Pricing of Interest Rate Derivatives

Chapter 14Pricing of Interest Rate Derivatives

In this chapter we consider the pricing of fixed income derivatives such ascaplets, caps, and swaptions, using change of numraire and forward swapmeasures.

14.1 Forward Measures and Tenor Structure

The maturity dates are arranged according to a discrete tenor structure

{0 = T0 < T1 < T2 < < Tn}.

An example of forward interest rate curve data is given in the table of Fig-ure 14.1, which contains the values of (T1, T2, . . . , T23) and of {f(t, t+Ti, t+Ti + )}i=1,2,...,23, with t = 07/05/2003 and = six months.

2D 1W 1M 2M 3M 1Y 2Y 3Y 4Y 5Y 6Y 7Y2.55 2.53 2.56 2.52 2.48 2.34 2.49 2.79 3.07 3.31 3.52 3.718Y 9Y 10Y 11Y 12Y 13Y 14Y 15Y 20Y 25Y 30Y3.88 4.02 4.14 4.23 4.33 4.40 4.47 4.54 4.74 4.83 4.86

Fig. 14.1: Forward rates arranged according to a tenor structure.

Recall that by definition of P (t, Ti) and absence of arbitrage the discountedbond price process

t 7 er t

0 rsdsP (t, Ti), 0 6 t 6 Ti, i = 1, 2, . . . , n,

is an Ft-martingale under the probability measure P = P, hence it satisfiesthe Assumption (A) page 386. As a consequence the bond price process canbe taken as a numraire

Nt := P (t, Ti), 0 6 t 6 Ti,

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in the definitiondPidP

= 1P (0, Ti)

er Ti

0 rsds (14.1)

of the forward measure Pi. The following proposition will allow us to pricecontingent claims using the forward measure Pi, it is a direct consequence ofProposition 12.3, noting that here we have P (Ti, Ti) = 1.Proposition 14.1. For all sufficiently integrable random variables F wehave

IE[F e

r Tit rsds

Ft] = P (t, Ti)IEi[F | Ft], 0 6 t 6 T, i = 1, 2, . . . , n.(14.2)

Recall that for all Ti, Tj > 0, the deflated process

t 7 P (t, Tj)P (t, Ti)

, 0 6 t 6 min(Ti, Tj),

is an Ft-martingale under Pi, cf. Proposition 12.4.

In the sequel we assume that the dynamics of the bond price P (t, Ti) isgiven by

dP (t, Ti)P (t, Ti)

= rtdt+ i(t)dWt, (14.3)

for i = 1, 2, . . . , n, where (Wt)tR+ is a standard Brownian motion underP and (rt)tR+ and (i(t))tR+ are adapted processes with respect to thefiltration (Ft)tR+ generated by (Wt)tR+ , i.e.

P (t, Ti) = P (0, Ti) exp(w t

0rsds+

w t0i(s)dWs

12

w t0|i(s)|2ds

),

0 6 t 6 Ti, i = 1, 2, . . . , n.

Forward Brownian motions

Proposition 14.2. For all i = 1, 2, . . . , n, the process

W it := Wt w t

0i(s)ds, 0 6 t 6 Ti, (14.4)

is a standard Brownian motion under the forward measure Pi.

Proof. The Girsanov Theorem 12.5 applied to the numraire

Nt := P (t, Ti), 0 6 t 6 Ti,472

This version: December 22, 2017http://www.ntu.edu.sg/home/nprivault/indext.html

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http://www.ntu.edu.sg/home/nprivault/indext.html

Pricing of Interest Rate Derivatives

as in (12.11), shows that

dW it := dWt 1NtdNt dWt

= dWt 1

P (t, Ti)dP (t, Ti) dWt

= dWt 1

P (t, Ti)(P (t, Ti)rtdt+ i(t)P (t, Ti)dWt) dWt

= dWt i(t)dt,

is a standard Brownian motion under the forward measure Pi for all i =1, 2, . . . , n.

We havedW it = dWt i(t)dt, i = 1, 2, . . . , n,

and

dW jt = dWt j(t)dt = dW it + (i(t) j(t))dt, i, j = 1, 2, . . . , n,

which shows that (W jt )tR+ has drift (i(t) j(t))tR+ under Pi.

Bond price dynamics under the forward measure

In order to apply Proposition 14.1 and to compute the price

IE[

er TtrsdsC

Ft] = P (t, Ti)IEi[C | Ft],of a random claim C, it can be useful to determine the dynamics of the un-derlying processes rt, f(t, T, S), and P (t, T ) via their stochastic differentialequations written under the forward measure Pi.

As a consequence of (14.3) and Proposition 14.2, the dynamics of t 7P (t, Tj) under Pi is given by

dP (t, Tj)P (t, Tj)

= rtdt+ i(t)j(t)dt+ j(t)dW it , i, j = 1, 2, . . . , n, (14.5)

where (W it )tR+ is a standard Brownian motion under Pi, and we have

P (t, Tj) = P (0, Tj) exp(w t

0rsds+

w t0j(s)dWs

12

w t0|j(s)|2ds

)= P (0, Tj) exp

(w t0rsds+

w t0j(s)dW js +

12

w t0|j(s)|2ds

)

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N. Privault

= P (0, Tj) exp(w t

0rsds+

w t0j(s)dW is +

w t0j(s)i(s)ds

12

w t0|j(s)|2ds

)= P (0, Tj) exp

(w t0rsds+

w t0j(s)dW is

12

w t0|j(s) i(s)|2ds+

12

w t0|i(s)|2ds

),

t [0, Tj ], i, j = 1, 2, . . . , n. Consequently, the forward price P (t, Tj)/P (t, Ti)can be written as

P (t, Tj)P (t, Ti)

= P (0, Tj)P (0, Ti)

exp(w t

0(j(s) i(s))dW js +

12

w t0|j(s) i(s)|2ds

)= P (0, Tj)P (0, Ti)

exp(w t

0(j(s) i(s))dW is

12

w t0|i(s) j(s)|2ds

),

(14.6)

t [0,min(Ti, Tj)], i, j = 1, 2, . . . , n, which also follows from Proposition 12.6.

Short rate dynamics under the forward measure

In case the short rate process (rt)tR+ is given as the (Markovian) solutionto the stochastic differential equation

drt = (t, rt)dt+ (t, rt)dWt,

its dynamics will be given under Pi by

drt = (t, rt)dt+ (t, rt)i(t)dt+ (t, rt)dW it . (14.7)

In the case of the Vasicek model, by (13.18) we have

drt = (a brt)dt+ dWt,

andi(t) =

b

(1 eb(Tit)

), 0 6 t 6 Ti,

hence from (14.7) we have

drt = (a brt)dt2

b

(1 eb(Tit)

)dt+ dW it (14.8)

and we obtain

dP (t, Ti)P (t, Ti)

= rtdt+2

b2(1 eb(Tit)

)2dt

b

(1 eb(Tit)

)dW it ,

from (13.18).

