamma density
f(x) = limtrarrinfin
ft(x) = 1Γ (2αβσ2)
(2βσ2
)2αβσ2
xminus1+2αβσ2eminus2βxσ2
x gt 0
(134)
413
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
with shape parameter 2αβσ2 and scale parameter σ2(2β) which is also theinvariant distribution of rt
Other classical mean reverting models include the Courtadon (1982) model
drt = β(αminus rt)dt+ σrtdBt
where α β σ are nonnegative and the exponential Vasicek model
drt = rt(η minus a log rt)dt+ σrtdBt
where a η σ gt 0 cf Exercises 414 and 415
Constant Elasticity of Variance (CEV)
Constant Elasticity of Variance models are designed to take into accountnonconstant volatilities that can vary as a power of the underlying assetThe Marsh-Rosenfeld (1983) model
drt = (βrminus(1minusγ)t + αrt)dt+ σr
γ2t dBt (135)
where α β σ γ are constants and β is the variance (or diffusion) elasticitycoefficient covers most of the CEV models Denoting by v(r) = σrγ2 thediffusion coefficient in (135) constant elasticity refers to the constant ratio
dv(r)v(r)drr
= rvprime(r)v(r) = d log v(r)
d log r = d log rγ2d log r = γ
2
between the relative change dv(r)v(r) in the variance v(r) and the relativechange drr in r
For γ = 1 this is the CIR model and for β = 0 we get the standard CEVmodel
drt = αrtdt+ σrγ2t dBt
If γ = 2 this yields the Dothan [Dot78] model
drt = αrtdt+ σrtdBt
which is a version of geometric Brownian motion used for short term interestrate modeling
Time-dependent affine Models
The class of short rate interest rate models admits a number of generalizationsthat can be found in the references quoted in the introduction of this chapter
414
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
among which is the class of affine models of the form
Such models are called affine because the associated zero-coupon bonds canbe priced using an affine PDE of the type (1316) below as will be seen afterProposition 132
Affine models also include the Ho-Lee model
drt = θ(t)dt+ σdBt
where θ(t) is a deterministic function of time as an extension of the Mertonmodel drt = θdt+σdBt and the Hull-White model [HW90] cf Section 131
drt = (θ(t)minus α(t)rt)dt+ σ(t)dBt
which is itself a time-dependent extension of the Vasicek model
132 Calibration of the Vasicek model
The Vasicek equation (131) ie
drt = (aminus brt)dt+ σdBt
can be discretized according to a discrete-time sequence (tk)k=01n as
rtk+1 minus rtk = (aminus brtk)∆t+ σZk k isin N
where∆t = tk+1minustk and (Zk)kgt0 is a Gaussian white noise with variance∆tie a sequence of independent centered and identically distributed N (0 ∆t)Gaussian random variables
We find
rtk+1 = rtk + (aminus brtk)∆t+ σZk = a∆t+ (1minus b∆t)rtk + σZk k isin N
Based on a set (rtk)k=0n of market data we can minimize the residual
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2over a and b using Ordinary Least Square (OLS) regression For this compute
part
parta
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2 415
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minus2∆t(minusan∆t+
nminus1sumk=0
(rtk+1 minus (1minus b∆t)rtk
))= 0
and
part
partb
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2= ∆t
nminus1sumk=0
rtk(minusa∆t+ rtk+1 minus (1minus b∆t)rtk
)= ∆t
nminus1sumk=0
rtk
(rtk+1 minus (1minus b∆t)rtk + 1
n
nminus1suml=0
(rtl+1 minus (1minus b∆t)rtl
))= 0
This leads to an estimate the parameters a and b respectively as the empiricalmean and covariance of (rtk)k=01n ie
a∆t = 1n
nminus1sumk=0
rtk+1 minus1n
(1minus b∆t)nminus1sumk=0
rtk
and
1minus b∆t =
nminus1sumk=0
rtk rtk+1 minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl+1
nminus1sumk=0
rtk rtk minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl
=
nminus1sumk=0
(rtk minus
1n
nminus1suml=0
rtl
)(rtk+1 minus
1n
nminus1suml=0
rtl+1
)nminus1sumk=0
(rtk minus
1n
nminus1sumk=0
rtk
)2
This also yields
σ2∆t = Var[σZk] = Var[rtk+1 minus (1minus b∆t)rtk minus a∆t
] k isin N
hence σ can be estimated as
σ2∆t = 1n
nminus1sumk=0
(rtk+1 minus rtk(1minus b∆t)minus a∆t
)2
416
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Defining rtk = rtk minus ab k isin N we have
rtk+1 = rtk+1 minus ab= rtk minus ab+ (aminus brtk)∆t+ σZk
= rtk minus abminus b(rtk minus ab)∆t+ σZk
= rtk minus brtk∆t+ σZk
= (1minus b∆t)rtk + σZk k isin N
In other words the sequence (rtk)kisinisinN is modeled according to an autore-gressive AR(1) time series in which the current state Xn of the system isexpressed as the linear combination
The next Figure 133 displays the yield of the 10 Year Treasury Note on theChicago Board Options Exchange (CBOE) Treasury notes usually have amaturity between one and 10 years whereas treasury bonds have maturitiesbeyond 10 years)
The next code is generating Vasicek random samples according to the AR(1)time series (137)
for (i in 1100) arsimlt-arimasim(model=list(ar=c(b))nstart=100n)y=ratek[1]+ab+sigmaarsimtime lt- asPOSIXct(time(TNX) format = Y-m-d)yield lt- xts(x = y orderby = time)chartSeries(yieldupcol=bluetheme=white)Syssleep(05)
418
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
iert N
(r0 eminusbt + a
b(1minus eminusbt) σ
2
2b (1minus eminus2bt)) t gt 0
In large time t with b gt 0 we have
limtrarrinfin
IE[rt] = a
band lim
trarrinfinVar[rt] = σ2
2b
and this distribution converges to the Gaussian N (ab σ2(2b)) distributionwhich is also the invariant (or stationary) distribution of (rt)tisinR+ and theprocess tends to revert to its long term mean ab = limtrarrinfin IE[rt]
Figure 131 presents a random simulation of t 7minusrarr rt in the Vasicek modelwith r0 = 3 and shows the mean reverting property of the process withrespect to ab = 25
-2
-1
0
1
2
3
4
5
6
7
8
0 01 02 03 04 05 06 07 08 09 1
ab
rt
()
t
Fig 131 Graph of the Vasicek short rate t 7rarr rt with a = 25 b = 1 and σ = 01
As can be checked from the simulation of Figure 131 the value of rt in theVasicek model may become negative due to its Gaussian distribution Al-though real interest rates can sometimes fall below zero this can be regardedas a potential drawback of the Vasicek model
Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) [CIR85] model brings a solution to the posi-tivity problem encountered with the Vasicek model by the use the nonlinearstochastic differential equation
drt = β(αminus rt)dt+ σradicrtdBt α gt 0 β gt 0
The probability distribution of rt at time t gt 0 admits the noncentral Chisquare probability density function given by
412
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
ft(x) (133)
= 2βσ2(1minus eminusβt) exp
(minus2β(x+ r0 eminusβt)
σ2(1minus eminusβt)
)(x
r0 eminusβt
)αβσ2minus12I2αβσ2minus1
(4βradicr0x eminusβt
σ2(1minus eminusβt)
)
x gt 0 where
Iλ(z) =(z
2
)λ infinsumk=0
(z24)kkΓ (λ+ k + 1) z isin R
is the modified Bessel function of the first kind cf Corollary 24 in [AL05]Note that ft(x) is not defined at x = 0 if αβσ2 minus 12 lt 0 ie σ2 gt 2αβ inwhich case the probability distribution of rt admits a point mass at x = 0On the other hand rt remains almost surely strictly positive under the Fellercondition 2αβ gt σ2 cf the study of the associated probability density inLemma 4 of [Fel51]
Figure 132 presents a random simulation of t 7minusrarr rt in the CIR model inthe case σ2 gt 2αβ in which the process is mean reverting with respect toα = 25 and has a nonzero probability of hitting 0
0
1
2
3
4
5
6
7
8
0 01 02 03 04 05 06 07 08 09 1
α=25
r
t (
)
t
Fig 132 Graph of the CIR short rate t 7rarr rt with α = 25 β = 1 and σ = 13
In large time t using the asymptotics
Iλ(z) zrarr01
Γ (λ+ 1)
(z2
)λ
the density (133) becomes the Gamma density
f(x) = limtrarrinfin
ft(x) = 1Γ (2αβσ2)
(2βσ2
)2αβσ2
xminus1+2αβσ2eminus2βxσ2
x gt 0
(134)
413
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
with shape parameter 2αβσ2 and scale parameter σ2(2β) which is also theinvariant distribution of rt
Other classical mean reverting models include the Courtadon (1982) model
drt = β(αminus rt)dt+ σrtdBt
where α β σ are nonnegative and the exponential Vasicek model
drt = rt(η minus a log rt)dt+ σrtdBt
where a η σ gt 0 cf Exercises 414 and 415
Constant Elasticity of Variance (CEV)
Constant Elasticity of Variance models are designed to take into accountnonconstant volatilities that can vary as a power of the underlying assetThe Marsh-Rosenfeld (1983) model
drt = (βrminus(1minusγ)t + αrt)dt+ σr
γ2t dBt (135)
where α β σ γ are constants and β is the variance (or diffusion) elasticitycoefficient covers most of the CEV models Denoting by v(r) = σrγ2 thediffusion coefficient in (135) constant elasticity refers to the constant ratio
dv(r)v(r)drr
= rvprime(r)v(r) = d log v(r)
d log r = d log rγ2d log r = γ
2
between the relative change dv(r)v(r) in the variance v(r) and the relativechange drr in r
For γ = 1 this is the CIR model and for β = 0 we get the standard CEVmodel
drt = αrtdt+ σrγ2t dBt
If γ = 2 this yields the Dothan [Dot78] model
drt = αrtdt+ σrtdBt
which is a version of geometric Brownian motion used for short term interestrate modeling
Time-dependent affine Models
The class of short rate interest rate models admits a number of generalizationsthat can be found in the references quoted in the introduction of this chapter
414
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
among which is the class of affine models of the form
Such models are called affine because the associated zero-coupon bonds canbe priced using an affine PDE of the type (1316) below as will be seen afterProposition 132
Affine models also include the Ho-Lee model
drt = θ(t)dt+ σdBt
where θ(t) is a deterministic function of time as an extension of the Mertonmodel drt = θdt+σdBt and the Hull-White model [HW90] cf Section 131
drt = (θ(t)minus α(t)rt)dt+ σ(t)dBt
which is itself a time-dependent extension of the Vasicek model
132 Calibration of the Vasicek model
The Vasicek equation (131) ie
drt = (aminus brt)dt+ σdBt
can be discretized according to a discrete-time sequence (tk)k=01n as
rtk+1 minus rtk = (aminus brtk)∆t+ σZk k isin N
where∆t = tk+1minustk and (Zk)kgt0 is a Gaussian white noise with variance∆tie a sequence of independent centered and identically distributed N (0 ∆t)Gaussian random variables
We find
rtk+1 = rtk + (aminus brtk)∆t+ σZk = a∆t+ (1minus b∆t)rtk + σZk k isin N
Based on a set (rtk)k=0n of market data we can minimize the residual
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2over a and b using Ordinary Least Square (OLS) regression For this compute
part
parta
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2 415
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minus2∆t(minusan∆t+
nminus1sumk=0
(rtk+1 minus (1minus b∆t)rtk
))= 0
and
part
partb
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2= ∆t
nminus1sumk=0
rtk(minusa∆t+ rtk+1 minus (1minus b∆t)rtk
)= ∆t
nminus1sumk=0
rtk
(rtk+1 minus (1minus b∆t)rtk + 1
n
nminus1suml=0
(rtl+1 minus (1minus b∆t)rtl
))= 0
This leads to an estimate the parameters a and b respectively as the empiricalmean and covariance of (rtk)k=01n ie
a∆t = 1n
nminus1sumk=0
rtk+1 minus1n
(1minus b∆t)nminus1sumk=0
rtk
and
1minus b∆t =
nminus1sumk=0
rtk rtk+1 minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl+1
nminus1sumk=0
rtk rtk minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl
=
nminus1sumk=0
(rtk minus
1n
nminus1suml=0
rtl
)(rtk+1 minus
1n
nminus1suml=0
rtl+1
)nminus1sumk=0
(rtk minus
1n
nminus1sumk=0
rtk
)2
This also yields
σ2∆t = Var[σZk] = Var[rtk+1 minus (1minus b∆t)rtk minus a∆t
] k isin N
hence σ can be estimated as
σ2∆t = 1n
nminus1sumk=0
(rtk+1 minus rtk(1minus b∆t)minus a∆t
)2
416
