financial engineering

Zvi Wiener ContTimeFin - 10 slide 1

Financial Engineering

Interest Rate Models

Zvi [email protected]

tel: 02-588-3049

following Hull and White

Hull and White


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.


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.


Modeling Bond Prices

Hull and White

dzTtPTtvdtTtrPTtdP ),(),(),(),(

dzTtvdtv

rTtPd ),(2

),(log2


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



),(

1

),( 21

))(,,( 1221

TtPTtP

e TTTTtF

),(

),(

2

1))(,,( 1221

TtP

TtPe TTTTtF

12

2121

),(log),(log),,(

TT

TtPTtPTTtF



12

2121

),(log),(log),,(

TT

TtPTtPTTtF

0,12 handhTT

T

TtPTtF

),(log

),(

),()( ttFtr


Forward Rates and Bond Prices))(,(),( tTTtReTtP

Hull and White

))(,(),(log tTTtRTtP

),()(),(),(log

TtRtTT

TtR

T

TtP

T

tTTtRtTTTTtRTtR

))(,())(,(

),(


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


Modeling Forward Rates

Hull and White

dzTtvdtv

rTtPd ),(2

),(log2

dzTtvdtTtvTtvTtdF TT ),(),(),(),(

Applying Ito’s lemma we obtain for F(t,T):


HJM Result

Hull and White

dzTtsdtTtmTtdF ),(),(),( Suppose

We know that for some v

),(),(),( TtvTtvTtm T

),(),( TtvTts T

Hence T

t

dtsTtsTtm ),(),(),(


Two Factor HJM Result

Hull and White

2211 ),(),(),(),( dzTtsdzTtsdtTtmTtdF

T

t

T

t

dtsTtsdtsTtsTtm ),(),(),(),(),( 2211


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


Short Rate

Non-Markov type of dynamic - path dependence.

Hull and White

t

tdFtFttFtr0

),(),0(),()(

dzdttdr )(

In a one factor model this process is


Ho and Lee Model

Rates are normally distributed.

All rates have the same variability.

The model has an analytic solution.

dzdtttdr )()(

Hull and White


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


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


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


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.


Black-Derman-Toy

dztdtrt

ttrd )(log

)(

)(')(log

Hull and White

Black-Karasinski

dztdtrtatrd )(log)()(log


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.


Hull and White

att e

ataFtFt 2

2

12

),0(),0()(

Hull and White

dzdtartdr )(


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


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


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


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


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


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


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(


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(


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


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.


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


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


Two Factor HW Model

Where x = f(r) and the correlation between

dz1 and dz2 is .

Hull and White

22

11)(

dzbudtdu

dzdtaxutdx


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.


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 ),(),(),(


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.


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 ),(),(),(


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


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.


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


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).


Modeling Bond Prices with One-Factor

Hull and White

vdzrdtP

dP

tjiviiPjiP

jiPjiP

),(1)1,(

1

),(

),(),1(

tjiv

iiPjiPjiP ),(

)1,(

1),(),1(


Modeling Bond Prices withTwo-Factors

Hull and White

21 ydzxdzrdtP

dP

tjiytjixiiP

jiP

jiPjiP

21 ),(),(1)1,(

1

),(

),(),1(

tjiytjixjiPiiP

jiPjiP

21 ),(),(),(

)1,(

),(),1(


Modeling Forward Rates with One-Factor

Hull and White

tjisjimjiFjiF ),(),(),(),1(

tkisjisjimj

ik

),(),(),(

The Heath-Jarrow-Morton result shows that


Euler Scheme

Hull and White

ttt dZXtbdtXtadX ),(),(

The order of convergence is 0.5

tXtbtXtaXX ttttt ),(),(


Milstein Scheme

Hull and White

ttt dZXtbdtXtadX ),(),(

The order of convergence is 1

tXtbtXtaXX ttttt ),(),(

tXtb t 1),(2

1 22


Trees

Hull and White

up

down


Trees

Hull and White

is big

is small


Trees

Hull and White


Other Topics

The first two factors

– duration

– twist

Hedging

Monotonicity in one-factor

Multi currency TS models


Other Topics

Credit spread

Model Risk

Path Dependent Securities

Binomial Trees with Barriers

financial engineering

Documents

maturity t

t volatility

t instantaneous forward

bond pricesforward rates

whitegeneralized hull

following hull

bond pricert

bond price volatility