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14.2 Bond Options

The next proposition can be obtained as an application of the Margrabe for-mula (12.29) of Proposition 12.11 by taking Xt = P (t, Tj), Nt = P (t, Ti),and Xt = Xt/Nt = P (t, Tj)/P (t, Ti). In the Vasicek model, this formula hasbeen first obtained in [Jam89].

We work with a standard Brownian motion (Wt)tR+ under P, generatingthe filtration (Ft)tR+ , and an (Ft)tR+ -adapted short rate process (rt)tR+ .Proposition 14.3. Let 0 6 Ti 6 Tj and assume that the dynamics of thebond prices P (t, Ti), P (t, Tj) under P are given by

dP (t, Ti)P (t, Ti)

= rtdt+ i(t)dWt,dP (t, Tj)P (t, Tj)

= rtdt+ j(t)dWt,

where (i(t))tR+ and (j(t))tR+ are deterministic volatility functions. Thenthe price of a bond call option on P (Ti, Tj) with payoff

C := (P (Ti, Tj) )+

can be written as

IE[

er Tit rsds(P (Ti, Tj) )+

Ft] (14.9)= P (t, Tj)

(v(t, Ti)

2 +1

v(t, Ti)log P (t, Tj)

P (t, Ti)

)P (t, Ti)

(v(t, Ti)2 +

1v(t, Ti)

log P (t, Tj)P (t, Ti)

),

where v2(t, Ti) :=w Tit|i(s) j(s)|2ds and

(x) := 12

w x

ey2/2dy, x R,

is the Gaussian cumulative distribution function.Proof. First, we note that using Nt := P (t, Ti) as a numraire the price of abond call option on P (Ti, Tj) with payoff F = (P (Ti, Tj))+ can be writtenfrom Proposition 12.3 using the forward measure Pi, or directly by (12.5), as

IE[


Ft] = P (t, Ti)IEi [(P (Ti, Tj) )+ Ft] .(14.10)

Next, by (14.6) or by solving (12.13) in Proposition 12.6 we can writeP (Ti, Tj) as the geometric Brownian motion" 475



N. Privault

P (Ti, Tj) =P (t, Tj)P (t, Ti)

exp(w Ti

t(j(s) i(s))dW is

12

w Tit|i(s) j(s)|2ds

),

under the forward measure Pi, and rewrite (14.10) as

IE[


Ft]= P (t, Ti)IEi

[(P (t, Tj)P (t, Ti)

er Tit (j(s)i(s))dW

is 12

r Tit |i(s)j(s)|

2ds )+ Ft]

= IEi[(P (t, Tj) e

r Tit (j(s)i(s))dW

is 12

r Tit |i(s)j(s)|

2ds P (t, Ti))+ Ft] .

Since (i(s))s[0,Ti] and (j(s))s[0,Tj ] in (14.3) are deterministic volatilityfunctions, P (Ti, Tj) is a lognormal random variable given Ft under Pi andwe can use Lemma 6.5 to price the bond option by the Black-Scholes formula

Bl(P (t, Tj), P (0, Ti), v(t, Ti)/

Ti t, 0, Ti t

)with underlying P (t, Tj), strike price P (t, Ti), volatility parameter v(t, Ti)/

Ti t,

time to maturity Ti t, and zero interest rate, which yields (14.9).

Note that from Corollary 12.13 the decomposition (14.9) gives the self-financing portfolio in the assets P (t, Ti) and P (t, Tj) for the claim with payoff(P (Ti, Tj) )+.

In the Vasicek case the above bond option price could also be computedfrom the joint law of

(rT ,

r Ttrsds

), which is Gaussian, or from the dynamics

(14.5)-(14.8) of P (t, T ) and rt under Pi, cf. 7.3 of [Pri12], and [Kim02] forthe CIR and other short rate models with correlated Brownian motions.

14.3 Caplet Pricing

A caplet is an option contract that offers protection against the fluctationsof a variable (or floating) rate with respect to a fixed rate . The payoff of acaplet on the yield (or spot forward rate) f(Ti, Ti, Ti+1) with strike price can be written as

(f(Ti, Ti, Ti+1) )+,

priced at time t [0, Ti] from Proposition 12.3 using the forward measure Pias

IE[

er Ti+1t rsds(f(Ti, Ti, Ti+1) )+

Ft] (14.11)= P (t, Ti+1)IEi+1

[(f(Ti, Ti, Ti+1) )+ | Ft

],

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by taking Nt = P (t, Ti+1) as a numraire. Next, we consider the caplet withpayoff

(L(Ti, Ti, Ti+1) )+

on the LIBOR rate

L(t, Ti, Ti+1) =1

Ti+1 Ti

(P (t, Ti)P (t, Ti+1)

1), 0 6 t 6 Ti < Ti+1,

which is a martingale under the probability measure Pi+1 defined in (14.1),from Proposition 12.4. The caplet on L(Ti, Ti, Ti+1) can be priced at timet [0, Ti] as

IE[

er Ti+1t rsds(L(Ti, Ti, Ti+1) )+

Ft] (14.12)= IE

[e

r Ti+1t rsds

(1

Ti+1 Ti

(P (t, Ti)P (t, Ti+1)

1) )+ Ft] .

The next formula (14.14) is known as the Black caplet formula. It allows usto price and hedge a caplet using a portfolio based on the bonds P (t, Ti)and P (t, Ti+1), cf. (14.15) below, when L(t, Ti, Ti+1) is modeled in the BGMmodel of Section 13.8.Proposition 14.4. Assume that L(t, Ti, Ti+1) is modeled as

dL(t, Ti, Ti+1)L(t, Ti, Ti+1)

= i(t)dBi+1t , (14.13)

0 6 t 6 Ti, i = 1, 2, . . . , n 1, where t 7 i(t) is a deterministic function,i = 1, 2, . . . , n 1. The caplet on L(Ti, Ti, Ti+1) is priced at time t [0, Ti]as

IE[


Ft] (14.14)= P (t, Ti+1)L(t, Ti, Ti+1)(d+(t, Ti)) P (t, Ti+1)(d(t, Ti)),

0 6 t 6 Ti+1, where

d+(t, Ti) =log(L(t, Ti, Ti+1)/) + 2i (Ti t)/2

i(t, Ti)Ti t

,

andd(t, Ti) =

log(L(t, Ti, Ti+1)/) 2i (Ti t)/2i(t, Ti)

Ti t

,

and" 477



N. Privault

|i(t, Ti)|2 =1

Ti t

w Tit|i|2(s)ds.