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Defining rtk = rtk minus ab k isin N we have
rtk+1 = rtk+1 minus ab= rtk minus ab+ (aminus brtk)∆t+ σZk
= rtk minus abminus b(rtk minus ab)∆t+ σZk
= rtk minus brtk∆t+ σZk
= (1minus b∆t)rtk + σZk k isin N
In other words the sequence (rtk)kisinisinN is modeled according to an autore-gressive AR(1) time series in which the current state Xn of the system isexpressed as the linear combination
The next Figure 133 displays the yield of the 10 Year Treasury Note on theChicago Board Options Exchange (CBOE) Treasury notes usually have amaturity between one and 10 years whereas treasury bonds have maturitiesbeyond 10 years)
The next code is generating Vasicek random samples according to the AR(1)time series (137)
for (i in 1100) arsimlt-arimasim(model=list(ar=c(b))nstart=100n)y=ratek[1]+ab+sigmaarsimtime lt- asPOSIXct(time(TNX) format = Y-m-d)yield lt- xts(x = y orderby = time)chartSeries(yieldupcol=bluetheme=white)Syssleep(05)
418
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
ft(x) (133)
= 2βσ2(1minus eminusβt) exp
(minus2β(x+ r0 eminusβt)
σ2(1minus eminusβt)
)(x
r0 eminusβt
)αβσ2minus12I2αβσ2minus1
(4βradicr0x eminusβt
σ2(1minus eminusβt)
)
x gt 0 where
Iλ(z) =(z
2
)λ infinsumk=0
(z24)kkΓ (λ+ k + 1) z isin R
is the modified Bessel function of the first kind cf Corollary 24 in [AL05]Note that ft(x) is not defined at x = 0 if αβσ2 minus 12 lt 0 ie σ2 gt 2αβ inwhich case the probability distribution of rt admits a point mass at x = 0On the other hand rt remains almost surely strictly positive under the Fellercondition 2αβ gt σ2 cf the study of the associated probability density inLemma 4 of [Fel51]
Figure 132 presents a random simulation of t 7minusrarr rt in the CIR model inthe case σ2 gt 2αβ in which the process is mean reverting with respect toα = 25 and has a nonzero probability of hitting 0
0
1
2
3
4
5
6
7
8
0 01 02 03 04 05 06 07 08 09 1
α=25
r
t (
)
t
Fig 132 Graph of the CIR short rate t 7rarr rt with α = 25 β = 1 and σ = 13
In large time t using the asymptotics
Iλ(z) zrarr01
Γ (λ+ 1)
(z2
)λ
the density (133) becomes the Gamma density
f(x) = limtrarrinfin
ft(x) = 1Γ (2αβσ2)
(2βσ2
)2αβσ2
xminus1+2αβσ2eminus2βxσ2
x gt 0
(134)
413
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
with shape parameter 2αβσ2 and scale parameter σ2(2β) which is also theinvariant distribution of rt
Other classical mean reverting models include the Courtadon (1982) model
drt = β(αminus rt)dt+ σrtdBt
where α β σ are nonnegative and the exponential Vasicek model
drt = rt(η minus a log rt)dt+ σrtdBt
where a η σ gt 0 cf Exercises 414 and 415
Constant Elasticity of Variance (CEV)
Constant Elasticity of Variance models are designed to take into accountnonconstant volatilities that can vary as a power of the underlying assetThe Marsh-Rosenfeld (1983) model
drt = (βrminus(1minusγ)t + αrt)dt+ σr
γ2t dBt (135)
where α β σ γ are constants and β is the variance (or diffusion) elasticitycoefficient covers most of the CEV models Denoting by v(r) = σrγ2 thediffusion coefficient in (135) constant elasticity refers to the constant ratio
dv(r)v(r)drr
= rvprime(r)v(r) = d log v(r)
d log r = d log rγ2d log r = γ
2
between the relative change dv(r)v(r) in the variance v(r) and the relativechange drr in r
For γ = 1 this is the CIR model and for β = 0 we get the standard CEVmodel
drt = αrtdt+ σrγ2t dBt
If γ = 2 this yields the Dothan [Dot78] model
drt = αrtdt+ σrtdBt
which is a version of geometric Brownian motion used for short term interestrate modeling
Time-dependent affine Models
The class of short rate interest rate models admits a number of generalizationsthat can be found in the references quoted in the introduction of this chapter
414
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
among which is the class of affine models of the form
Such models are called affine because the associated zero-coupon bonds canbe priced using an affine PDE of the type (1316) below as will be seen afterProposition 132
Affine models also include the Ho-Lee model
drt = θ(t)dt+ σdBt
where θ(t) is a deterministic function of time as an extension of the Mertonmodel drt = θdt+σdBt and the Hull-White model [HW90] cf Section 131
drt = (θ(t)minus α(t)rt)dt+ σ(t)dBt
which is itself a time-dependent extension of the Vasicek model
132 Calibration of the Vasicek model
The Vasicek equation (131) ie
drt = (aminus brt)dt+ σdBt
can be discretized according to a discrete-time sequence (tk)k=01n as
rtk+1 minus rtk = (aminus brtk)∆t+ σZk k isin N
where∆t = tk+1minustk and (Zk)kgt0 is a Gaussian white noise with variance∆tie a sequence of independent centered and identically distributed N (0 ∆t)Gaussian random variables
We find
rtk+1 = rtk + (aminus brtk)∆t+ σZk = a∆t+ (1minus b∆t)rtk + σZk k isin N
Based on a set (rtk)k=0n of market data we can minimize the residual
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2over a and b using Ordinary Least Square (OLS) regression For this compute
part
parta
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2 415
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minus2∆t(minusan∆t+
nminus1sumk=0
(rtk+1 minus (1minus b∆t)rtk
))= 0
and
part
partb
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2= ∆t
nminus1sumk=0
rtk(minusa∆t+ rtk+1 minus (1minus b∆t)rtk
)= ∆t
nminus1sumk=0
rtk
(rtk+1 minus (1minus b∆t)rtk + 1
n
nminus1suml=0
(rtl+1 minus (1minus b∆t)rtl
))= 0
This leads to an estimate the parameters a and b respectively as the empiricalmean and covariance of (rtk)k=01n ie
a∆t = 1n
nminus1sumk=0
rtk+1 minus1n
(1minus b∆t)nminus1sumk=0
rtk
and
1minus b∆t =
nminus1sumk=0
rtk rtk+1 minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl+1
nminus1sumk=0
rtk rtk minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl
=
nminus1sumk=0
(rtk minus
1n
nminus1suml=0
rtl
)(rtk+1 minus
1n
nminus1suml=0
rtl+1
)nminus1sumk=0
(rtk minus
1n
nminus1sumk=0
rtk
)2
This also yields
σ2∆t = Var[σZk] = Var[rtk+1 minus (1minus b∆t)rtk minus a∆t
] k isin N
hence σ can be estimated as
σ2∆t = 1n
nminus1sumk=0
(rtk+1 minus rtk(1minus b∆t)minus a∆t
)2
416
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Defining rtk = rtk minus ab k isin N we have
rtk+1 = rtk+1 minus ab= rtk minus ab+ (aminus brtk)∆t+ σZk
= rtk minus abminus b(rtk minus ab)∆t+ σZk
= rtk minus brtk∆t+ σZk
= (1minus b∆t)rtk + σZk k isin N
In other words the sequence (rtk)kisinisinN is modeled according to an autore-gressive AR(1) time series in which the current state Xn of the system isexpressed as the linear combination
The next Figure 133 displays the yield of the 10 Year Treasury Note on theChicago Board Options Exchange (CBOE) Treasury notes usually have amaturity between one and 10 years whereas treasury bonds have maturitiesbeyond 10 years)
The next code is generating Vasicek random samples according to the AR(1)time series (137)
for (i in 1100) arsimlt-arimasim(model=list(ar=c(b))nstart=100n)y=ratek[1]+ab+sigmaarsimtime lt- asPOSIXct(time(TNX) format = Y-m-d)yield lt- xts(x = y orderby = time)chartSeries(yieldupcol=bluetheme=white)Syssleep(05)
418
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
with shape parameter 2αβσ2 and scale parameter σ2(2β) which is also theinvariant distribution of rt
Other classical mean reverting models include the Courtadon (1982) model
drt = β(αminus rt)dt+ σrtdBt
where α β σ are nonnegative and the exponential Vasicek model
drt = rt(η minus a log rt)dt+ σrtdBt
where a η σ gt 0 cf Exercises 414 and 415
Constant Elasticity of Variance (CEV)
Constant Elasticity of Variance models are designed to take into accountnonconstant volatilities that can vary as a power of the underlying assetThe Marsh-Rosenfeld (1983) model
drt = (βrminus(1minusγ)t + αrt)dt+ σr
γ2t dBt (135)
where α β σ γ are constants and β is the variance (or diffusion) elasticitycoefficient covers most of the CEV models Denoting by v(r) = σrγ2 thediffusion coefficient in (135) constant elasticity refers to the constant ratio
dv(r)v(r)drr
= rvprime(r)v(r) = d log v(r)
d log r = d log rγ2d log r = γ
2
between the relative change dv(r)v(r) in the variance v(r) and the relativechange drr in r
For γ = 1 this is the CIR model and for β = 0 we get the standard CEVmodel
drt = αrtdt+ σrγ2t dBt
If γ = 2 this yields the Dothan [Dot78] model
drt = αrtdt+ σrtdBt
which is a version of geometric Brownian motion used for short term interestrate modeling
Time-dependent affine Models
The class of short rate interest rate models admits a number of generalizationsthat can be found in the references quoted in the introduction of this chapter
414
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
among which is the class of affine models of the form
Such models are called affine because the associated zero-coupon bonds canbe priced using an affine PDE of the type (1316) below as will be seen afterProposition 132
Affine models also include the Ho-Lee model
drt = θ(t)dt+ σdBt
where θ(t) is a deterministic function of time as an extension of the Mertonmodel drt = θdt+σdBt and the Hull-White model [HW90] cf Section 131
drt = (θ(t)minus α(t)rt)dt+ σ(t)dBt
which is itself a time-dependent extension of the Vasicek model
132 Calibration of the Vasicek model
The Vasicek equation (131) ie
drt = (aminus brt)dt+ σdBt
can be discretized according to a discrete-time sequence (tk)k=01n as
rtk+1 minus rtk = (aminus brtk)∆t+ σZk k isin N
where∆t = tk+1minustk and (Zk)kgt0 is a Gaussian white noise with variance∆tie a sequence of independent centered and identically distributed N (0 ∆t)Gaussian random variables
We find
rtk+1 = rtk + (aminus brtk)∆t+ σZk = a∆t+ (1minus b∆t)rtk + σZk k isin N
Based on a set (rtk)k=0n of market data we can minimize the residual
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2over a and b using Ordinary Least Square (OLS) regression For this compute
part
parta
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2 415
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minus2∆t(minusan∆t+
nminus1sumk=0
(rtk+1 minus (1minus b∆t)rtk
))= 0
and
part
partb
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2= ∆t
nminus1sumk=0
rtk(minusa∆t+ rtk+1 minus (1minus b∆t)rtk
)= ∆t
nminus1sumk=0
rtk
(rtk+1 minus (1minus b∆t)rtk + 1
n
nminus1suml=0
(rtl+1 minus (1minus b∆t)rtl
))= 0
This leads to an estimate the parameters a and b respectively as the empiricalmean and covariance of (rtk)k=01n ie
a∆t = 1n
nminus1sumk=0
rtk+1 minus1n
(1minus b∆t)nminus1sumk=0
rtk
and
1minus b∆t =
nminus1sumk=0
rtk rtk+1 minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl+1
nminus1sumk=0
rtk rtk minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl
=
nminus1sumk=0
(rtk minus
1n
nminus1suml=0
rtl
)(rtk+1 minus
1n
nminus1suml=0
rtl+1
)nminus1sumk=0
(rtk minus
1n
nminus1sumk=0
rtk
)2
This also yields
σ2∆t = Var[σZk] = Var[rtk+1 minus (1minus b∆t)rtk minus a∆t
] k isin N
hence σ can be estimated as
σ2∆t = 1n
nminus1sumk=0
(rtk+1 minus rtk(1minus b∆t)minus a∆t
)2
416
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Defining rtk = rtk minus ab k isin N we have
rtk+1 = rtk+1 minus ab= rtk minus ab+ (aminus brtk)∆t+ σZk
= rtk minus abminus b(rtk minus ab)∆t+ σZk
= rtk minus brtk∆t+ σZk
= (1minus b∆t)rtk + σZk k isin N
In other words the sequence (rtk)kisinisinN is modeled according to an autore-gressive AR(1) time series in which the current state Xn of the system isexpressed as the linear combination
