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Zvi Wiener ContTimeFin - 10 slide 1
Financial Engineering
Interest Rate Models
Zvi Wienermswiener@mscc.huji.ac.il
tel: 02-588-3049
following Hull and White
Hull and White
Zvi Wiener ContTimeFin - 10 slide 2
Definitions
Hull and White
P(t,T) zero-coupon bond price maturing at time T as seen at time t.
r - instantaneous short rate
v(t,T) volatility of the bond price
R(t,T) rate for maturity T as seen at time t
F(t,T) instantaneous forward rate as seen at time t for maturity T.
Zvi Wiener ContTimeFin - 10 slide 3
Modeling Bond Prices
Hull and White
Suppose there is only one factor.
The process followed by a zero-coupon bond price in a risk-neutral world takes the form:
dP(t,T) = rP(t,T)dt + v(t,T)P(t,T)dZ
the bond price volatility must satisfy v(t,t)=0, for all t.
Zvi Wiener ContTimeFin - 10 slide 4
Modeling Bond Prices
Hull and White
dzTtPTtvdtTtrPTtdP ),(),(),(),(
dzTtvdtv
rTtPd ),(2
),(log2
Zvi Wiener ContTimeFin - 10 slide 5
Forward Rates and Bond Prices
$1 today grows by time T1 to
This amount invested at forward rate F(t,T1,T2)
for an additional time T2-T1 will grow to
),(
1
1TtP
),( 1
))(,,( 1221
TtP
e TTTTtF
By no arbitrage this must be equal),(
1
2TtP
Zvi Wiener ContTimeFin - 10 slide 6
Forward Rates and Bond Prices
),(
1
),( 21
))(,,( 1221
TtPTtP
e TTTTtF
),(
),(
2
1))(,,( 1221
TtP
TtPe TTTTtF
12
2121
),(log),(log),,(
TT
TtPTtPTTtF
Zvi Wiener ContTimeFin - 10 slide 7
Forward Rates and Bond Prices
12
2121
),(log),(log),,(
TT
TtPTtPTTtF
0,12 handhTT
T
TtPTtF
),(log
),(
),()( ttFtr
Zvi Wiener ContTimeFin - 10 slide 8
Forward Rates and Bond Prices))(,(),( tTTtReTtP
Hull and White
))(,(),(log tTTtRTtP
),()(),(),(log
TtRtTT
TtR
T
TtP
T
tTTtRtTTTTtRTtR
))(,())(,(
),(
Zvi Wiener ContTimeFin - 10 slide 9
Modeling Forward Rates
Hull and White
dzTtvdtv
rTtPd ),(2
),(log2
What is the dynamic of F(t,T)?
T
TtPTtF
),(log
),(
T
TtPddF
),(log
),(log TtPdT
Zvi Wiener ContTimeFin - 10 slide 10
Modeling Forward Rates
Hull and White
dzTtvdtv
rTtPd ),(2
),(log2
dzTtvdtTtvTtvTtdF TT ),(),(),(),(
Applying Ito’s lemma we obtain for F(t,T):
Zvi Wiener ContTimeFin - 10 slide 11
HJM Result
Hull and White
dzTtsdtTtmTtdF ),(),(),( Suppose
We know that for some v
),(),(),( TtvTtvTtm T
),(),( TtvTts T
Hence T
t
dtsTtsTtm ),(),(),(
Zvi Wiener ContTimeFin - 10 slide 12
Two Factor HJM Result
Hull and White
2211 ),(),(),(),( dzTtsdzTtsdtTtmTtdF
T
t
T
t
dtsTtsdtsTtsTtm ),(),(),(),(),( 2211
Zvi Wiener ContTimeFin - 10 slide 13
Volatility Structure
The HJM result shows that once we have identified the forward rate volatilites we have defined the drifts of the forward rates as well.
We have therefore fully defined the model!
Hull and White
Zvi Wiener ContTimeFin - 10 slide 14
Short Rate
Non-Markov type of dynamic - path dependence.
Hull and White
t
tdFtFttFtr0
),(),0(),()(
dzdttdr )(
In a one factor model this process is
Zvi Wiener ContTimeFin - 10 slide 15
Ho and Lee Model
Rates are normally distributed.
All rates have the same variability.
The model has an analytic solution.
dzdtttdr )()(
Hull and White
Zvi Wiener ContTimeFin - 10 slide 16
Ho and Lee Model
Where F(t,T) is the instantaneous forward rate as seen at time t for maturity T.
ttFt t2),0()(
Hull and White
Zvi Wiener ContTimeFin - 10 slide 17
Bond Prices under Ho and Lee
Where
)(),(),( tTreTtATtP
Hull and White
22 )(2
1
),0(log)(
),0(
),0(log),(log
tTt
t
tPtT
tP
TPTtA
Zvi Wiener ContTimeFin - 10 slide 18
Option Prices under Ho and Lee
A discount bond matures at s, a call option matures at T
)(),()(),( PhNTtXPhNstPCall
Hull and White
)(),()(),( hNstPhNTtXPPut P
2),(
),(log
1
)(
P
P
P
XTtP
stPh
tTTs
Zvi Wiener ContTimeFin - 10 slide 19
Lognormal Ho and Lee
dzdttrd )(log
Hull and White
Is like Black-Derman-Toy without mean reversion.