Proof. Taking P (t, Ti+1) as a numraire, the forward price

Xt :=P (t, Ti)P (t, Ti+1)

= 1 + (Ti+1 Ti)L(Ti, Ti, Ti+1)

and the forward rate process (L(t, Ti, Ti+1)t[0,Ti] are martingales under Pi+1by Proposition 12.4. More precisely, by (14.13) we have

L(Ti, Ti, Ti+1) = L(t, Ti, Ti+1) exp(w Ti

ti(s)dBi+1s

12

w Tit|i(s)|2ds

),

0 6 t 6 Ti, i.e. t 7 L(t, Ti, Ti+1) is a geometric Brownian motion withvolatility i(t) under Pi+1. Hence by (14.11), since NTi+1 = 1, we have

IE[


Ft]= P (t, Ti+1)IEi+1

[(L(Ti, Ti, Ti+1) )+ | Ft

]= P (t, Ti+1)

(L(t, Ti, Ti+1)(d+(t, Ti)) er(d(t, Ti))

)= P (t, Ti+1)Bl(L(t, Ti, Ti+1), , i(t, Ti), 0, Ti t),

t [0, Ti], where

Bl(x, , , r, ) = x(d+(t, Ti)) er(d(t, Ti))

is the Black-Scholes function with

|i(t, Ti)|2 =1

Ti t

w Tit|i|2(s)ds.

In general we may also write (14.14) as

(Ti+1 Ti) IE[


Ft]= P (t, Ti+1)

(P (t, Ti)P (t, Ti+1)

1)(d+(t, Ti)) (Ti+1 Ti)P (t, Ti+1)(d(t, Ti))

= (P (t, Ti) P (t, Ti+1))(d+(t, Ti)) (Ti+1 Ti)P (t, Ti+1)(d(t, Ti)),

and by Corollary 12.13 this gives the self-financing portfolio strategy

((d+(t, Ti)),(d+(t, Ti)) (Ti+1 Ti)(d(t, Ti))) (14.15)

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in the bonds (P (t, Ti), P (t, Ti+1)) with maturities Ti and Ti+1, cf. Corol-lary 12.14 and [PT12].

The formula (14.14) is also known as the Black (1976) formula when appliedto options on underlying futures or forward contracts on commodities, whichare modeled according to (14.13). In this case, the bond price P (t, Ti+1) canbe simply modeled as P (t, Ti+1) = er(Ti+1t) and (14.14) becomes

er(Ti+1t)L(t, Ti, Ti+1)(d+(t, Ti)) er(Ti+1t)(d(t, Ti)),

where L(t, Ti, Ti+1) is the underlying future price.

Floorlets

Similarly, a floorlet on L(Ti, Ti, Ti+1) with strike price is a contract withpayoff ( L(Ti, Ti, Ti+1))+, priced at time t [0, Ti] as

IE[

er Ti+1t rsds( L(Ti, Ti, Ti+1))+

Ft]= P (t, Ti+1)IEi+1

[( L(Ti, Ti, Ti+1))+ | Ft

]= P (t, Ti+1)

(( d(Ti t)

) L(t, Ti, Ti+1)

( d+(Ti t)

)), (14.16)

0 6 t 6 Ti+1. Floorlets are analog to put options and can be similarly pricedby the call/put parity in the Black-Scholes formula.

Cap Pricing

More generally one can consider caps that are relative to a given tenor struc-ture {T1, T2, . . . , Tn}, with discounted payoff

n1k=1

(Tk+1 Tk) er Tk+1t rsds(L(Tk, Tk, Tk+1) )+.

Pricing formulas for caps are easily deduced from analog formulas for caplets,since the payoff of a cap can be decomposed into a sum of caplet payoffs. Thusthe price of a cap at time t [0, T1] is given by

IE[n1k=1

(Tk+1 Tk) er Tk+1t rsds(L(Tk, Tk, Tk+1) )+

Ft]

=n1k=1

(Tk+1 Tk) IE[

er Tk+1t rsds(L(Tk, Tk, Tk+1) )+

Ft]

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N. Privault

=n1k=1

(Tk+1 Tk)P (t, Tk+1)IEk+1[(L(Tk, Tk, Tk+1) )+

Ft].(14.17)

In the BGM model (14.13) the cap with payoff

n1k=1

(Tk+1 Tk)(L(Tk, Tk, Tk+1) )+

can be priced at time t [0, T1] by the Black formula

n1k=1

(Tk+1 Tk)P (t, Tk+1)Bl(L(t, Tk, Tk+1), , k(t, Tk), 0, Tk t).

14.4 Forward Swap Measures

In this section we introduce the forward measures to be used for the pricingof swaptions, and we study their properties. We start with the definition ofthe annuity numraire

Nt := P (t, Ti, Tj) =j1k=i

(Tk+1 Tk)P (t, Tk+1), 0 6 t 6 Ti, (14.18)

with in particular, when j = i+ 1,

P (t, Ti, Ti+1) = (Ti+1 Ti)P (t, Ti+1), 0 6 t 6 Ti.

1 6 i < n. The annuity numraire can be also used to price a bond ladder. Itsatisfies the following martingale property, which can be proved by linearityand the fact that t 7 e

r t0 rsdsP (t, Tk) is a martingale for all k = 1, 2, . . . , n,

under Assumption (A).

Remark 14.5. The discounted annuity numraire

t 7 er t

0 rsdsP (t, Ti, Tj) = er t

0 rsds

j1k=i

(Tk+1Tk)P (t, Tk+1), 0 6 t 6 Ti,

is a martingale under P.

The forward swap measure Pi,j is defined, according to Definition 12.1, by

dPi,jdP

:= er Ti

0 rsdsP (Ti, Ti, Tj)P (0, Ti, Tj)

, (14.19)

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1 6 i < j 6 n. We have

IE[dPi,jdP

Ft] = 1P (0, Ti, Tj)

IE[

er Ti

0 rsdsP (Ti, Ti, Tj) Ft]

= 1P (0, Ti, Tj)

IE[

er Ti

0 rsds

j1k=i

(Tk+1 Tk)P (Ti, Tk+1) Ft]

= 1P (0, Ti, Tj)

j1k=i

(Tk+1 Tk) IE[

er Ti

0 rsdsP (Ti, Tk+1) Ft]

= 1P (0, Ti, Tj)

er t

0 rsds

j1k=i

(Tk+1 Tk)P (t, Tk+1)

= P (t, Ti, Tj)P (0, Ti, Tj)

er t

0 rsds,

0 6 t 6 Ti, by Remark 14.5, and

dPi,j|FtdP|Ft

= er Tit rsds

P (Ti, Ti, Tj)P (t, Ti, Tj)

, 0 6 t 6 Ti+1, (14.20)

by Relation (12.3) in Lemma 12.2. We also know that the deflated process

t 7 vi,jk (t) :=P (t, Tk)P (t, Ti, Tj)

, i, j, k = 1, 2, . . . , n,

is an Ft-martingale under Pi,j by Proposition 12.4. It follows that the LIBORswap rate

S(t, Ti, Tj) :=P (t, Ti) P (t, Tj)

P (t, Ti, Tj)= vi,ji (t) v

i,jj (t), 0 6 t 6 Ti,

defined in Proposition 13.9 is also a martingale under Pi,j .