The next Figure 133 displays the yield of the 10 Year Treasury Note on theChicago Board Options Exchange (CBOE) Treasury notes usually have amaturity between one and 10 years whereas treasury bonds have maturitiesbeyond 10 years)
The next code is generating Vasicek random samples according to the AR(1)time series (137)
for (i in 1100) arsimlt-arimasim(model=list(ar=c(b))nstart=100n)y=ratek[1]+ab+sigmaarsimtime lt- asPOSIXct(time(TNX) format = Y-m-d)yield lt- xts(x = y orderby = time)chartSeries(yieldupcol=bluetheme=white)Syssleep(05)
418
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
among which is the class of affine models of the form
Such models are called affine because the associated zero-coupon bonds canbe priced using an affine PDE of the type (1316) below as will be seen afterProposition 132
Affine models also include the Ho-Lee model
drt = θ(t)dt+ σdBt
where θ(t) is a deterministic function of time as an extension of the Mertonmodel drt = θdt+σdBt and the Hull-White model [HW90] cf Section 131
drt = (θ(t)minus α(t)rt)dt+ σ(t)dBt
which is itself a time-dependent extension of the Vasicek model
132 Calibration of the Vasicek model
The Vasicek equation (131) ie
drt = (aminus brt)dt+ σdBt
can be discretized according to a discrete-time sequence (tk)k=01n as
rtk+1 minus rtk = (aminus brtk)∆t+ σZk k isin N
where∆t = tk+1minustk and (Zk)kgt0 is a Gaussian white noise with variance∆tie a sequence of independent centered and identically distributed N (0 ∆t)Gaussian random variables
We find
rtk+1 = rtk + (aminus brtk)∆t+ σZk = a∆t+ (1minus b∆t)rtk + σZk k isin N
Based on a set (rtk)k=0n of market data we can minimize the residual
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2over a and b using Ordinary Least Square (OLS) regression For this compute
part
parta
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2 415
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minus2∆t(minusan∆t+
nminus1sumk=0
(rtk+1 minus (1minus b∆t)rtk
))= 0
and
part
partb
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2= ∆t
nminus1sumk=0
rtk(minusa∆t+ rtk+1 minus (1minus b∆t)rtk
)= ∆t
nminus1sumk=0
rtk
(rtk+1 minus (1minus b∆t)rtk + 1
n
nminus1suml=0
(rtl+1 minus (1minus b∆t)rtl
))= 0
This leads to an estimate the parameters a and b respectively as the empiricalmean and covariance of (rtk)k=01n ie
a∆t = 1n
nminus1sumk=0
rtk+1 minus1n
(1minus b∆t)nminus1sumk=0
rtk
and
1minus b∆t =
nminus1sumk=0
rtk rtk+1 minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl+1
nminus1sumk=0
rtk rtk minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl
=
nminus1sumk=0
(rtk minus
1n
nminus1suml=0
rtl
)(rtk+1 minus
1n
nminus1suml=0
rtl+1
)nminus1sumk=0
(rtk minus
1n
nminus1sumk=0
rtk
)2
This also yields
σ2∆t = Var[σZk] = Var[rtk+1 minus (1minus b∆t)rtk minus a∆t
] k isin N
hence σ can be estimated as
σ2∆t = 1n
nminus1sumk=0
(rtk+1 minus rtk(1minus b∆t)minus a∆t
)2
416
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Defining rtk = rtk minus ab k isin N we have
rtk+1 = rtk+1 minus ab= rtk minus ab+ (aminus brtk)∆t+ σZk
= rtk minus abminus b(rtk minus ab)∆t+ σZk
= rtk minus brtk∆t+ σZk
= (1minus b∆t)rtk + σZk k isin N
In other words the sequence (rtk)kisinisinN is modeled according to an autore-gressive AR(1) time series in which the current state Xn of the system isexpressed as the linear combination
The next Figure 133 displays the yield of the 10 Year Treasury Note on theChicago Board Options Exchange (CBOE) Treasury notes usually have amaturity between one and 10 years whereas treasury bonds have maturitiesbeyond 10 years)
The next code is generating Vasicek random samples according to the AR(1)time series (137)
for (i in 1100) arsimlt-arimasim(model=list(ar=c(b))nstart=100n)y=ratek[1]+ab+sigmaarsimtime lt- asPOSIXct(time(TNX) format = Y-m-d)yield lt- xts(x = y orderby = time)chartSeries(yieldupcol=bluetheme=white)Syssleep(05)
418
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
= minus2∆t(minusan∆t+
nminus1sumk=0
(rtk+1 minus (1minus b∆t)rtk
))= 0
and
part
partb
nminus1sumk=0
(rtk+1 minus a∆tminus (1minus b∆t)rtk
)2= ∆t
nminus1sumk=0
rtk(minusa∆t+ rtk+1 minus (1minus b∆t)rtk
)= ∆t
nminus1sumk=0
rtk
(rtk+1 minus (1minus b∆t)rtk + 1
n
nminus1suml=0
(rtl+1 minus (1minus b∆t)rtl
))= 0
This leads to an estimate the parameters a and b respectively as the empiricalmean and covariance of (rtk)k=01n ie
a∆t = 1n
nminus1sumk=0
rtk+1 minus1n
(1minus b∆t)nminus1sumk=0
rtk
and
1minus b∆t =
nminus1sumk=0
rtk rtk+1 minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl+1
nminus1sumk=0
rtk rtk minus1n
nminus1sumk=0
rtk
nminus1suml=0
rtl
=
nminus1sumk=0
(rtk minus
1n
nminus1suml=0
rtl
)(rtk+1 minus
1n
nminus1suml=0
rtl+1
)nminus1sumk=0
(rtk minus
1n
nminus1sumk=0
rtk
)2
This also yields
σ2∆t = Var[σZk] = Var[rtk+1 minus (1minus b∆t)rtk minus a∆t
] k isin N
hence σ can be estimated as
σ2∆t = 1n
nminus1sumk=0
(rtk+1 minus rtk(1minus b∆t)minus a∆t
)2
416
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Defining rtk = rtk minus ab k isin N we have
rtk+1 = rtk+1 minus ab= rtk minus ab+ (aminus brtk)∆t+ σZk
= rtk minus abminus b(rtk minus ab)∆t+ σZk
= rtk minus brtk∆t+ σZk
= (1minus b∆t)rtk + σZk k isin N
In other words the sequence (rtk)kisinisinN is modeled according to an autore-gressive AR(1) time series in which the current state Xn of the system isexpressed as the linear combination
The next Figure 133 displays the yield of the 10 Year Treasury Note on theChicago Board Options Exchange (CBOE) Treasury notes usually have amaturity between one and 10 years whereas treasury bonds have maturitiesbeyond 10 years)
The next code is generating Vasicek random samples according to the AR(1)time series (137)
for (i in 1100) arsimlt-arimasim(model=list(ar=c(b))nstart=100n)y=ratek[1]+ab+sigmaarsimtime lt- asPOSIXct(time(TNX) format = Y-m-d)yield lt- xts(x = y orderby = time)chartSeries(yieldupcol=bluetheme=white)Syssleep(05)
418
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
Defining rtk = rtk minus ab k isin N we have
rtk+1 = rtk+1 minus ab= rtk minus ab+ (aminus brtk)∆t+ σZk
= rtk minus abminus b(rtk minus ab)∆t+ σZk
= rtk minus brtk∆t+ σZk
= (1minus b∆t)rtk + σZk k isin N
In other words the sequence (rtk)kisinisinN is modeled according to an autore-gressive AR(1) time series in which the current state Xn of the system isexpressed as the linear combination
The next Figure 133 displays the yield of the 10 Year Treasury Note on theChicago Board Options Exchange (CBOE) Treasury notes usually have amaturity between one and 10 years whereas treasury bonds have maturitiesbeyond 10 years)
The next code is generating Vasicek random samples according to the AR(1)time series (137)
for (i in 1100) arsimlt-arimasim(model=list(ar=c(b))nstart=100n)y=ratek[1]+ab+sigmaarsimtime lt- asPOSIXct(time(TNX) format = Y-m-d)yield lt- xts(x = y orderby = time)chartSeries(yieldupcol=bluetheme=white)Syssleep(05)
418
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
The next Figure 133 displays the yield of the 10 Year Treasury Note on theChicago Board Options Exchange (CBOE) Treasury notes usually have amaturity between one and 10 years whereas treasury bonds have maturitiesbeyond 10 years)
The next code is generating Vasicek random samples according to the AR(1)time series (137)
for (i in 1100) arsimlt-arimasim(model=list(ar=c(b))nstart=100n)y=ratek[1]+ab+sigmaarsimtime lt- asPOSIXct(time(TNX) format = Y-m-d)yield lt- xts(x = y orderby = time)chartSeries(yieldupcol=bluetheme=white)Syssleep(05)
418
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
A zero-coupon bond is a contract priced P (t T ) at time t lt T to deliverP (T T ) = $1 at time T In addition to its value at maturity a bond mayyield a periodic coupon payment at regular time intervals until the maturitydate
Fig 135 Five dollar Louisiana bond of 1875 with 75 biannual coupons
The computation of the arbitrage price P0(t T ) of a zero-coupon bond basedon an underlying short term interest rate process (rt)tisinR+ is a basic andimportant issue in interest rate modeling
Constant short rate
In case the short term interest rate is a constant rt = r t isin R+ a standardarbitrage argument shows that the price P (t T ) of the bond is given by
P (t T ) = eminusr(Tminust) 0 6 t 6 T
419
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
Indeed if P (t T ) gt eminusr(Tminust) we could issue a bond at the price P (t T ) andinvest this amount at the compounded risk free rate r which would yieldP (t T ) er(Tminust) gt 1 at time T
On the other hand if P (t T ) lt eminusr(Tminust) we could borrow P (t T ) at the rater to buy a bond priced P (t T ) At maturity time T we would receive $1 andrefund only P (t T ) er(Tminust) lt 1
Deterministic short rates
Similarly to the above when the short term interest rate process (rt)tisinR+ isa deterministic function of time a similar argument shows that
P (t T ) = eminusr Ttrsds 0 6 t 6 T (138)
Stochastic short rates
In case (rt)tisinR+ is an (Ft)tisinR+ -adapted random process the formula (138)is no longer valid as it relies on future information and we replace it with
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T (139)
under a risk-neutral measure Plowast It is natural to write P (t T ) as a conditionalexpectation under a martingale measure as the use of conditional expectationhelps to ldquofilter outrdquo the future information past time t contained in
w T
trsds
The expression (139) makes sense as the ldquobest possible estimaterdquo of thefuture quantity eminus
r Ttrsds in mean square sense given information known up
to time t
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general theamount and periodicity of coupons are deterministiclowast In the case of a con-stant continuous-time coupon yield at the rate c gt 0 another application ofthe above absence of arbitrage argument shows that the price Pc(t T ) of thecoupon bond is given by
Pc(t T ) = ec(Tminust)P0(t T ) 0 6 t 6 T
see also Figure 139 below In the sequel we will mostly consider zero-couponbonds priced as P (t T ) = P0(t T ) 0 6 t 6 T lowast However coupon default cannot be excluded
420
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is sat-isfied in other words the bond price process t 7minusrarr P (t T ) can be used as anumeacuteraireProposition 131 The discounted bond price process
t 7minusrarr P (t T ) = eminusr t
0 rsdsP (t T )
is a martingale under PlowastProof By (139) we have
eminusr t
0 rsdsP (t T ) = eminusr t
0 rsds IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r t0 rsds eminus
r Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r T0 rsds
∣∣∣ Ft] and this suffices to conclude since by the ldquotower propertyrdquo (1739) of condi-tional expectations any process (Xt)tisinR+ of the form t 7minusrarr Xt = IElowast[F | Ft]F isin L1(Ω) is a martingale cf Relation (61)
Path integrals
In physics the Feynman path integral
ψ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(i
~S(x(middot))
)where ~ is the Planck constant and S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 minus V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 minus V (x(timinus1)))∆ti
solves the Schroumldinger equation
i~partψ
partt(x t) = minus ~2
2mpart2ψ
partx2 (x t) + V (x(t))ψ(x t)
After the Wick rotation t 7rarr minusit the function
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
) 421
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
where S(x(middot)) is the action
S(x(middot)) =w t
0L(x(s) x(s) s)ds =
w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
solves the heat equation
~partφ
partt(x t) = minus ~2