Short rate is lognormally distributed.
No analytic tractability.
Zvi Wiener ContTimeFin - 10 slide 20
Black-Derman-Toy
dztdtrt
ttrd )(log
)(
)(')(log
Hull and White
Black-Karasinski
dztdtrtatrd )(log)()(log
Zvi Wiener ContTimeFin - 10 slide 21
Hull and White
dzdtartdr )(
Hull and White
Similar to Ho and Lee but with mean reversion or an extension of Vasicek.
All rates are normal, but long rates are less variable than short rates.
Is analytic tractability.
Zvi Wiener ContTimeFin - 10 slide 22
Hull and White
att e
ataFtFt 2
2
12
),0(),0()(
Hull and White
dzdtartdr )(
Zvi Wiener ContTimeFin - 10 slide 23
Bond Prices in Hull and White
Where
rTtBeTtATtP ),(),(),(
Hull and White
)1()(4
1
),0(log),(
),0(
),0(log),(log
1),(
2223
)(
atataT
tTa
eeea
t
tPTtB
tP
TPTtA
a
eTtB
Zvi Wiener ContTimeFin - 10 slide 24
Option Prices in Hull and WhiteA discount bond matures at s, a call option matures at T
)(),()(),( PhNTtXPhNstPCall
Hull and White
)(),()(),( hNstPhNTtXPPut P
a
eTtv
XTtP
stPh
TtBTtv
tTa
P
P
P
2
1),(
2),(
),(log
1
),(),(
)(222
Zvi Wiener ContTimeFin - 10 slide 25
Generalized Hull and White
f(r) follows the same process as r in the HW model.
When f(r) is log(r) the model is similar to Black-Karasinski model.
Analytic solution is only when f(r)=r.
dzdtraftdr )()(
Hull and White
Zvi Wiener ContTimeFin - 10 slide 26
Options on coupon bearing bond
In a one-factor model an option on a bond can be expressed as a sum of options on the discount bonds that comprise the coupon bearing bond.
Let T be the bond’s maturity,
s - option’s maturity.
Suppose C=Pi - bond’s price.
Hull and White
Zvi Wiener ContTimeFin - 10 slide 27
Options on coupon bearing bond
The first step is to find the critical r at time T
for which C=X, where X is the strike price.
Suppose this is r*.
The correct strike price for each Pi is the value
it has at time T when r=r*.
Hull and White
0,max *ii PP Pi(r) is monotonic in r!
0,max XPi 0,max *ii PP
Zvi Wiener ContTimeFin - 10 slide 28
Example
Suppose that in HW model a=0.1, =0.015.
We wish to value a 3-month European option
on a 15-month bond where there is a 12%
semiannual coupon.
Strike price is =100, bond principal =100.
Assume that the yield curve is linear
y(t) = 0.09 + 0.02 t
Hull and White
Zvi Wiener ContTimeFin - 10 slide 29
ExampleIn this case
Hull and White
4877.01.0
1)75.0,25.0(
5.01.0
e
B
9516.01.0
1)25.1,25.0(
11.0
e
B
AlsottetP )02.009.0(),0(
Zvi Wiener ContTimeFin - 10 slide 30
ExampleThus
Hull and White
Substituting into the equation for logA(t,T)
9733.0)25.1,25.0(
9926.0)75.0,25.0(
A
A
ttettPt
)02.009.0()04.009.0(),0(
Zvi Wiener ContTimeFin - 10 slide 31
ExampleThe bond price equals the strike price of 100
after 0.25 year when
Hull and White
This can be solved, the solution is r = 0.0943.
The option is a sum of two options on
discount bonds.
The first one is on a bond paying 6 at time
0.75 and strike 6x0.9926e-0.4877*0.0943=5.688.
1009733.01069926.06 9516.04877.0 rr ee
Zvi Wiener ContTimeFin - 10 slide 32
Example
Hull and White
This can be solved, the solution is r = 0.0943.
The option is a sum of two options on discount
bonds.
The first one is on a bond paying 6 at time 0.75
and strike 6x0.9926e-0.4877*0.0943=5.688.
1009733.01069926.06 9516.04877.0 rr ee
The second is on a bond paying 106 at time 1.25
and strike 106x0.9733e-0.9516*0.0943=94.315.