Using the forward swap measure we obtain the following pricing formulafor a given integrable claim with payoff of the form P (Ti, Ti, Tj)F :

IE[

er Tit rsdsP (Ti, Ti, Tj)F

Ft] = P (t, Ti, Tj) IE [F dPi,j|FtdP|Ft

Ft]= P (t, Ti, Tj)IEi,j

[F Ft], (14.21)

after applying (14.19) and (14.20) on the last line, or Proposition 12.3.

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14.5 Swaption Pricing on the LIBOR

A swaption on the forward rate f(T1, Tk, Tk+1) is a contract meant for pro-tection against a risk based on an interest rate swap, and has payoff(

j1k=i

(Tk+1 Tk) er Tk+1Ti

rsds(f(Ti, Tk, Tk+1) ))+

,

at time Ti. This swaption can be priced at time t [0, Ti] under the risk-neutral measure P as

IE(j1

k=i(Tk+1 Tk) e

r Tk+1t rsds(f(Ti, Tk, Tk+1) )

)+ Ft (14.22)

= IE e r Tit rsds(j1

k=i(Tk+1 Tk) e

r Tk+1Ti

rsds(f(Ti, Tk, Tk+1) ))+ Ft

.In the sequel and in practice the price (14.22) of the swaption will be evalu-ated as

IE e r Tit rsds(j1

k=i(Tk+1 Tk)P (Ti, Tk+1)(f(Ti, Tk, Tk+1) )

)+ Ft ,

(14.23)t [0, Ti], i.e. the discount factor e

r Tk+1Ti

rsds is replaced with the bondprice P (Ti, Tk+1), which is its conditional expectation given FTi .

Using the inequality

(x1 + x2 + + xm)+ 6 x+1 + x+2 + + x+m, x1, . . . , xm R,

the above term (14.23) can be upper bounded by the cap price (14.17) writtenas

IE e r Tit rsds(j1

k=i(Tk+1 Tk)P (Ti, Tk+1)(f(Ti, Tk, Tk+1) )

)+ Ft

6 IE[

er Tit rsds

j1k=i

(Tk+1 Tk)P (Ti, Tk+1) (f(Ti, Tk, Tk+1) )+ Ft]

=j1k=i

(Tk+1 Tk) IE[

er Tit rsdsP (Ti, Tk+1) (f(Ti, Tk, Tk+1) )+

Ft]

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=j1k=i

(Tk+1 Tk) IE[

er Tit rsds IE

[e

r Tk+1Ti

rsds FTi] (f(Ti, Tk, Tk+1) )+ Ft]

=j1k=i

(Tk+1 Tk) IE[IE[

er Tk+1t rsds (f(Ti, Tk, Tk+1) )+

FTi] Ft]

=j1k=i

(Tk+1 Tk) IE[

er Tk+1t rsds (f(Ti, Tk, Tk+1) )+

Ft]

= IE[j1k=i

(Tk+1 Tk) er Tk+1t rsds (f(Ti, Tk, Tk+1) )+

Ft] ,0 6 t 6 Ti. In addition, when j = i+ 1, the swaption price (14.23) coincideswith the price at time t of a caplet on [Ti, Ti+1] since

IE[

er Tit rsds ((Ti+1 Ti)P (Ti, Ti+1)(f(Ti, Ti, Ti+1) ))+

Ft]= (Ti+1 Ti) IE

[e

r Tit rsdsP (Ti, Ti+1) (f(Ti, Ti, Ti+1) )+

Ft]= (Ti+1 Ti) IE

[e

r Tit rsds IE

[e

r Ti+1Ti

rsds FTi] (f(Ti, Ti, Ti+1) )+ Ft]

= (Ti+1 Ti) IE[IE[

er Tit rsds e

r Ti+1Ti

rsds (f(Ti, Ti, Ti+1) )+ FTi] Ft]

= (Ti+1 Ti) IE[

er Ti+1t rsds (f(Ti, Ti, Ti+1) )+

Ft] ,0 6 t 6 Ti, which coincides with the caplet price (14.11) up to the factorTi+1 Ti.

In case the forward rate f(t, T, S) is replaced with the LIBOR rateL(t, T, S) defined in Proposition 13.9, the payoff of the swaption can berewritten as in the following lemma which is a direct consequence of thedefinition of the swap rate S(Ti, Ti, Tj).Lemma 14.6. The payoff of the swaption in (14.23) can be rewritten as(

j1k=i

(Tk+1 Tk)P (Ti, Tk+1)(L(Ti, Tk, Tk+1) ))+

= (P (Ti, Ti) P (Ti, Tj) P (Ti, Ti, Tj))+ (14.24)= P (Ti, Ti, Tj) (S(Ti, Ti, Tj) )+ . (14.25)

Proof. The relation

j1k=i

(Tk+1 Tk)P (t, Tk+1)(L(t, Tk, Tk+1) S(t, Ti, Tj)) = 0

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N. Privault

that defines the forward swap rate S(t, Ti, Tj) shows that

j1k=i

(Tk+1 Tk)P (t, Tk+1)L(t, Tk, Tk+1)

= S(t, Ti, Tj)j1k=i


= P (t, Ti, Tj)S(t, Ti, Tj)= P (t, Ti) P (t, Tj),

cf. Proposition 13.9, hence by the definition (14.18) of P (t, Ti, Tj) we have

j1k=i

(Tk+1 Tk)P (t, Tk+1)(L(t, Tk, Tk+1) )

= P (t, Ti) P (t, Tj) P (t, Ti, Tj)= P (t, Ti, Tj) (S(t, Ti, Tj) ) ,

and for t = Ti we get(j1k=i


= P (Ti, Ti, Tj) (S(Ti, Ti, Tj) )+ .

The next proposition simply states that a swaption on the LIBOR rate canbe priced as a European call option on the swap rate S(Ti, Ti, Tj) under theforward swap measure Pi,j .

Proposition 14.7. The price (14.23) of the European swaption with payoff(j1k=i


(14.26)

on the LIBOR market can be written under the forward swap measure Pi,j as

P (t, Ti, Tj)IEi,j[(S(Ti, Ti, Tj) )+

Ft] , 0 6 t 6 Ti.Proof. As a consequence of (14.21) and Lemma 14.6 we find

IE e r Tit rsds(j1

k=i(Tk+1 Tk)P (Ti, Tk+1)(L(Ti, Tk, Tk+1) )

)+ Ft

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= IE[

er Tit rsds (P (Ti, Ti) P (Ti, Tj) P (Ti, Ti, Tj))+

Ft](14.27)

= IE[

er Tit rsdsP (Ti, Ti, Tj) (S(Ti, Ti, Tj) )+

Ft]= P (t, Ti, Tj) IE

[dPi,j|FtdP|Ft

(S(Ti, Ti, Tj) )+ Ft]

= P (t, Ti, Tj)IEi,j[(S(Ti, Ti, Tj)

)+ Ft]. (14.28)

In the next Proposition 14.8, we price a swaption with payoff (14.26) orequivalently (14.25), by modeling the swap rate (S(t, Ti, Tj))06t6Ti usingstandard Brownian motion (W i,jt )06t6Ti under the forward measure Pi,j , seeExercise 14 for swaption pricing without the Black-Scholes formula.