2mpart2φ
partx2 (x t) + V (x(t))φ(x t)
Given the action
S(x(middot)) =w t
0
(12m(x(s))2 + V (x(s))
)ds
Nsumi=1
((x(ti)minus x(timinus1))2
2(ti minus timinus1)2 + V (x(timinus1)))∆ti
we can rewrite the Euclidean path integral as
φ(y t) =wx(0)=x x(t)=y
Dx(middot) exp(minus1~S(x(middot))
)=
wx(0)=x x(t)=y
Dx(middot) exp(minus 1
2~
Nsumi=1
(x(ti)minus x(timinus1))2
2∆timinus 1
~
Nsumi=1
V (x(timinus1)))
= IElowast[exp
(minus1~
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
]
This type of path integral computation
φ(y t) = IElowast[exp
(minus
w t
0V (Bs)ds
) ∣∣∣B0 = xBt = y
] (1310)
is particularly useful for bond pricing as (1310) can be interpreted as theprice of a bond with short term interest rate process (rt)tisinR+ = (V (Bt)))tisinR+
conditionally to the value of the endpoint Bt = y cf (1331) below It can alsobe useful for exotic option pricing cf Chapter 10 and for risk managementThe path integral (1310) can be estimated either by closed-form expressionsusing Partial Differential Equations (PDEs) or probability densities by ap-proximations such as (conditional) Moment matching or by Monte Carloestimation from the paths of a Brownian bridge as shown in Figure 136
422
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
Fig 136 Brownian bridge
Bond pricing PDE
We assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = micro(t rt)dt+ σ(t rt)dBt (1311)
where (Bt)tisinR+ is a standard Brownian motion under Plowast Note that specify-ing the dynamics of (rt)tisinR+ under the historical probability measure P willalso lead to a notion of market price of risk (MPoR) for the modeling of shortrates
Since all solutions of stochastic differential equations such as (1311) havethe Markov property cf eg Theorem V-32 of [Pro04] the arbitrage priceP (t T ) can be rewritten as a function F (t rt) of rt ie
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] = IElowast[
eminusr Ttrsds
∣∣∣ rt] = F (t rt)
and depends on rt only instead of depending on all information available inFt up to time t meaning that the pricing problem can now be formulated asa search for the function F (t x)
Proposition 132 (Bond pricing PDE) The bond pricing PDE for P (t T ) =F (t rt) is written as
xF (t x) = partF
partt(t x) + micro(t x)partF
partx(t x) + 1
2σ2(t x)part
2F
partx2 (t x) (1312)
423
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
t isin R+ x isin R subject to the terminal condition
F (T x) = 1 x isin R (1313)
Proof By Itocircrsquos formula we have
d(
eminusr t
0 rsdsP (t T ))
= minusrt eminusr t
0 rsdsP (t T )dt+ eminusr t
0 rsdsdP (t T )
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdsdF (t rt)
= minusrt eminusr t
0 rsdsF (t rt)dt+ eminusr t
0 rsdspartF
partx(t rt)(micro(t rt)dt+ σ(t rt)dBt)
+ eminusr t
0 rsds
(12σ
2(t rt)part2F
partx2 (t rt) + partF
partt(t rt)
)dt
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
+ eminusr t
0 rsds
(minusrtF (t rt) + micro(t rt)
partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
)dt
(1314)
Given that t 7minusrarr eminusr t
0 rsdsP (t T ) is a martingale the above expression(1314) should only contain terms in dBt (cf Corollary II-1 page 72 of[Pro04]) and all terms in dt should vanish inside (1314) This leads to theidentities
rtF (t rt)
= micro(t rt)partF
partx(t rt) + 1
2σ2(t rt)
part2F
partx2 (t rt) + partF
partt(t rt)
d(
eminusr t
0 rsdsP (t T ))
= eminusr t
0 rsdsσ(t rt)partF
partx(t rt)dBt
(1315a)
(1315b)
which recover (1312) Condition (1313) is due to the fact that P (T T ) = $1
In the case of an interest rate process modeled by (136) we have
hence (1312) yields the (time dependent) affine PDE
xF (t x) = partF
partt(t x) + (η(t) + λ(t)x)partF
partx(t x) + 1
2(δ(t) + γ(t)x)part2F
partx2 (t x)
(1316)
t isin R+ x isin R By (1315b) the above proposition also shows that
424
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
dP (t T )P (t T ) = 1
P (t T )d(
er t
0 rsds eminusr t
0 rsdsP (t T ))
= 1P (t T )
(rtP (t T )dt+ e
r t0 rsdsd
(eminus
r t0 rsdsP (t T )
))= rtdt+ 1
P (t T ) er t
0 rsdsd(
eminusr t
0 rsdsP (t T ))
= rtdt+ 1F (t rt)
partF
partx(t rt)σ(t rt)dBt
= rtdt+ σ(t rt)part logFpartx
(t rt)dBt (1317)
In the Vasicek casedrt = (aminus brt)dt+ σdWt
the bond price takes the form
F (t rt) = P (t T ) = eA(Tminust)+rtC(Tminust)
where A(middot) and C(middot) are functions of time cf (1321) below and (1317)yields
dP (t T )P (t T ) = rtdtminus
σ
b(1minus eminusb(Tminust))dWt (1318)
since F (t x) = eA(Tminust)+xC(Tminust)
Note that more generally all affine short rate models as defined in Rela-tion (136) including the Vasicek model will yield a bond pricing formula ofthe form
P (t T ) = eA(Tminust)+rtC(Tminust)
cf eg sect 324 of [BM06]
Probabilistic solution of the Vasicek PDE
Next we solve the PDE (1312) written with micro(t x) = aminusbx and σ(t x) = σin the Vasicek [Vaš77] model
drt = (aminus brt)dt+ σdBt
as xF (t x) = partF
partt(t x) + (aminus bx)partF
partx(t x) + σ2
2part2F
partx2 (t x)
F (T x) = 1(1319)
by a direct computation of the conditional expectation
425
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
F (t rt) = P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] (1320)
Recall that in this model the short rate (rt)tisinR+ has the expression
rt = g(t) +w t
0h(t s)dBs = r0 eminusbt + a
b(1minus eminusbt) + σ
w t
0eminusb(tminuss)dBs
whereg(t) = r0 eminusbt + a
b(1minus eminusbt) t isin R+
andh(t s) = σ eminusb(tminuss) 0 6 s 6 t
are deterministic functions
Letting uort = max(u t) using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function we have
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft]= IElowast
[eminus
r Tt
(g(s)+r s
0 h(su)dBu)ds∣∣∣ Ft]
= exp(minus
w T
tg(s)ds
)IElowast[
eminusr Tt
r s0 h(su)dBuds
∣∣∣ Ft]= exp
(minus
w T
tg(s)ds
)IElowast[
eminusr T
0r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
uorth(s u)dsdBu
)IElowast[
eminusr Tt
r Tuort h(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
∣∣∣ Ft]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu
)IElowast[
eminusr Tt
r Tuh(su)dsdBu
]= exp
(minus
w T
tg(s)dsminus
w t
0
w T
th(s u)dsdBu + 1
2w T
t
(w T
uh(s u)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
w t
0
w T
teminusb(sminusu)dsdBu
)times exp
(σ2
2w T
t
(w T
ueminusb(sminusu)ds
)2du
)
= exp(minus
w T
t(r0 eminusbs + a
b(1minus eminusbs))dsminus σ
b(1minus eminusb(Tminust))
w t
0eminusb(tminusu)dBu
)times exp
(σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)
426
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
= exp(minusrtb
(1minus eminusb(Tminust)) + 1b
(1minus eminusb(Tminust))(r0 eminusbt + a
b(1minus eminusbt)
))times exp
(minus
w T
t
(r0 eminusbs + a
b(1minus eminusbs)
)ds+ σ2
2w T
te2bu
(eminusbu minus eminusbT
b
)2
du
)= eA(Tminust)+rtC(Tminust) (1321)
whereC(T minus t) = minus1
b(1minus eminusb(Tminust)) (1322)
and
A(T minus t) = 4abminus 3σ2
4b3 + σ2 minus 2ab2b2 (T minus t) + σ2 minus ab
b3eminusb(Tminust)minus σ2
4b3 eminus2b(Tminust)
(1323)
Analytical solution of the Vasicek PDE
In order to solve the PDE (1319) analytically we may look for a solution ofthe form
F (t x) = eA(Tminust)+xC(Tminust) (1324)
where A(middot) and C(middot) are functions to be determined under the conditionsA(0) = 0 and C(0) = 0 Substituting (1324) into the PDE (1312) with theVasicek coefficients micro(t x) = (aminus bx) and σ(t x) = σ shows that
x eA(Tminust)+xC(Tminust) = minus(Aprime(T minus t)minus xC prime(T minus t)) eA(Tminust)+xC(Tminust)
+(aminus bx)C(T minus t) eA(Tminust)+xC(Tminust)
+12σ
2C2(T minus t) eA(Tminust)+xC(Tminust)
ie
x = minusAprime(T minus t) + xC prime(T minus t) + (aminus bx)C(T minus t) + 12σ
2C2(T minus t)
By identification of terms for x = 0 and x 6= 0 this yields the system ofRiccati and linear differential equations
Aprime(s) = aC(s) + σ2
2 C2(s)
C prime(s) = 1 + bC(s)
which can be solved to recover the above value of P (t T ) = F (t rt)
427
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model in which the short rate(rt)tisinR+ is solution to (131) Figure 137 presents a random simulation oft 7minusrarr P (t T ) in the same Vasicek model The graph of the correspondingdeterministic zero coupon bond price obtained for a = b = σ = 0 is alsoshown on the Figure 137
03
04
05
06
07
08
09
1
11
0 5 10 15 20
Fig 137 Graphs of t 7rarr F (t rt) = P (t T ) vs t 7rarr eminusr0(Tminust)
Figure 138 presents a random simulation of t 7minusrarr P (t T ) for a (non-zero)coupon bond with price Pc(t T ) = ec(Tminust)P (t T ) and coupon rate c gt 00 6 t 6 T
10000
10200
10400
10600
10800
0 5 10 15 20
Fig 138 Graph of t 7rarr F (t rt) = P (t T ) for a bond with a 23 coupon
The simulation of Figure 138 can be compared to the coupon bond marketdata of Figure 139 below
428
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
Fig 139 Bond price graph with maturity 011808 and coupon rate 625
See Exercise 133 for a bond pricing formula in the CIR model
Zero coupon bond price and yield data
The following zero coupon bond price was downloaded at EMMA fromthe Municipal Securities Rulemaking Board
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLE-REF-SER A (CA)CUSIP 68428LBB9Dated Date 06121996 (June 12 1996)Maturity Date 09012016 (September 1st 2016)Interest Rate 00 Principal Amount at Issuance $26056000Initial Offering Price 19465
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
The next Figure 1311 plots the bond yield y(t T ) defined as
y(t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)y(tT ) 0 6 t 6 T
2
4
6
8
yield [2005minus01minus262016minus01minus13]
Last 1666
Jan 262005
Aug 032006
May 132008
Feb 082010
Mar 012011
Nov 092012
Dec 042014
Fig 1311 Orange Cnty Calif bond yields
Bond pricing in the Dothan model
In the Dothan [Dot78] model the short term interest rate process (rt)tisinR+ ismodeled according to a geometric Brownian motion
drt = micrortdt+ σrtdBt (1325)430
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
where the volatility σ gt 0 and the drift micro isin R are constant parameters and(Bt)tisinR+ is a standard Brownian motion In this model the short term inter-est rate rt remains always positive while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt
On the other hand the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
For convenience of notation we let p = 1minus 2microσ2 and rewrite (1325) as
drt = (1minus p)σ2
2 rtdt+ σrtdBt
with solutionrt = r0 eσBtminuspσ
2t2 t isin R+ (1326)
By the Markov property of (rt)tisinR+ the bond price P (t T ) is a functionF (t rt) of rt and time t isin [0 T ]
P (t T ) = F (t rt) = IElowast[
eminusr Ttrsds
∣∣∣ rt] 0 6 t 6 T (1327)
By computation of the conditional expectation (1327) using (106) we easilyobtain the following result cf Proposition 12 of [PP11] where the functionθ(v t) is defined in (104)
Proposition 133 The zero-coupon bond price P (t T ) = F (t rt) is givenfor all p isin R by
F (t x) (1328)
= eminusσ2p2(Tminust)8
winfin0
winfin0
eminusux exp(minus2(1 + z2)σ2u
)θ
(4zσ2u
σ2(T minus t)
4
)du
u
dz
zp+1
x gt 0
Proof By Proposition 101 cf [Yor92] Proposition 2 the probability distri-bution of the time integral
w Tminust
0eσBsminuspσ
2s2ds is given by
P(w Tminust
0eσBsminuspσ
2s2ds isin dy)
=winfinminusinfin
P(w t
0eσBsminuspσ
2s2ds isin dy Bt minus pσt2 isin dz)
431
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
= σ
2winfinminusinfin
eminuspσz2minusp2σ2t8 exp
(minus21 + eσz
σ2y
)θ
(4 eσz2σ2y
σ2t
4
)dy
ydz
= eminusp2σ2(Tminust)8
winfin0
exp(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y y gt 0