Zvi Wiener ContTimeFin - 10 slide 33
Example
The first option is worth 0.01
The second option is worth 0.41
The value of the put option on the bond is
0.01+0.41=0.42
Hull and White
Zvi Wiener ContTimeFin - 10 slide 34
Interest Rates in Two Currencies
Model each currency separately (by building
a corresponding binomial tree).
Combine them into a three-dimensional tree.
Include correlations by changing
probabilities.
Hull and White
Zvi Wiener ContTimeFin - 10 slide 35
Two Factor HW Model
Where x = f(r) and the correlation between
dz1 and dz2 is .
Hull and White
22
11)(
dzbudtdu
dzdtaxutdx
Zvi Wiener ContTimeFin - 10 slide 36
Discount Bond Prices
When f(r) = r, discount bond prices are
Hull and White
),(),(),( TtuCTtrBeTtA
Where A(t,T), B(t,T), C(t,T) are given in HW
paper in Journal of Derivatives, Winter 1994.
Zvi Wiener ContTimeFin - 10 slide 37
General HJM Model
Hull and White
is
i dzTtsdtTtmTtdF ),(),(),(
In addition of being functions of t, T and m, the si can depend on past and present term structures.
But we must have:
T
t
is
i dtsTtsTtm ),(),(),(
Zvi Wiener ContTimeFin - 10 slide 38
General HJM Model
Hull and White
Once volatilities for all instantaneous forward
rates have been specified, their drifts can be
calculated and the term structure has been
defined.
Zvi Wiener ContTimeFin - 10 slide 39
One-factor HJM
Hull and White
The model is not Markov in r.
The behavior of r between times t and t+t
depends on the whole history of the term
structure prior to time t.
dzTtsdtTtmTtdF ),(),(),(
Zvi Wiener ContTimeFin - 10 slide 40
Specific Cases of One-factor HJM
Hull and White
s(t,T) is constant: Ho and Lee
s(t,T) = e-a(T-t): Hull and White
Zvi Wiener ContTimeFin - 10 slide 41
Cheyette Model
Hull and White
s(t,T) = (r)e-a(T-t): Cheyette
dzrdtQetaFtartFtdr att )(),0()(),0()( 2
t
a derQ0
22 )( where
There are two state variables r and Q.
Zvi Wiener ContTimeFin - 10 slide 42
Cheyette Model
Hull and White
Discount bond prices in the Cheyette model are
Qeea
tFtrea
TtP attTatTa 22)(2
)( 12
1),0()(1
1),(log
Zvi Wiener ContTimeFin - 10 slide 43
Simualtions
Hull and White
P(i,j) price at time it of discount bond maturing at time jt.
F(i,j) price at time it of a forward contract lasting between jt and (j+1)t.
v(i,j) volatility of P(i,j)
s(i,j) standard deviation of F(i,j)
m(i,j) drift of F(i,j)
random sample from N(0,1).
Zvi Wiener ContTimeFin - 10 slide 44
Modeling Bond Prices with One-Factor
Hull and White
vdzrdtP
dP
tjiviiPjiP
jiPjiP
),(1)1,(
1
),(
),(),1(
tjiv
iiPjiPjiP ),(
)1,(
1),(),1(
Zvi Wiener ContTimeFin - 10 slide 45
Modeling Bond Prices withTwo-Factors
Hull and White
21 ydzxdzrdtP
dP
tjiytjixiiP
jiP
jiPjiP
21 ),(),(1)1,(
1
),(
),(),1(
tjiytjixjiPiiP
jiPjiP
21 ),(),(),(
)1,(
),(),1(
Zvi Wiener ContTimeFin - 10 slide 46
Modeling Forward Rates with One-Factor
Hull and White
tjisjimjiFjiF ),(),(),(),1(
tkisjisjimj
ik
),(),(),(
The Heath-Jarrow-Morton result shows that
Zvi Wiener ContTimeFin - 10 slide 47
Euler Scheme
Hull and White
ttt dZXtbdtXtadX ),(),(
The order of convergence is 0.5
tXtbtXtaXX ttttt ),(),(
Zvi Wiener ContTimeFin - 10 slide 48
Milstein Scheme
Hull and White
ttt dZXtbdtXtadX ),(),(
The order of convergence is 1
tXtbtXtaXX ttttt ),(),(
tXtb t 1),(2
1 22
Zvi Wiener ContTimeFin - 10 slide 49
Trees
Hull and White
up
down
Zvi Wiener ContTimeFin - 10 slide 50
Trees
Hull and White
is big
is small
Zvi Wiener ContTimeFin - 10 slide 51
Trees
Hull and White
Zvi Wiener ContTimeFin - 10 slide 52
Other Topics
The first two factors
– duration
– twist
Hedging
Monotonicity in one-factor
Multi currency TS models
Zvi Wiener ContTimeFin - 10 slide 53
Other Topics
Credit spread
Model Risk
Path Dependent Securities
Binomial Trees with Barriers