Proposition 14.8. Assume that the LIBOR swap rate (13.46) is modeled asa geometric Brownian motion under Pi,j, i.e.

dS(t, Ti, Tj) = S(t, Ti, Tj)(t)dW i,jt ,

where ((t))tR+ is a deterministic function. Then the swaption with payoff

(P (T, Ti) P (T, Tj) P (Ti, Ti, Tj))+ = P (Ti, Ti, Tj) (S(Ti, Ti, Tj) )+

can be priced using the Black-Scholes formula as

IE[


Ft]= (P (t, Ti) P (t, Tj))+(d+(Ti t))

(d(Ti t))j1k=i

(Tk+1 Tk)P (t, Tk+1),

whered+(Ti t) =

log(S(t, Ti, Tj)/) + 2i,j(t, Ti)/2i,j(t, Ti)

Ti t

,

andd(Ti t) =

log(S(t, Ti, Tj)/) 2i,j(t, Ti)/2i,j(t, Ti)

Ti t

,

and|i,j(t, Ti)|2 =

1Ti t

w Tit

((s))2ds, 0 6 t 6 T.

Proof. Since S(t, Ti, Tj) is a geometric Brownian motion with volatility func-tion ((t))tR+ under Pi,j , by (14.24)-(14.25) and (14.27)-(14.28) we have

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N. Privault

IE[


Ft]= IE

[e

r Ttrsds(P (T, Ti) P (T, Tj) P (Ti, Ti, Tj))+

Ft]= P (t, Ti, Tj)IEi,j

[(S(Ti, Ti, Tj) )+

Ft]= P (t, Ti, Tj)Bl(S(Ti, Ti, Tj), , i,j(t, Ti), 0, Ti t)= P (t, Ti, Tj) (S(t, Ti, Tj)+(t, S(t, Ti, Tj)) (t, S(t, Ti, Tj)))=(P (t, Ti) P (t, Tj)

)+(t, S(t, Ti, Tj)) P (t, Ti, Tj)(t, S(t, Ti, Tj))

=(P (t, Ti) P (t, Tj)

)+(t, S(t, Ti, Tj))

(t, S(t, Ti, Tj))j1k=i

(Tk+1 Tk)P (t, Tk+1).

In addition, the hedging strategy

(+(t, S(t, Ti, Tj)),(t, S(t, Ti, Tj))(Ti+1 Ti), . . .. . . ,(t, S(t, Ti, Tj))(Tj1 Tj2),+(t, S(t, Ti, Tj)))

based on the assets (P (t, Ti), . . . , P (t, Tj)) is self-financing by Corollary 12.14,cf. also [PT12].

Swaption prices can also be computed by an approximation formula, from theexact dynamics of the swap rate S(t, Ti, Tj) under Pi,j , based on the bondprice dynamics of the form (14.3), cf. [Sch05], page 17.

Swaption volatilities can be estimated from swaption prices as impliedvolatilities from the Black pricing formula:

0 1 2 3 4 5 6 7 8 9j 0

1 2

3 4

5 6

7 8

9

i 0.08

0.1

0.12

0.14

0.16

0.18

0.2

Fig. 14.2: Implied swaption volatilities.

486


"



Implied swaption volatilities can then be used to calibrate the BGMmodel,cf. [Sch05], [PW09], 11.4 of [Pri12].

Bermudan swaption pricing in Quantlib

The Bermudan swaption on the tenor structure {Ti, . . . , Tj} is priced as thesupremum

sup{Ti,...,Tj1}

IE e r t rsds(j1

k=(Tk+1 Tk)P (, Tk+1)(L(, Tk, Tk+1) )

)+ Ft

= sup{Ti,...,Tj1}

IE[

er trsds (P (, ) P (, Tj) P (, , Tj))+

Ft]= sup

{Ti,...,Tj1}IE[

er trsdsP (, , Tj) (S(, , Tj) )+

Ft] ,where the supremum is over all stopping times taking values in {Ti, . . . , Tj}.

Bermudan swaptions can be priced using this in (R)quantlib, with thefollowing output:

Summary of pricing results for Bermudan Swaption

Price (in bp) of Bermudan swaption is 24.92137Stike is NULL (ATM strike is 0.05 )Model used is: Hull-White using analytic formulasCalibrated model parameters are:a = 0.04641sigma = 0.005869

This modified can be used in particular the pricing of ordinary swap-tions with the output:

Summary of pricing results for Bermudan Swaption

Price (in bp) of Bermudan swaption is 22.45436Stike is NULL (ATM strike is 0.05 )Model used is: Hull-White using analytic formulasCalibrated model parameters are:a = 0.07107sigma = 0.006018 Click to open or download. Click to open or download.

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# https://cran.r-project.org/web/packages/RQuantLib/RQuantLib.pdfinstall.packages("RQuantLib")library(RQuantLib)params

N. Privault

Exercises

Exercise 14.1 Consider a floorlet on a three-month LIBOR rate in ninemonths time, with a notional principal amount of $1 million. The term struc-ture is flat at 4.95% with continuous compounding, and the volatility of theforward LIBOR rate in nine months is 10%. The annual floor rate with simplecompounding is 4.5% and the floorplet price is quoted in basis points (onebasis point = 0.01%).a) What are the key assumptions on the LIBOR rate in nine month in order

to apply Blacks formula to price this floorlet?b) Compute the price of this floorlet using Blacks formula as an application

of Proposition 14.4 and (14.16), using the functions (d+) and (d).

Exercise 14.2 Consider a European swaption giving its holder the right toenter into a 3-year annual pay swap in four years, where a fixed rate of 5%is paid and the LIBOR rate is received. Assume that the yield curve is flatat 5% per annum with annual compounding and the volatility of the swaprate is 20%. The notional principal is $10 millions and the swaption price isquoted in basis points.a) What are the key assumptions in order to apply Blacks formula to value

this swaption?b) Compute the price of this swaption using Blacks formula as an applica-

tion of Proposition 14.8, using the functions (d+) and (d).

Exercise 14.3 Consider two bonds with maturities T1 and T2, T1 < T2, whichfollow the stochastic differential equations

dP (t, T1) = rtP (t, T1)dt+ 1(t)P (t, T1)dWt

anddP (t, T2) = rtP (t, T2)dt+ 2(t)P (t, T2)dWt.

a) Using It calculus, show that the forward process P (t, T2)/P (t, T1) is adriftless geometric Brownian motion driven by dWt := dWt1(t)dt underthe T1-forward measure P.

b) Compute the price IE[

er T1

0 rsds(K P (T1, T2))+]of a bond put option

using change of numeraire and the Black-Scholes formula.