where the exchange of integrals is justified by the Fubini theorem and thenonnegativity of integrands Hence by (106) and (1326) we find
F (t rt) = P (t T )
= IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minusrt
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
) ∣∣∣ Ft]= IElowast
[exp
(minusx
w T
teσ(BsminusBt)minusσ2p(sminust)2ds
)]x=rt
= IElowast[exp
(minusx
w Tminust
0eσBsminusσ
2ps2ds
)]x=rt
=winfin
0eminusrty P
(w Tminust
0eσBsminuspσ
2s2ds isin dy)
= eminusp2σ2(Tminust)8
winfin0
eminusrtywinfin
0exp
(minus21 + z2
σ2y
)θ
(4zσ2y
σ2(T minus t)
4
)dz
zp+1dy
y
The zero-coupon bond price P (t T ) = F (t rt) in the Dothan model can alsobe written for all p isin R as
F (t x) = (2x)p22π2σp
winfin0ueminusσ
2(p2+u2)t8 sinh(πu)∣∣∣Γ (minusp2 + i
u
2
)∣∣∣2Kiu
(radic8xσ
)du
+ (2x)p2σp
infinsumk=0
2(pminus 2k)+
k(pminus k) eσ2k(kminusp)t2Kpminus2k
(radic8xσ
) x gt 0 t gt 0
cf Corollary 22 of [PP10] see also [PU13] for numerical computations Zero-coupon bond prices in the Dothan model can also be computed by the con-ditional expression
IE[exp
(minus
w T
0rtdt
)]=
winfin0
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]dP(rT 6 z)
(1329)where rT has the lognormal distribution
dP(rT 6 z) = dP(r0eσBTminuspσ2T2 6 z) = 1
zradic
2πσ2Teminus(pσ2T2+log(zr0))2(2σ2T )
432
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
In Proposition 134 we note that the conditional Laplace transform
IE[exp
(minus
w T
0rtdt
) ∣∣∣rT = z
]cf (1310) above can be computed by a closed-form integral expression basedon the modified Bessel function of the second kind
Kζ(z) = zζ
2ζ+1
winfin0
exp(minusuminus z2
4u
)du
uζ+1 ζ isin R z isin C (1330)
cf eg [Wat95] page 183 provided that the real part R(z2) of z2 isin C ispositiveProposition 134 [PY16] Proposition 41 Taking r0 = 1 for all λ z gt 0we have
IE[exp
(minusλ
w T
0rsds
) ∣∣∣ rT = z
]= 4 eminusσ2T8
π32σ2p(z)
radicλ
T(1331)
timeswinfin
0e2(π2minusξ2)(σ2T ) sin
(4πξσ2T
)sinh(ξ)
K1
(radic8λradic
1 + 2radicz cosh ξ + zσ
)radic
1 + 2radicz cosh ξ + z
dξ
Note however that (1331) fails for small values of T and for this reasonthe integral can be estimated by a gamma approximation cf (1332) belowUnder the Gamma approximation we can approximate the conditional bondprice on the Dothan short rate rt as
IE[exp
(minusλ
w T
0rtdt
) ∣∣∣rT = z
] (1 + λθ(z))minusν(z)
where the parameters ν(z) and θ(z) are determined by conditional momentfitting to a gamma distribution as
θ(z) = Var[ΛT | ST = z]IE[ΛT | ST = z] ν(z) = (IE[ΛT | ST = z])2
Var[ΛT | ST = z] = IE[ΛT | ST = z]θ
cf [PY16] which yields
IE[exp
(minusλ
w T
0rsds
)]
winfin0
(1 + λθ(z))minusν(z)dP(rT 6 z) (1332)
Note that θ(z) is known in physics as the Fano factor which measures the dis-persion of the probability distribution of ΛT given that ST = z Figures 1312shows that the stratified gamma approximation (1332) matches the MonteCarlo estimate while the use of the integral expressions (1329) and (1331)leads to numerical instabilities
433
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
0
02
04
06
08
1
0 1 2 3 4 5 6 7 8 9 10T=
F(x
t)
t
stratified gammaMonte Carlo
integral expression
Fig 1312 Approximation of Dothan bond prices t 7rarr F (t x) with σ = 03 and T = 10
Related computations for yield options in the CIR model can also be foundin [PP17]
134 Forward Rates
A forward interest rate contract (or Forward Rate Agreement FRA) givesto its holder the possibility to lock an interest rate denoted by f(t T S) atpresent time t for a loan to be delivered over a future period of time [T S]with t 6 T 6 S The rate f(t T S) is called a forward interest rate WhenT = t the spot forward rate f(t t T ) is also called the yield
Figure 1313 presents a typical yield curve on the LIBOR (London InterbankOffered Rate) market with t =07 May 2003
Fig 1313 Forward rate graph T 7minusrarr f(t t T )
Maturity transformation ie the ability to transform short term borrowing(debt with short maturities such as deposits) into long term lending (credits
434
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
with very long maturities such as loans) is among the roles of banks Prof-itability is then dependent on the difference between long rates and shortrates
Another example of market data is given in the next Figure 1314 in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011
Fig 1314 Market example of yield curves cf (1335)
Forward rates from bond prices
Let us determine the arbitrage or ldquofairrdquo value of the forward interest ratef(t T S) by implementing the Forward Rate Agreement using the instru-ments available in the market which are bonds priced at P (t T ) for variousmaturity dates T gt t
The loan can be realized using the available instruments (here bonds) on themarket by proceeding in two steps
1) At time t borrow the amount P (t S) by issuing (or short selling) onebond with maturity S which means refunding $1 at time S
2) Since the money is only needed at time T the rational investor willinvest the amount P (t S) over the period [t T ] by buying a (possibly frac-tional) quantity P (t S)P (t T ) of a bond with maturity T priced P (t T )at time t This will yield the amount
435
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
$1times P (t S)P (t T )
at time T gt 0
As a consequence the investor will actually receive P (t S)P (t T ) at timeT to refund $1 at time S
The corresponding forward rate f(t T S) is then given by the relation
P (t S)P (t T ) exp ((S minus T )f(t T S)) = $1 0 6 t 6 T 6 S (1333)
where we used exponential compounding which leads to the following defi-nition (1334)Definition 135 The forward rate f(t T S) at time t for a loan on [T S]is given by
f(t T S) = logP (t T )minus logP (t S)S minus T
(1334)
The spot forward rate f(t t T ) coincides with the yield given by
f(t t T ) = minus logP (t T )T minus t
or P (t T ) = eminus(Tminust)f(ttT ) 0 6 t 6 T(1335)
The instantaneous forward rate f(t T ) = f(t T T ) is defined by taking thelimit of f(t T S) as S T ie
f(t T ) = limST
f(t T S)
= minus limST
logP (t S)minus logP (t T )S minus T
= minus limε0
logP (t T + ε)minus logP (t T )ε
= minuspart logP (t T )partT
= minus 1P (t T )
partP (t T )partT
(1336)
The above equation (1336) can be viewed as a differential equation to besolved for logP (t T ) under the initial condition P (T T ) = 1 which yieldsthe following proposition436
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
Proposition 136 We have
P (t T ) = exp(minus
w T
tf(t s)ds
) 0 6 t 6 T (1337)
Proof We check that
logP (t T ) = logP (t T )minus logP (t t) =w T
t
part logP (t s)parts
ds = minusw T
tf(t s)ds
Proposition 136 also shows that
f(t t) = part
partT
w T
tf(t s)ds|T=t
= minus part
partTlogP (t T )|T=t
= minus 1P (t T ) |T=t
part
partTP (t T )|T=t
= minus part
partTIElowast[
eminusr Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rT eminus
r Ttrsds
∣∣∣ Ft]|T=t
= IElowast[rt | Ft]= rt
ie the short rate rt can be recovered from the instantaneous forward rateas
rt = f(t t) = limTt
f(t T )
As a consequence of (1333) and (1337) the forward rate f(t T S) 0 6 t 6T 6 S can be recovered from (1334) and the instantaneous forward ratef(t s) as
f(t T S) = logP (t T )minus logP (t S)S minus T
= minus 1S minus T
(w T
tf(t s)dsminus
w S
tf(t s)ds
)= 1S minus T
w S
Tf(t s)ds 0 6 t 6 T lt S (1338)
In particular the spot forward rate or yield f(t t T ) can be written as
437
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
f(t t T ) = minus logP (t T )T minus t
= 1T minus t
w T
tf(t s)ds 0 6 t lt T (1339)
Differentiation with respect to T of the above relation shows that the yieldf(t t T ) and the instantaneous forward rate f(t s) are linked by the relation
partf
partT(t t T ) = minus 1
(T minus t)2
w T
tf(t s)ds+ 1
T minus tf(t T ) 0 6 t lt T
from which it follows that
f(t T ) = 1T minus t
w T
tf(t s)ds+ (T minus t) partf
partT(t t T )
= f(t t T ) + (T minus t) partfpartT
(t t T ) 0 6 t lt T
Forward Swap Rates
The first interest rate swap occurred in 1981 between IBM and the WorldBank The vanilla interest rate swap makes it possible to exchange a se-quence of variable forward rates f(t Tk Tk+1) k = 1 2 nminus 1 against afixed rate κ over a time period [T1 Tn] Over the succession of time intervals[T1 T2) [T2 T3) [Tnminus1 Tn] defining a tenor structure see Section 141for details the combination of such exchanges will generate a cumulativediscounted cash flow(nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsdsf(t Tk Tk+1)
)minus
(nminus1sumk=1
κ(Tk+1 minus Tk) eminusr Tk+1t rsds
)
=nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
at time t = T0 in which we used simple (or linear) interest rate compoundingThis cash flow is used to make the contract fair and it can be priced at timet as
IElowast[nminus1sumk=1
(Tk+1 minus Tk) eminusr Tk+1t rsds(f(t Tk Tk+1)minus κ)
∣∣∣ Ft]
=nminus1sumk=1
(Tk+1 minus Tk)(f(t Tk Tk+1)minus κ) IElowast[
eminusr Tk+1t rsds
∣∣∣ Ft]438
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus κ
)
The swap rate S(t T1 Tn) is by definition the value of the rate κ that makesthe contract fair by making this cash flow vanish The next Proposition 137makes use of the annuity numeacuteraire
P (t T1 Tn) =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1) 0 6 t 6 T1 (1340)
which represents the present value at time t of future $1 receipts at timesT1 T2 Tn weighted by the time intervals Tk+1 minus Tk k = 1 2 nminus 1
Proposition 137 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
(1341)
Proof By definition S(t T1 Tn) is the fixed rate over [T1 Tn] that willbe agreed in exchange for the family of forward rates f(t Tk Tk+1) k =1 2 nminus 1 and it solves
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)(f(t Tk Tk+1)minus S(t T1 Tn)
)= 0 (1342)
ie
0 =nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)
minusS(t T1 Tn)nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)
=nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)f(t Tk Tk+1)minus P (t T1 Tn)S(t T1 Tn)
which shows (1341) by solving for S(t T1 Tn)
The time intervals (Tk+1 minus Tk)k=12nminus1 in the definition (1340) of theannuity numeacuteraire can be replaced by coupon payments (ck+1)k=12nminus1occurring at times (Tk+1)k=12nminus1 in which case the annuity numeacuterairebecomes
439
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
P (t T1 Tn) =nminus1sumk=1
ck+1P (t Tk+1) 0 6 t 6 T1 (1343)
which represents the value at time t of the future coupon payments discountedaccording to the bond prices P (t Tk+1)k=12nminus1 This expression can alsobe used for amortizing swaps in which the value of the notional decreasesover time or for accreting swaps in which the value of the notional increasesover time
LIBOR Rates
Recall that the forward rate f(t T S) 0 6 t 6 T 6 S is defined usingexponential compounding from the relation
f(t T S) = minus logP (t S)minus logP (t T )S minus T
(1344)
In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (1344)
The forward LIBOR L(t T S) for a loan on [T S] is defined using linearcompounding ie by replacing (1344) with the relation
1 + (S minus T )L(t T S) = P (t T )P (t S) 0 6 t 6 T
which yields the following definition
Definition 138 The forward LIBOR rate L(t T S) at time t for a loan on[T S] is given by
L(t T S) = 1S minus T
(P (t T )P (t S) minus 1
) 0 6 t 6 T lt S (1345)
Note that (1345) above yields the same formula for the (LIBOR) instanta-neous forward rate
L(t T ) = limST
L(t T S)
= limST
P (t S)minus P (t T )(S minus T )P (t S)
= limε0
P (t T + ε)minus P (t T )εP (t T + ε)
440
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
= 1P (t T ) lim
ε0
P (t T + ε)minus P (t T )ε
= minus 1P (t T )
partP (t T )partT
= minuspart logP (t T )partT
as (1336)
In addition Relation (1345) shows that the LIBOR rate can be viewedas a forward price Xt = XtNt with numeacuteraire Nt = (S minus T )P (t S) andXt = P (t T ) minus P (t S) according to Relation (127) of Chapter 12 As aconsequence from Proposition 124 the LIBOR rate (L(t T S))tisin[TS] is amartingale under the forward measure P defined by
dPdPlowast
= 1P (0 S) eminus
r S0 rtdt
LIBOR Swap Rates
The LIBOR swap rate S(t T1 Tn) satisfies the same relation as (1342) withthe forward rate f(t Tk Tk+1) replaced with the LIBOR rate L(t Tk Tk+1)ie
Proposition 139 The LIBOR swap rate S(t T1 Tn) is given by
S(t T1 Tn) = P (t T1)minus P (t Tn)P (t T1 Tn) 0 6 t 6 T1 (1346)
Proof By (1341) (1345) and a telescoping sum we have
S(t T1 Tn) = 1P (t T1 Tn)
nminus1sumk=1
(Tk+1 minus Tk)P (t Tk+1)L(t Tk Tk+1)
= 1P (t T1 Tn)
nminus1sumk=1
P (t Tk+1)(
P (t Tk)P (t Tk+1) minus 1
)
= 1P (t T1 Tn)
nminus1sumk=1
(P (t Tk)minus P (t Tk+1))
= P (t T1)minus P (t Tn)P (t T1 Tn) (1347)
441
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
Clearly a simple expression for the swap rate such as that of Proposition 139cannot be obtained using the standard (ie non-LIBOR) rates defined in(1344) Similarly it will not be available for amortizing or accreting swapsbecause the telescoping summation argument does not apply to the expression(1343) of the annuity numeraire
When n = 2 the swap rate S(t T1 T2) coincides with the forward rateL(t T1 T2)
S(t T1 T2) = L(t T1 T2) (1348)
and the bond prices P (t T1) can be recovered from the forward swap ratesS(t T1 Tn)
Similarly to the case of LIBOR rates Relation (1346) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeacuteraireNt = P (t T1 Tn) and Xt = P (t T1) minus P (t Tn) Consequently the LIBORswap rate (S(t T1 Tn)tisin[TS] is a martingale under the forward measure Pdefined from (121) by
dPdPlowast
= P (T1 T1 Tn)P (0 T1 Tn) eminus
r T10 rtdt
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package ldquoYieldCurverdquo [Gui15]for the following code and further details on yield curve and interest ratemodeling using Rinstallpackages(YieldCurve)require(YieldCurve)data(FedYieldCurve)first(FedYieldCurve3 month)last(FedYieldCurve3 month)matFed=c(025051235710)n=50plot(matFed FedYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest rates
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
Jan1982
Jan1984
Jan1986
Jan1988
Jan1990
Jan1992
Jan1994
Jan1996
Jan1998 Jan
2000 Jan2002 Jan
2004 Jan2006 Jan
2008 Jan2010 Jan
2012 Jan2012
R_3MR_6MR_1YR_2YR_3YR_5YR_7YR_10Y0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Fig 1315 Federal Reserve yield curves from 1982 to 2012
European Central Bank (ECB) data can be similarly obtained
data(ECBYieldCurve)first(ECBYieldCurve3 month)last(ECBYieldCurve3 month)matECBlt-c(312 05 123456789101112131415161718192021222324252627282930)for (n in 200400) plot(matECB ECBYieldCurve[n] type=oxlab=Maturities structure in years ylab=Interest
rates valuesylim=c(3151))title(main=paste(European Central Bank yield curve observed attime(ECBYieldCurve[n] sep= )
))grid()Syssleep(05)
The next Figure 1316 represents the output of the above script
Fig 1316 European Central Bank yield curveslowast
lowast The animation works in Acrobat Reader on the entire pdf file
443
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
Decreasing yield curves can occur when central banks attempts to limitinflation by tightening interest rates In the next section we turn to the mod-eling of the market curves observed in Figure 1316
135 The HJM Model
From the beginning of this chapter we have started with the modeling ofthe short rate (rt)tisinR+ followed by its consequences on the pricing of bondsP (t T ) and on the expressions of the forward rates f(t T S) and L(t T S)
In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t T ) The graphgiven in Figure 1317 presents a possible random evolution of a forward in-terest rate curve using the Musiela convention ie we will write
g(x) = f(t t+ x) = f(t T )
under the substitution x = T minus t x gt 0 and represent a sample of theinstantaneous forward curve x 7minusrarr f(t t+ x) for each t isin R+
0 5
10 15
20 0
5
10
15
20
05 1
15 2
25 3
35 4
45 5
Forward rate
x
t
Fig 1317 Stochastic process of forward curves
In the Heath-Jarrow-Morton (HJM) model the instantaneous forward ratef(t T ) is modeled under Plowast by a stochastic differential equation of the form
dtf(t T ) = α(t T )dt+ σ(t T )dBt 0 6 t 6 T (1349)
where t 7minusrarr α(t T ) and t 7minusrarr σ(t T ) 0 6 t 6 T are allowed to be ran-dom (adapted) processes In the above equation the date T is fixed and thedifferential dt is with respect to t
Under basic Markovianity assumptions a HJM model with deterministiccoefficients α(t T ) and σ(t T ) will yield a short rate process (rt)tisinR+ of theform
444
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
drt = (a(t)minus b(t)rt)dt+ σ(t)dBt
cf sect 66 of [Pri12] which is the Hull-White model [HW90] with explicitsolution
rt = rs eminusr tsb(τ)dτ +
w t
seminus
r tub(τ)dτa(u)du+
w t
sσ(u) eminus
r tub(τ)dτdBu
0 6 s 6 t
The HJM Condition
How to ldquoencoderdquo absence of arbitrage in the defining HJM Equation (1349)is an important question Recall that under absence of arbitrage the bondprice P (t T ) has been constructed as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] = exp(minus
w T
tf(t s)ds
) (1350)
cf Proposition 136 hence the discounted bond price process is given by
t 7minusrarr exp(minus
w t
0rsds
)P (t T ) = exp
(minus
w t
0rsdsminus
w T
tf(t s)ds
)(1351)
is a martingale under Plowast by Proposition 131 and Relation (1337) in Propo-sition 136 This shows that Plowast is a risk-neutral measure and by the firstfundamental Theorem 57 of asset pricing we conclude that the market iswithout arbitrage opportunities
Proposition 1310 (HJM Condition [HJM92]) Under the condition
α(t T ) = σ(t T )w T
tσ(t s)ds t isin [0 T ] (1352)
which is known as the HJM absence of arbitrage condition the discountedbond price process (1351) is a martingale and the measure Plowast is risk-neutral
Proof Consider the spot forward rate or yield given from (1339) as
f(t t T ) = 1T minus t
w T
tf(t s)ds
and letXt =
w T
tf(t s)ds = minus logP (t T ) 0 6 t 6 T
with the relation
445
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
f(t t T ) = 1T minus t
w T
tf(t s)ds = Xt
T minus t 0 6 t 6 T (1353)
where the dynamics of t 7minusrarr f(t s) is given by (1349) We note that whenf(t s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation
dtw T
tg(t)h(s)ds = minusg(t)h(t)dt+ gprime(t)
w T
th(s)dsdt
which extends to f(t s) as
dtw T
tf(t s)ds = minusf(t t)dt+
w T
tdtf(t s)ds
which can be seen as a form of the Leibniz integral rule Therefore we have
dtXt = dtw T
tf(t s)ds
= minusf(t t)dt+w T
tdtf(t s)ds
= minusf(t t)dt+w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
= minusrtdt+(w T
tα(t s)ds
)dt+
(w T
tσ(t s)ds
)dBt
hence we have|dtXt|2 =
(w T
tσ(t s)ds
)2dt
Hence by Itocircrsquos calculus we have
dtP (t T ) = dt eminusXt
= minus eminusXtdtXt + 12 eminusXt(dtXt)2
= minus eminusXtdtXt + 12 eminusXt
(w T
tσ(t s)ds
)2dt
= minus eminusXt(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 eminusXt(w T
tσ(t s)ds
)2dt
and the discounted bond price satisfies
dt
(exp
(minus
w t
0rsds
)P (t T )
)
446
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
= minusrt exp(minus
w t
0rsdsminusXt
)dt+ exp
(minus
w t
0rsds
)dtP (t T )
= minusrt exp(minus
w t
0rsdsminusXt
)dtminus exp
(minus
w t
0rsdsminusXt
)dtXt
+12 exp
(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minusrt exp(minus
w t
0rsdsminusXt
)dt
minus exp(minus
w t
0rsdsminusXt
)(minusrtdt+
w T
tα(t s)dsdt+
w T
tσ(t s)dsdBt
)+1
2 exp(minus
w t
0rsdsminusXt
)(w T
tσ(t s)ds
)2dt
= minus exp(minus
w t
0rsdsminusXt
)w T
tσ(t s)dsdBt
minus exp(minus
w t
0rsdsminusXt
)(w T
tα(t s)dsdtminus 1
2
(w T
tσ(t s)ds
)2)dt
Thus the discounted bond price process
t 7minusrarr exp(minus
w t
0rsds
)P (t T )
will be a martingale provided that
w T
tα(t s)dsminus 1
2
(w T
tσ(t s)ds
)2= 0 0 6 t 6 T (1354)
Differentiating the above relation with respect to T we get
α(t T ) = σ(t T )w T
tσ(t s)ds
which is in fact equivalent to (1354)
136 Forward Vasicek Rates
In this section we consider the Vasicek model in which the short rate processis the solution (132) of (131) as illustrated in Figure 131
In the Vasicek model the forward rate is given by
f(t T S) = minus logP (t S)minus logP (t T )S minus T
447
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
= minusrt(C(S minus t)minus C(T minus t)) +A(S minus t)minusA(T minus t))S minus T
= minusσ2 minus 2ab
2b2
minus 1S minus T
((rtb
+ σ2 minus abb3
)(eminusb(Sminust) minus eminusb(Tminust)
)minus σ
2
4b3(
eminus2b(Sminust) minus eminus2b(Tminust))) and the spot forward rate or yield satisfies
f(t t T ) = minus logP (t T )T minus t
= minusrtC(T minus t) +A(T minus t)T minus t
= minusσ2 minus 2ab
2b2 + 1T minus t
((rtb
+ σ2 minus abb3
)(1minus eminusb(Tminust)) minus σ
2
4b3 (1minus eminus2b(Tminust)))
In this model the forward rate t 7minusrarr f(t T S) can be represented as inFigure 1318 with here ba gt r0
0005
00055
0006
00065
0007
00075
0008
00085
0009
00095
001
0 2 4 6 8 10
t
f(tTS)
Fig 1318 Forward rate process t 7minusrarr f(t T S)
Note that the forward rate cure t 7minusrarr f(t T S) appears flat for small valuesof t ie longer rates are more stable while shorter rates show higher volatilityor risk Similar features can be observed in Figure 1319 for the instantaneousshort rate given by
f(t T ) = minuspart logP (t T )partT
(1355)
= rt eminusb(Tminust) + a
b
(1minus eminusb(Tminust)
)minus σ2
2b2(1minus eminusb(Tminust)
)2
from which the relation limTt f(t T ) = rt can be easily recovered
The instantaneous forward rate t 7minusrarr f(t T ) can be represented as in Fig-ure 1319 with ba gt r0
448
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
0
002
004
006
008
01
012
014
0 2 4 6 8 10 12 14 16 18 20
t
f(tT)
Fig 1319 Instantaneous forward rate process t 7minusrarr f(t T )
The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have
and σ(t T ) = σ eminusb(Tminust) and the HJM condition reads
α(t T ) = σ2 eminusb(Tminust)w T
teb(tminuss)ds = σ(t T )
w T
tσ(t s)ds (1356)
Random simulations of the Vasicek instantaneous forward rates are providedin Figures 1320 and 1321
Fig 1320 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek modellowast
449
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
Fig 1321 Forward instantaneous curve x 7minusrarr f(0 x) in the Vasicek modellowast
For x = 0 the first ldquoslicerdquo of this surface is actually the short rate Vasicekprocess rt = f(t t) = f(t t + 0) which is represented in Figure 1322 usinganother discretization
003
0035
004
0045
005
0055
006
0065
007
0 5 10 15 20
Fig 1322 Short term interest rate curve t 7minusrarr rt in the Vasicek model
137 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curvesare parametrized by 4 coefficients z1 z2 z3 z4 as
g(x) = z1 + (z2 + z3x) eminusxz4 x gt 0
An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 1323 for z1 = 1 z2 = minus10 z3 = 100 z4 = 10lowast The animation works in Acrobat Reader on the entire pdf file
450
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
-10
-8
-6
-4
-2
0
2
4
0 02 04 06 08 1
z1+(z2+xz3)exp(-xz4)
Fig 1323 Graph of x 7minusrarr g(x) in the Nelson-Siegel model