Hint: Given X a centered Gaussian random variable with variance v2, wehave:

IE[(x eXv

2/2)+]

= (v/2+(log(/x))/v)x(v/2+(log(/x))/v).488


"



Exercise 14.4 Given two bonds with maturities T , S and prices P (t, T ),P (t, S), consider the LIBOR rate

L(t, T, S) := P (t, T ) P (t, S)(S T )P (t, S)

at time t [0, T ], modeled as

dL(t, T, S) = tL(t, T, S)dt+ L(t, T, S)dWt, 0 6 t 6 T, (14.29)

where (Wt)t[0,T ] is a standard Brownian motion under the risk-neutral mea-sure P, > 0 is a constant, and (t)t[0,T ] is an adapted process. Let

Ft = IE[

er Strsds( L(T, T, S))+

Ft]denote the price at time t of a floorlet option with strike price , maturityT , and payment date S.

a) Rewrite the value of Ft using the forward measure PS with maturity S.b) What is the dynamics of L(t, T, S) under the forward measure PS?c) Write down the value of Ft using the Black-Scholes formula.

Hint. Given X a centered Gaussian random variable with variance v2 wehave

IE[( em+X)+] = ((m log )/v) em+v2/2(v (m log )/v),

where denotes the Gaussian cumulative distribution function.

Exercise 14.5 Jamshidians trick. Consider a European swaption with payoff

(P (Ti, Ti) P (Ti, Tj) P (Ti, Ti, Tj)

)+ = (1 j1k=i

ck+1P (Ti, Tk+1))+

,

where cj contains the final capital payment. We assume that the bond pricesare functions P (Ti, Tk+1) = Fk(Ti, rTi) of rTi that are increasing in the vari-able rTi , for all k = i, i+ 1, . . . , j 1, where (rt)tR+ denotes the short rateprocess. Show, choosing such that

j1k=i

ck+1Fk+1(Ti, ) = 1,

that the swaption can be priced as a sum of weighted bond put options underthe forward measure Pi with numeraire Nt := P (t, Ti).

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N. Privault

Exercise 14.6 We work in the short rate model

drt = dBt,

where (Bt)tR+ is a standard Brownian motion under P, and P2 is the for-ward measure defined by

dP2dP

= 1P (0, T2)

er T2

0 rsds.

a) State the expressions of 1(t) and 2(t) in

dP (t, Ti)P (t, Ti)

= rtdt+ i(t)dBt, i = 1, 2,

and the dynamics of the P (t, T1)/P (t, T2) under P2, where P (t, T1) andP (t, T2) are bond prices with maturities T1 and T2.

b) State the expression of the forward rate f(t, T1, T2).c) Compute the dynamics of f(t, T1, T2) under the forward measure P2 with

dP2dP

= 1P (0, T2)

er T2

0 rsds.

d) Compute the price

(T2 T1) IE[

er T2t rsds(f(T1, T1, T2) )+

Ft]of a cap at time t [0, T1], using the expectation under the forwardmeasure P2.

e) Compute the dynamics of the swap rate process

S(t, T1, T2) =P (t, T1) P (t, T2)(T2 T1)P (t, T2)

, t [0, T1],

under P2.f) Compute the swaption price

(T2 T1) IE[

er T1t rsdsP (T1, T2)(S(T1, T1, T2) )+

Ft]on the swap rate S(T1, T1, T2) using the expectation under the forwardswap measure P1,2.

Exercise 14.7 Consider three zero-coupon bonds P (t, T1), P (t, T2) andP (t, T3) with maturities T1 = , T2 = 2 and T3 = 3 respectively, andthe forward LIBOR L(t, T1, T2) and L(t, T2, T3) defined by

490


"



L(t, Ti, Ti+1) =1

(P (t, Ti)P (t, Ti+1)

1), i = 1, 2.

Assume that L(t, T1, T2) and L(t, T2, T3) are modeled in the BGM model by

dL(t, T1, T2)L(t, T1, T2)

= eatdW 2t , 0 6 t 6 T1, (14.30)

and L(t, T2, T3) = b, 0 6 t 6 T2, for some constants a, b > 0, where W 2t is astandard Brownian motion under the forward rate measure P2 defined by

dP2dP

= e

r T20 rsds

P (0, T2).

a) Compute L(t, T1, T2), 0 6 t 6 T2 by solving Equation (14.30).b) Show that the price at time t of the caplet with strike price can be

written as

IE[

er T2t rsds(L(T1, T1, T2) )+

Ft] = P (t, T2)IE2 [(L(T1, T1, T2) )+ | Ft] ,where IE2 denotes the expectation under the forward measure P2.

c) Using the hint below, compute the price at time t of the caplet with strikeprice on L(T1, T1, T2).

d) Compute

P (t, T1)P (t, T1, T3)

, 0 6 t 6 T1, andP (t, T3)

P (t, T1, T3), 0 6 t 6 T2,

in terms of b and L(t, T1, T2), where P (t, T1, T3) is the annuity numraire

P (t, T1, T3) = P (t, T2) + P (t, T3), 0 6 t 6 T2.

e) Compute the dynamics of the swap rate

t 7 S(t, T1, T3) =P (t, T1) P (t, T3)

P (t, T1, T3), 0 6 t 6 T1,

i.e. show that we have

dS(t, T1, T3) = 1,3(t)S(t, T1, T3)dW 2t ,

where 1,3(t) is a process to be determined.f) Using the Black-Scholes formula, compute an approximation of the swap-

tion price

IE[

er T1t rsdsP (T1, T1, T3)(S(T1, T1, T3) )+

Ft]" 491



N. Privault

= P (t, T1, T3)IE2[(S(T1, T1, T3) )+ | Ft

],

at time t [0, T1]. You will need to approximate 1,3(s), s > t, by freez-ing all random terms at time t.


IE[( em+X )+

]= em+v

2/2(v + (m log )/v) ((m log )/v),


Exercise 14.8 Consider a portfolio allocation (Tt , St )t[0,T ] made of twobonds with maturities T , S, and value

Vt = Tt P (t, T ) + St P (t, S), 0 6 t 6 T,

at time t. We assume that the portfolio is self-financing, i.e.

dVt = Tt dP (t, T ) + St dP (t, S), 0 6 t 6 T, (14.31)

and that it hedges the claim (P (T, S) )+, so that

Vt = IE[

er Ttrsds (P (T, S) )+

Ft]= P (t, T ) IET

[(P (T, S) )+

Ft] , 0 6 t 6 T.a) Show that

IE[

er Ttrsds (P (T, S)K)+

Ft]= P (0, T ) IET

[(P (T, S)K)+

]+

w t0Ts dP (s, T ) +

w t0Ss dP (s, S).

b) Show that under the self-financing condition (14.31), the discounted port-folio price Vt = e

r t0 rsdsVt satisfies

dVt = Tt dP (t, T ) + St dP (t, S),

where P (t, T ) = er t

0 rsdsP (t, T ) and P (t, S) = er t

0 rsdsP (t, S) denotethe discounted bond prices.

c) Show that

IET[(P (T, S)K)+ |Ft

]= IET

[(P (T, S)K)+

]+

w t0

C

x(Xu, T u, v(u, T ))dXu.