The Svensson parametrization has the advantage to reproduce two humps in-stead of one the location and height of which can be chosen via 6 parametersz1 z2 z3 z4 z5 z6 as
A typical graph of a Svensson parametrization is given in Figure 1324 forz1 = 7 z2 = minus5 z3 = minus100 z4 = 10 z5 = minus12 z6 = minus1
2
25
3
35
4
45
5
0 5 10 15 20 25 30
lambda
x-gtz1+(z2+z3x)exp(-xz4)+z5xexp(-z6x)
Fig 1324 Graph of x 7minusrarr g(x) in the Svensson model
Figure 1325 presents a fit of the market data of Figure 1313 using a Svenssoncurve
451
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
2
25
3
35
4
45
5
0 5 10 15 20 25 30
years
Market dataSvensson curve
Fig 1325 Comparison of market data vs a Svensson curve
It can be shown cf sect 35 of [Bjouml04b] that the forward yield curves producedby the Vasicek model are included neither in the Nelson-Siegel space nor inthe Svensson space In addition the Vasicek yield curves do not appear tocorrectly model the market forward curves cf also Figure 1313 aboveIn the Vasicek model we have
partf
partT(t T ) =
(minusbrt + aminus σ2
b+ σ2
beminusb(Tminust)
)eminusb(Tminust)
and one can check that the sign of the derivatives of f can only change onceat most As a consequence the possible forward curves in the Vasicek modelare limited to one change of ldquoregimerdquo per curve as illustrated in Figure 1326for various values of rt and in Figure 1327
0
001
002
003
004
005
006
007
008
009
0 5 10 15 20
Fig 1326 Graphs of forward rates
452
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
0 2
4 6
8 10
x
0 5
10 15
20
t
002
003
004
005
006
007
008
009
Fig 1327 Forward instantaneous curve (t x) 7minusrarr f(t t+ x) in the Vasicek model
One may think of constructing an instantaneous rate process taking values inthe Svensson space however this type of modelization is not consistent withabsence of arbitrage and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces cf sect35 of [Bjouml04b]
Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve such as the initial curve at t = 0cf eg sect 82 of [Pri12]
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forwardrate curves are parametrized by four coefficients z1 z2 z3 z4 as
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
The correlation problem is another issue of concern when using the affinemodels considered so far Let us compare three bond price simulations withmaturity T1 = 10 T2 = 20 and T3 = 30 based on the same Brownian pathas given in Figure 1329 Clearly the bond prices F (rt T1) = P (t T1) andlowast The animation works in Acrobat Reader on the entire pdf file454
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
F (rt T2) = P (t T2) with maturities T1 and T2 are linked by the relation
P (t T2) = P (t T1) exp(A(t T2)minusA(t T1)+rt(C(t T2)minusC(t T1))
) (1358)
meaning that bond prices with different maturities could be deduced fromeach other which is unrealistic
03
04
05
06
07
08
09
1
0 5 10 15 20 25 30
t
P(tT1)P(tT2)P(tT3)
Fig 1329 Graph of t 7minusrarr P (t T1)
In affine short rates models by (1358) logP (t T1) and logP (t T2) are linkedby the linear relationship
with constant coefficients which yields the perfect (positive or negative) cor-relation
Cor(logP (t T1) logP (t T2)) = plusmn1
depending on the sign of the coefficient 1 + (C(t T2)minusC(t T1))A(t T1) cfsect 83 of [Pri12]
A solution to the correlation problem is to consider a two-factor modelbased on two control processes (Xt)tisinR+ (Yt)tisinR+ which are solution of
dXt = micro1(tXt)dt+ σ1(tXt)dB(1)t
dYt = micro2(t Yt)dt+ σ2(t Yt)dB(2)t
(1359)
455
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
where (B(1)t )tisinR+ (B(2)
t )tisinR+ have correlated Brownian motion with
Cov(B(1)s B
(2)t ) = ρmin(s t) s t isin R+ (1360)
anddB
(1)t middot dB
(2)t = ρdt (1361)
for some correlation parameter ρ isin [minus1 1] In practice (B(1))tisinR+ and(B(2))tisinR+ can be constructed from two independent Brownian motions(W (1))tisinR+ and (W (2))tisinR+ by letting
B(1)t = W
(1)t
B(2)t = ρW
(1)t +
radic1minus ρ2W
(2)t t isin R+
and Relations (1360) and (1361) are easily satisfied from this construction
In two-factor models one chooses to build the short term interest rate rt via
rt = Xt + Yt t isin R+
By the previous standard arbitrage arguments we define the price of a bondwith maturity T as
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft]= IElowast
[exp
(minus
w T
trsds
) ∣∣∣ Xt Yt
]= IElowast
[exp
(minus
w T
t(Xs + Ys)ds
) ∣∣∣ Xt Yt
]= F (tXt Yt) (1362)
since the couple (Xt Yt)tisinR+ is Markovian Applying the Itocirc formula with
two variables to
t 7minusrarr F (tXt Yt) = P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] and using the fact that the discounted process
t 7minusrarr eminusr t
0 rsdsP (t T ) = IElowast[exp
(minus
w T
0rsds
) ∣∣∣ Ft]is an Ft-martingale under Plowast we can derive a PDE
456
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
minus(x+ y)F (t x y) + micro1(t x)partFpartx
(t x y) + micro2(t y)partFparty
(t x y)
+12σ
21(t x)part
2F
partx2 (t x y) + 12σ
22(t y)part
2F
party2 (t x y)
+ρσ1(t x)σ2(t y) part2F
partxparty(t x y) + partF
partt(tXt Yt) = 0 (1363)
on R2 for the bond price P (t T ) In the Vasicek modeldXt = minusaXtdt+ σdB
(1)t
dYt = minusbYtdt+ ηdB(2)t
this yields the solution F (t x y) of (1363) as
P (t T ) = F (tXt Yt) = F1(tXt)F2(t Yt) exp (Uρ(t T )) (1364)
where F1(tXt) and F2(t Yt) are the bond prices associated to Xt and Yt inthe Vasicek model and
Uρ(t T ) = ρση
ab
(T minus t+ eminusa(Tminust) minus 1
a+ eminusb(Tminust) minus 1
bminus eminus(a+b)(Tminust) minus 1
a+ b
)is a correlation term which vanishes when (B(1)
t )tisinR+ and (B(2)t )tisinR+ are in-
dependent ie when ρ = 0 cf [BM06] Chapter 4 Appendix A and sect 84 of[Pri12]
Partial differentiation of logP (t T ) with respect to T leads to the instanta-neous forward rate
where f1(t T ) f2(t T ) are the instantaneous forward rates corresponding toXt and Yt respectively cf sect 84 of [Pri12]
An example of a forward rate curve obtained in this way is given in Fig-ure 1330
457
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
018
019
02
021
022
023
024
0 5 10 15 20 25 30 35 40
T
Fig 1330 Graph of forward rates in a two-factor model
Next in Figure 1331 we present a graph of the evolution of forward curvesin a two factor model
0 1
2 3
4 5
6 7
8
x 0
02 04
06 08
1 12
14
t
0215
022
0225
023
0235
024
Fig 1331 Random evolution of forward rates in a two-factor model
138 The BGM Model
The models (HJM affine etc) considered in the previous chapter sufferfrom various drawbacks such as nonpositivity of interest rates in Vasicekmodel and lack of closed-form solutions in more complex models The BGM[BGM97] model has the advantage of yielding positive interest rates and topermit to derive explicit formulas for the computation of prices for interestrate derivatives such as caps and swaptions on the LIBOR market
In the BGM model we consider two bond prices P (t T1) P (t T2) with ma-turities T1 T2 and the forward measure
dP2
dPlowast2= eminus
r T20 rsds
P (0 T2)
458
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
with numeraire P (t T2) cf (126) The forward LIBOR rate L(t T1 T2) ismodeled as a geometric Brownian motion under P2 ie
dL(t T1 T2)L(t T1 T2) = γ1(t)dB(2)
t (1366)
0 6 t 6 T1 i = 1 2 n minus 1 for some deterministic function γ1(t) withsolution
L(u T1 T2) = L(t T1 T2) exp(w u
tγ1(s)dB(2)
s minus12
w u
t|γ1|2(s)ds
)
ie for u = T1
L(T1 T1 T2) = L(t T1 T2) exp(w T1
tγ1(s)dB(2)
s minus12
w T1
t|γ1|2(s)ds
)
Since L(t T1 T2) is a geometric Brownian motion under P2 standard capletscan be priced at time t isin [0 T1] from the Black-Scholes formula
The following Graph 1332 summarizes the notions introduced in this chapter
459
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
Short rate1 rt
Short ratert = f(t t) = f(t t t)
Bond price2
P (t T ) = IElowast[eminus
r Tt rsds | Ft
]
LIBOR rate3
L(t T S) = P (tT )minusP (tS)(SminusT )P (tS)
Forward rate3
f(t T S) = logP (tT )minuslogP (tS)SminusT
Instantaneous forward rate4
f(t T ) = L(t T ) = limST f(t T S)= limST L(t T S)
Bond price
P (t T ) = eminusr Tt f(ts)ds
Bond priceP (t T ) = eminus(Tminust)f(ttT )
Instantaneous forward rate4
f(t T ) = L(t T ) = minuspart logP (tT )partT
Spot forward rate (yield)
f(t t T ) =r Tt f(t s)ds(T minus t)
1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t T )P (t T )3Can be modeled in the BGM model4Can be modeled in the HJM model
Fig 1332 Roadmap of stochastic interest rate modeling
460
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
Exercises
Exercise 131 Consider a tenor structure T1 T2 and a bond with maturityT2 and price given at time t isin [0 T2] by
P (t T2) = exp(minus
w T2
tf(t s)ds
) t isin [0 T2]
where the instantaneous yield curve f(t s) is parametrized as
f(t s) = r11[0T1](s) + r21[T1T2](s) s isin [t T2]
Find a formula to estimate the values of r1 and r2 from the data of P (0 T2)and P (T1 T2)
Exercise 132 Let (Bt)tisinR+ denote a standard Brownian motion started at0 under the risk-neutral measure Plowast We consider a short term interest rateprocess (rt)tisinR+ in a Ho-Lee model with constant deterministic volatilitydefined by
drt = adt+ σdBt
where a isin R and σ gt 0 Let P (t T ) will denote the arbitrage price of azero-coupon bond in this model
P (t T ) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ Ft] 0 6 t 6 T (1367)
a) State the bond pricing PDE satisfied by the function F (t x) defined via
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣ rt = x
] 0 6 t 6 T
b) Compute the arbitrage price F (t rt) = P (t T ) from its expression (1367)as a conditional expectation
Hint One may use the integration by parts argumentw T
tBsds = TBT minus tBt minus
w T
tsdBs
= (T minus t)Bt + T (BT minusBt)minusw T
tsdBs
461
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
= (T minus t)Bt +w T
t(T minus s)dBs
and the Laplace transform identity IE[ eλX ] = eλ2η22 for X N (0 η2)c) Check that the function F (t x) computed in question (b) does satisfy the
PDE derived in question (a)d) Compute the forward rate f(t T S) in this model
From now on we let a = 0e) Compute the instantaneous forward rate f(t T ) in this modelf) Derive the stochastic equation satisfied by the instantaneous forward ratef(t T )
g) Check that the HJM absence of arbitrage condition is satisfied in thisequation
Exercise 133 Consider the CIR process (rt)tisinR+ solution of
drt = minusartdt+ σradicrtdBt
where a σ gt 0 are constants (Bt)tisinR+ is a standard Brownian motion startedat 0
a) Write down the bond pricing PDE for the function F (t x) given by
F (t x) = IElowast[exp
(minus
w T
trsds
) ∣∣∣rt = x
] 0 6 t 6 T
Hint Use Itocirc calculus and the fact that the discounted bond price is amartingale
b) Show that the PDE of Question (a) admits a solution of the formF (t x) = eA(Tminust)+xC(Tminust) where the functions A(s) and C(s) satisfyordinary differential equations to be also written down together with thevalues of A(0) and C(0)
Exercise 134 Convertible bonds Consider an underlying stock price process(St)tisinR+ given by
dSt = rStdt+ σStdB(1)t
and a short term interest rate process (rt)tisinR+ given by
drt = γ(t rt)dt+ η(t rt)dB(2)t
where (B(1)t )tisinR+ and (B(2)
t )tisinR+ are two correlated Brownian motions underthe risk-neutral measure Plowast with dB
(1)t middot dB(2)
t = ρdt A convertible bondis a corporate bond that can be exchanged into a quantity α gt 0 of the
462
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