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Hint: use the martingale property and the It formula.d) Show that the discounted portfolio price Vt = Vt/P (t, T ) satisfies

dVt =C

x(Xt, T t, v(t, T ))dXt

= P (t, S)P (t, T )

C

x(Xt, T t, v(t, T ))(St Tt )dBTt .

e) Show that

dVt = P (t, S)C

x(Xt, T t, v(t, T ))(St Tt )dBt + VtdP (t, T ).

f) Show that

dVt = P (t, S)C

x(Xt, T t, v(t, T ))(St Tt )dBt + VtdP (t, T ).

g) Compute the hedging strategy (Tt , St )t[0,T ] of the bond option.h) Show that

C

x(x, , v) =

(log(x/K) + v2/2

v

).

Exercise 14.9 Consider a LIBOR rate L(t, T, S), t [0, T ], modeled asdL(t, T, S) = tL(t, T, S)dt+(t)L(t, T, S)dWt, 0 6 t 6 T , where (Wt)t[0,T ]is a standard Brownian motion under the risk-neutral measure P, (t)t[0,T ]is an adapted process, and (t) > 0 is a deterministic function.a) What is the dynamics of L(t, T, S) under the forward measure P with

numeraire Nt := P (t, S)?b) Rewrite the price

IE[

er Strsds(L(T, T, S))

Ft] (14.32)at time t [0, T ] of an option with payoff function using the forwardmeasure P.

c) Write down the above option price (14.32) using an integral.

Exercise 14.10 Given n bonds with maturities T1, T2, . . . , Tn, consider theannuity numeraire P (t, Ti, Tj) =

j1k=i(Tk+1 Tk)P (t, Tk+1) and the swap

rateS(t, Ti, Tj) =

P (t, Ti) P (t, Tj)P (t, Ti, Tj)

at time t [0, Ti], modeled as

dS(t, Ti, Tj) = tS(t, Ti, Tj)dt+ S(t, Ti, Tj)dWt, 0 6 t 6 Ti, (14.33)" 493



N. Privault

where (Wt)t[0,Ti] is a standard Brownian motion under the risk-neutral mea-sure P, (t)t[0,T ] is an adapted process and > 0 is a constant. Let

IE[

er Tit rsdsP (Ti, Ti, Tj)(S(Ti, Ti, Tj))

Ft] (14.34)at time t [0, Ti] of an option with payoff function .

a) Rewrite the option price (14.34) at time t [0, Ti] using the forward swapmeasure Pi,j defined from the annuity numeraire P (t, Ti, Tj).

b) What is the dynamics of S(t, Ti, Tj) under the forward swap measure Pi,j?c) Write down the above option price (14.32) using a Gaussian integral.d) Apply the above to the computation at time t [0, Ti] of the put swaption

priceIE[

er Tit rsdsP (Ti, Ti, Tj)( S(Ti, Ti, Tj))+

Ft]with strike price , using the Black-Scholes formula.


IE[( em+X)+] = ((m log )/v) em+v2/2(v (m log )/v),


Exercise 14.11 Consider a bond market with two bonds with maturities T1,T2, whose prices P (t, T1), P (t, T2) at time t are given by

dP (t, T1)P (t, T1)

= rtdt+ 1(t)dBt,dP (t, T2)P (t, T2)

= rtdt+ 2(t)dBt,

where (rt)tR+ is a short term interest rate process, (Bt)tR+ is a standardBrownian motion generating a filtration (Ft)tR+ , and 1(t), 2(t) are volatil-ity processes. The LIBOR rate L(t, T1, T2) is defined by

L(t, T1, T2) =P (t, T1) P (t, T2)

P (t, T2).

Recall that a caplet on the LIBOR market can be priced at time t [0, T1]as

IE[

er T2t rsds (L(T1, T1, T2) )+

Ft] (14.35)= P (t, T2)IE

[(L(T1, T1, T2)

)+ Ft],under the forward measure P defined by

494


"



dPdP

= er T1

0 rsdsP (T1, T2)P (0, T2)

,

under whichBt := Bt

w t02(s)ds, t R+, (14.36)

is a standard Brownian motion.

In the sequel we let Lt = L(t, T1, T2) for simplicity of notation.

a) Using It calculus, show that the LIBOR rate satisfies

dLt = Lt(t)dBt, 0 6 t 6 T1, (14.37)

where the LIBOR rate volatility is given by

(t) = P (t, T1)(1(t) 2(t))P (t, T1) P (t, T2)

.

b) Solve the equation (14.37) on the interval [t, T1], and compute LT1 fromthe initial condition Lt.

c) Assuming that (t) in (14.37) is a deterministic function, show that theprice

P (t, T2)IE[(LT1

)+ Ft]of the caplet can be written as P (t, T2)C(Lt, v(t, T1)), where v2(t, T1) =w T1t|(s)|2ds, and C(t, v(t, T1)) is a function of Lt and v(t, T1).

d) Consider a portfolio allocation (1t , 2t )t[0,T1] made of bonds with matu-rities T1, T2 and value

Vt = 1tP (t, T1) + 2tP (t, T2),

at time t [0, T1]. We assume that the portfolio is self-financing, i.e.

dVt = 1t dP (t, T1) + 2t dP (t, T2), 0 6 t 6 T1, (14.38)

and that it hedges the claim (LT1 )+, so that

Vt = IE[

er T1t rsds (P (T1, T2)(LT1 ))

+ Ft]

= P (t, T2)IE[(LT1

)+ Ft],0 6 t 6 T1. Show that we have

IE[

er T1t rsds

(P (T1, T2)(LT1 )

)+ Ft]= P (0, T2)IE

[(LT1

)+]+ w t01sdP (s, T1) +

w t02sdP (s, T1),

" 495



N. Privault

0 6 t 6 T1.e) Show that under the self-financing condition (14.38), the discounted port-

folio price Vt = er t

0 rsdsVt satisfies

dVt = 1t dP (t, T1) + 2t dP (t, T2),

where P (t, T1) := er t

0 rsdsP (t, T1) and P (t, T2) := er t

0 rsdsP (t, T2) de-note the discounted bond prices.

f) Show that

IE[(LT1

)+ Ft] = IE[(LT1 )+]+ w t0 Cx (Lu, v(u, T1))dLu,and that the discounted portfolio price Vt = Vt/P (t, T2) satisfies

dVt =C

x(Lt, v(t, T1))dLt = Lt

C

x(Lt, v(t, T1))(t)dBt.

Hint: use the martingale property and the It formula.g) Show that

dVt = (P (t, T1) P (t, T2))C

x(Lt, v(t, T1))(t)dBt + VtdP (t, T2).

h) Show that

dVt =C

x(Lt, v(t, T1))d(P (t, T1) P (t, T2))

+(Vt Lt

C

x(Lt, v(t, T1))

)dP (t, T2),

and deduce the values of the hedging portfolio allocation (1t , 2t )tR+ .

Exercise 14.12 Consider a bond market with tenor structure {Ti, . . . , Tj} andbonds with maturities Ti, . . . , Tj , whose prices P (t, Ti), . . . P (t, Tj) at time tare given by

dP (t, Tk)P (t, Tk)

= rtdt+ k(t)dBt, k = i, . . . , j,

where (rt)tR+ is a short term interest rate process and (Bt)tR+ de-notes a standard Brownian motion generating a filtration (Ft)tR+ , andi(t), . . . , j(t) are volatility processes.

The swap rate S(t, Ti, Tj) is defined by

496


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S(t, Ti, Tj) =P (t, Ti) P (t, Tj)

P (t, Ti, Tj),

where

P (t, Ti, Tj) =j1k=i


is the annuity numraire. Recall that a swaption on the LIBOR market canbe priced at time t [0, Ti] as

IE e r Tit rsds(j1

k=i(Tk+1 Tk)P (Ti, Tk+1)(S(Ti, Tk, Tk+1) )

)+ Ft

= P (t, Ti, Tj) IEi,j[(S(Ti, Ti, Tj)

)+ Ft], (14.39)under the forward swap measure Pi,j defined by

dPi,jdP

= er Ti

0 rsdsP (Ti, Ti, Tj)P (0, Ti, Tj)

, 1 6 i < j 6 n,

under which

Bi,jt := Bt j1k=i

(Tk+1 Tk)P (t, Tk+1)P (t, Ti, Tj)

k+1(t)dt (14.40)

is a standard Brownian motion. Recall that the swap rate can be modeled as

dS(t, Ti, Tj) = S(t, Ti, Tj)i,j(t)dBi,jt , 0 6 t 6 Ti, (14.41)

where the swap rate volatilities are given by

i,j(t) =j1l=i

(Tl+1 Tl)P (t, Tl+1)P (t, Ti, Tj)

(i(t) l+1(t)) (14.42)

+ P (t, Tj)P (t, Ti) P (t, Tj)

(i(t) j(t))

1 6 i, j 6 n, cf. e.g. Proposition 10.8 of [Pri09]. In the sequel we denoteSt = S(t, Ti, Tj) for simplicity of notation.

a) Solve the equation (14.41) on the interval [t, Ti], and compute S(Ti, Ti, Tj)from the initial condition S(t, Ti, Tj).

b) Assuming that i,j(t) is a deterministic function of t for 1 6 i, j 6 n, showthat the price (14.28) of the swaption can be written as

P (t, Ti, Tj)C(St, v(t, Ti)),

" 497



N. Privault

wherev2(t, Ti) =

w Tit|i,j(s)|2ds,

and C(x, v) is a function to be specified using the Black-Scholes formulaBl(x,K, , r, ), with

IE[(x em+X K)+] = (v+ (m+ log(x/K))/v)K((m+ log(x/K))/v),

where X is a centered Gaussian random variable with meanm = rv2/2and variance v2.

c) Consider a portfolio allocation (it, . . . , jt )t[0,Ti] made of bonds with ma-

turities Ti, . . . , Tj and value

Vt =jk=i

kt P (t, Tk),

at time t [0, Ti]. We assume that the portfolio is self-financing, i.e.

dVt =jk=i

kt dP (t, Tk), 0 6 t 6 Ti, (14.43)

and that it hedges the claim (S(Ti, Ti, Tj) )+, so that

Vt = IE e r Tit rsds(j1


)+ Ft

= P (t, Ti, Tj) IEi,j[(S(Ti, Ti, Tj) )+

Ft] ,0 6 t 6 Ti. Show that

IE e r Tit rsds(j1


)+ Ft

= P (0, Ti, Tj) IEi,j[(S(Ti, Ti, Tj) )+

]+

jk=i

w t0ks dP (s, Ti),

0 6 t 6 Ti.d) Show that under the self-financing condition (14.43), the discounted port-

folio price Vt = er t

0 rsdsVt satisfies

dVt =jk=i

kt dP (t, Tk),

498


"



where P (t, Tk) = er t

0 rsdsP (t, Tk), k = i, i + 1 . . . , j, denote the dis-counted bond prices.

e) Show that

IEi,j[(S(Ti, Ti, Tj) )+

Ft]= IEi,j

[(S(Ti, Ti, Tj) )+

]+

w t0

C

x(Su, v(u, Ti))dSu.

Hint: use the martingale property and the It formula.f) Show that the discounted portfolio price Vt = Vt/P (t, Ti, Tj) satisfies

dVt =C

x(St, v(t, Ti))dSt = St

C

x(St, v(t, Ti))i,jt dB

i,jt .

g) Show that

dVt = (P (t, Ti) P (t, Tj))C

x(St, v(t, Ti))i,jt dBt + VtdP (t, Ti, Tj).

h) Show that

dVt = Sti(t)C

x(St, v(t, Ti))

j1k=i

(Tk+1 Tk)P (t, Tk+1)dBt

+(Vt StC

x(St, v(t, Ti)))

j1k=i

(Tk+1 Tk)P (t, Tk+1)k+1(t)dBt

+Cx

(St, v(t, Ti))P (t, Tj)(i(t) j(t))dBt.

i) Show that

dVt =C

x(St, v(t, Ti))d(P (t, Ti) P (t, Tj))

+(Vt StC

x(St, v(t, Ti)))dP (t, Ti, Tj).

j) Show that

C

x(x, v(t, Ti)) =

(log(x/K)v(t, Ti)

+ v(t, Ti)2

).

k) Show that we have

dVt = (

log(St/K)v(t, Ti)

+ v(t, Ti)2

)d(P (t, Ti) P (t, Tj))

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N. Privault

(

log(St/K)v(t, Ti)

v(t, Ti)2

)dP (t, Ti, Tj).

l) Show that the hedging strategy is given by

it = (

log(St/K)v(t, Ti)

+ v(t, Ti)2

),

jt = (

log(St/K)v(t, Ti)

+ v(t, Ti)2

)(TjTj1)

(log(St/K)v(t, Ti)

v(t, Ti)2

),

and

kt = (Tk+1 Tk)(

log(St/K)v(t, Ti)

v(t, Ti)2

), i 6 k 6 j 2.

500


"


Pricing of Interest Rate DerivativesForward Measures and Tenor StructureBond OptionsCaplet PricingForward Swap MeasuresSwaption Pricing on the LIBORExercises

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Pricing of Interest Rate Derivatives

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