underlying companyrsquos stock Sτ at a future time τ whichever has a highervalue where α is a conversion ratea) Find the payoff of the convertible bond at time τ b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ T ) and a call option payoff on Sτ whose strike price is to bedetermined
c) Write down the corporate bond price at time t isin [0 τ ] as a functionC(t St rt) of the underlying asset price and interest rate using a dis-counted conditional expectation and show that the discounted corporatebond price
eminusr t
0 rsdsC(t St rt) t isin [0 τ ]
is a martingaled) Write down d
(eminus
r t0 rsdsC(t St rt)
)using the Itocirc formula and derive the
pricing PDE satisfied by the function C(t x y) together with its terminalcondition
e) Taking the bond price P (t T ) as a numeraire price the convertible bondas a European option with strike price K = 1 on an underlying assetpriced Zt = StP (t T ) t isin [0 τ ] under the forward measure IET
f) Assuming the bond price dynamics dP (t T ) = rtP (t T )dt+σB(t)P (t T )dBtdetermine the dynamics of the process (Zt)tisinR+ under the forward mea-sure IET
g) Assuming that (Zt)tisinR+ can be modeled as a geometric Brownian motionprice the corporate bond option using the Black-Scholes formula
Exercise 135 Given (Bt)tisinR+ a standard Brownian motion consider a HJMmodel given by
dtf(t T ) = σ2
2 T (T 2 minus t2)dt+ σTdBt (1368)
a) Show that the HJM condition is satisfied by (1368)b) Compute f(t T ) by solving (1368)
Hint We have f(t T ) = f(0 T ) +r t0 dsf(s T ) = middot middot middot
c) Compute the short rate rt = f(t t) from the result of Question (b)d) Show that the short rate rt satisfies a stochastic differential equation of
the formdrt = η(t)dt+ (rt minus f(0 t))ψ(t)dt+ ξ(t)dBt
where η(t) ψ(t) ξ(t) are deterministic functions to be determined
Exercise 136 Let (rt)tisinR+ denote a short term interest rate process Forany T gt 0 let P (t T ) denote the price at time t isin [0 T ] of a zero couponbond defined by the stochastic differential equation 463
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
dP (t T )P (t T ) = rtdt+ σTt dBt 0 6 t 6 T (1369)
under the terminal condition P (T T ) = 1 where (σTt )tisin[0T ] is an adaptedprocess We define the forward measure PT by
IElowast[dPTdPlowast
∣∣∣ Ft] = P (t T )P (0 T ) eminus
r t0 rsds 0 6 t 6 T
Recall thatBTt = Bt minus
w t
0σTs ds 0 6 t 6 T
is a standard Brownian motion under PT
a) Solve the stochastic differential equation (1369)b) Derive the stochastic differential equation satisfied by the discounted bond
price processt 7minusrarr eminus
r t0 rsdsP (t T ) 0 6 t 6 T
and show that it is a martingalec) Show that
IElowast[
eminusr T
0 rsds∣∣∣ Ft] = eminus
r t0 rsdsP (t T ) 0 6 t 6 T
d) Show that
P (t T ) = IElowast[
eminusr Ttrsds
∣∣∣ Ft] 0 6 t 6 T
e) Compute P (t S)P (t T ) 0 6 t 6 T show that it is a martingale underPT and that
P (T S) = P (t S)P (t T ) exp
(w T
t(σSs minus σTs )dBTs minus
12
w T
t(σSs minus σTs )2ds
)
f) Assuming that (σTt )tisin[0T ] and (σSt )tisin[0S] are deterministic functionscompute the price
IElowast[
eminusr Ttrsds (P (T S)minus κ)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minus κ)+
∣∣∣ Ft]of a bond option with strike price κ
Recall that if X is a centered Gaussian random variable with mean mt
and variance v2t given Ft we have
IE[( eX minusK)+ | Ft] = emt+v2t 2Φ
(vt2 + 1
vt(mt + v2
t 2minus logK))
464
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
minusKΦ(minusvt2 + 1
vt(mt + v2
t 2minus logK))
where Φ(x) x isin R denotes the Gaussian cumulative distribution function
Exercise 137 (Exercise 413 continued) Bridge model Assume that theprice P (t T ) of a zero coupon bond is modeled as
P (t T ) = eminusmicro(Tminust)+XTt t isin [0 T ]
where micro gt 0a) Show that the terminal condition P (T T ) = 1 is satisfiedb) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
c) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
d) Show that the limit limTt
f(t T ) does not exist in L2(Ω)e) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
f) Show using the results of Exercise 136-(d) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣ Ft] where (rT
t )tisin[0T ] is a process to be determinedg) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣∣∣ Ft]
= P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowasth) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT i) Compute the dynamics of XS
t and P (t S) under PT Hint Show that
465
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
j) Compute the bond option price
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣ Ft] = P (t T ) IET[(P (T S)minusK)+
∣∣∣ Ft] 0 6 t lt T lt S
Exercise 138 (Exercise 416 continued) Write down the bond pricing PDEfor the function
F (t x) = IElowast[
eminusr Ttrsds
∣∣∣ rt = x]
and show that in case α = 0 the corresponding bond price P (t T ) equals
P (t T ) = eminusB(Tminust)rt 0 6 t 6 T
whereB(x) = 2( eγx minus 1)
2γ + (β + γ)( eγx minus 1)
with γ =radicβ2 + 2σ2
Exercise 139 Consider a short rate process (rt)tisinR+ of the form rt = h(t) +Xt where h(t) is a deterministic function and (Xt)R+ is a Vasicek processstarted at X0 = 0a) Compute the price P (0 T ) at time t = 0 of a bond with maturity T using
h(t) and the function A(T ) defined in (1323) for the pricing of Vasicekbonds
b) Show how the function h(t) can be estimated from the market data of theinitial instantaneous forward rate curve f(0 t)
Exercise 1310a) Given two LIBOR spot rates L(t t T ) and L(t t S) compute the corre-
sponding LIBOR forward rate L(t T S)b) Assuming that L(t t T ) = 2 L(t t S) = 25 and t = 0 T = 1
S = 2T = 2 would you buy a LIBOR forward contract over [T 2T ] withrate L(0 T 2T ) if L(T T 2T ) remained at L(T T 2T ) = L(0 0 T ) = 2
Exercise 1311 Black-Derman-Toy model Consider a two-step interest ratemodel in which the short term interest rate r0 on [0 1] can turn into twopossible values ru1 = r0 emicro∆t+σ
radic∆t and rd1 = r0 emicro∆tminusσ
radic∆t on [1 2] with
equal probabilities 12 at time ∆t = 1 year and σ = 22 per year and twozero coupon bonds with prices P (0 1) and P (0 2) at time t = 0466
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
a) Write down the value of P (1 2) using ru1 and rd1 b) Write down the value of P (0 2) using ru1 rd1 and r0c) Estimate the value of r0 from the market price P (0 1) = 9174d) Estimatethe values of ru1 and rd1 from the market price P (0 2) = 8340
Exercise 1312 Consider a yield curve (f(t t T ))06t6T and a bond payingcoupons c1 c2 cn at times T1 T2 Tn until maturity Tn and priced as
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTk) 0 6 t 6 T1
where cn is inclusive of the last coupon payment and the nominal $1 valueof the bond Let f(t t Tn) denote the compounded yield to maturity definedby equating
P (t Tn) =nsumk=1
ck eminus(Tkminust)f(ttTn) 0 6 t 6 T1 (1370)
ie f(t t Tn) solves the equation
F (t f(t t Tn)) = P (t Tn) 0 6 t 6 T1
withF (t x) =
nsumk=1
ck eminus(Tkminust)x 0 6 t 6 T1
The bond duration D(t Tn) is the relative sensitivity of P (t Tn) with respectto f(t t Tn) defined as
D(t Tn) = minus 1P (t Tn)
partF
partx(t f(t t Tn)) 0 6 t 6 T1
The bond convexity C(t Tn) is defined as
C(t Tn) = 1P (t Tn)
part2F
partx2 (t f(t t Tn)) 0 6 t 6 T1
a) Compute the bond duration in case n = 1b) Show that the bond duration D(t Tn) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffsc) Show that the bond convexity C(t Tn) satisfies
C(t Tn) = (D(t Tn))2 + (S(t Tn))2
where S(t Tn) measures the dispersion of the duration of the bond payoffsaround the portfolio duration D(t Tn)
467
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
d) Consider now the zero-coupon yield defined as
fα(t t Tn) = minus 1α(Tn minus t)
logP (t t+ α(Tn minus t))
where α isin (0 1) Compute the bond duration associated to the yieldfα(t t Tn) in affine bond pricing models of the form
P (t T ) = eA(Tminust)+rtB(Tminust) 0 6 t 6 T
e) [Wu00] Compute the bond duration associated to the yield fα(t t Tn) inthe Vasicek model in which B(T minus t) = (1minus eminusb(Tminust))b 0 6 t 6 T
Exercise 1313 Stochastic string model [SCS01] Consider an instantaneousforward rate f(t x) solution of
dtf(t x) = αx2dt+ σdtB(t x) (1371)
with a flat initial curve f(0 x) = r where x represents the time to maturityand (B(t x))(tx)isinR2
+is a standard Brownian sheet with covariance
IE[B(s x)B(t y)] = min(s t)timesmin(x y) s t x y isin R+
and initial conditions B(t 0) = B(0 x) = 0 for all t x isin R+
a) Solve the equation (1371) for f(t x)b) Compute the short term interest rate rt = f(t 0)c) Compute the value at time t isin [0 T ] of the bond price
P (t T ) = exp(minus
w Tminust
0f(t x)dx
)with maturity T
d) Compute the variance IE[(w Tminust
0B(t x)dx
)2]of the centered Gaussian
random variabler Tminust0 B(t x)dx
e) Compute the expected value IElowast[P (t T )]f) Find the value of α such that the discounted bond price
eminusrtP (t T ) = exp(minusrT minus α
3 t(T minus t)3 minus σ
w Tminust
0B(t x)dx
) t isin [0 T ]
satisfies eminusrt IElowast[P (t T )] = eminusrT
g) Compute the bond option price IElowast[exp
(minus
w T
0rsds
)(P (T S)minusK)+
]by the Black-Scholes formula knowing that
468
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
Forward Rate Modeling
IE[(x em+XminusK)+] = x em+ v22 Φ(v+(m+log(xK))v)minusKΦ((m+log(xK))v)
when X is a centered Gaussian random variable with mean m = rτminusv22and variance v2
Exercise 1314 (Exercise 137 continued)
a) Compute the forward rate
f(t T S) = minus 1S minus T
(logP (t S)minus logP (t T ))
b) Compute the instantaneous forward rate
f(t T ) = minus limST
1S minus T
(logP (t S)minus logP (t T ))
c) Show that the limit limTt
f(t T ) does not exist in L2(Ω)d) Show that P (t T ) satisfies the stochastic differential equation
dP (t T )P (t T ) = σdBt + 1
2σ2dtminus logP (t T )
T minus tdt t isin [0 T ]
e) Show using the results of Exercise 136-(c) that
P (t T ) = IElowast[
eminusr TtrTs ds
∣∣∣Ft] where (rTt )tisin[0T ] is a process to be determined
f) Compute the conditional density
IElowast[dPTdPlowast
∣∣∣Ft] = P (t T )P (0 T ) eminus
r t0 r
Ts ds
of the forward measure PT with respect to Plowastg) Show that the process
Bt = Bt minus σt 0 6 t 6 T
is a standard Brownian motion under PT h) Compute the dynamics of XS
t and P (t S) under PT
Hint Show that
minusmicro(S minus T ) + σ(S minus T )w t
0
1S minus s
dBs = S minus TS minus t
logP (t S)
i) Compute the bond option price
469
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
Forward Rate Modeling
Short Term Models and Mean Reversion
Calibration of the Vasicek model
Zero-Coupon and Coupon Bonds
Forward Rates
The HJM Model
Forward Vasicek Rates
Modeling Issues
The BGM Model
Exercises
anm19
anm20
20EndLeft
20StepLeft
20PauseLeft
20PlayLeft
20PlayPauseLeft
20PauseRight
20PlayRight
20PlayPauseRight
20StepRight
20EndRight
20Minus
20Reset
20Plus
anm21
21EndLeft
21StepLeft
21PauseLeft
21PlayLeft
21PlayPauseLeft
21PauseRight
21PlayRight
21PlayPauseRight
21StepRight
21EndRight
21Minus
21Reset
21Plus
anm22
22EndLeft
22StepLeft
22PauseLeft
22PlayLeft
22PlayPauseLeft
22PauseRight
22PlayRight
22PlayPauseRight
22StepRight
22EndRight
22Minus
22Reset
22Plus
anm23
23EndLeft
23StepLeft
23PauseLeft
23PlayLeft
23PlayPauseLeft
23PauseRight
23PlayRight
23PlayPauseRight
23StepRight
23EndRight
23Minus
23Reset
23Plus
N Privault
IElowast[
eminusr TtrTs ds(P (T S)minusK)+
∣∣∣Ft] = P (t T ) IET[(P (T S)minusK)+ ∣∣Ft]
0 6 t lt T lt S
470
This version December 22 2017httpwwwntuedusghomenprivaultindexthtml
