© 2011 Neil D. Pearson A Simulation Implementation of the Hull- White Model Neil D. Pearson.

© 2011 Neil D. Pearson

A Simulation Implementation of the Hull-White Model

Neil D. Pearson

Hull-White ModelIn the Hull-White model, under the risk-neutral probability the “short rate” follows the process

dr(t) = ((t) – ar(t))dt + dBQ(t),

or equivalently

where BQ is a Brownian motion under the risk-neutral probability.

Note that one models the movement of the short-rate under the risk-neutral probability, because this is what matters for pricing derivatives and other securities. The model does not specify the movement of the interest rate under the original probability.


),(dd)()(

)(d tBttrat

atr Q

Hull-White Model

Hull-White model:

Some properties:

• the short rate tends to revert to the level (t)/a

•parameter a determines the strength of mean reversion

• is the local volatility

• typically (t) is chosen to match the term structure of zero-coupon bond prices (or equivalently, yields)

Often a and are chosen to provide an approximate match to the prices (implied volatilities) of interest rate caps/floors or swaptions.


),(dd)()(

)(d tBttrat

atr Q


Partial Differential Equation

If the interest rate follows the process

dr(t) = ((t) – ar(t))dt + dBQ(t),

then the value V of a security satisfies the pde

Boundary conditions are determined by the security.

Note that, as usual, the differential equation has the interpretation that drift under the risk-neutral probability (computed using Ito’s formula) equals the riskless return rV

return riskless

yprobabilit neutral-riskunder drift

2

22 ))((

21

rVrV

artrV

tV


Partial Differential Equation for a Zero-Coupon Bond

If the security is a zero-coupon bond paying $1 at maturity T with current price P(t, T), then the pde is

with terminal boundary condition P(r, T,T) = 1.

The solution can be characterized as

rPrP

artrP

tP

))((21

2

22

)(1d)(exp),( truurETtP

T

t

Q

Calibration to Bond Prices

It can be shown that the function can be calculated from the initial (time 0) term structure*

Here F(0,t) is the instantaneous forward rate, observed at time 0, for a forward loan from time t to the next instant. If we let f(0,t, t+dt) be the forward rate, observed at time 0, for a loan from t to t + dt, then

Instantaneous forward rates are related to the bond price through


)1(2

),0(),0(

)( 22

atea

taFt

tFt

),,0(lim),0(0

dtttftFdt

T

uuFTP0

d),0(exp),0( *This is a difficult argument. If you are interested, see Hull, J. and A. White, “Pricing Interest-Rate Derivative Securities,” Review of Financial Studies, 1990, Wilmott IQF, sections 17.2 and 17.3

Calibration to Bond PricesThe term F(0,t)/t is the slope of the initial forward curve.

The term (2/2a)(1e2at) is a “convexity adjustment,” and is usually small. For the time being, let’s assume = 0 so that the convexity adjustment is zero. In this case the drift of the short rate r is

And we conclude that r follows the forward curve.

If > 0 then:

0 assume webecasue 0

22

)0( )(then 0 if because 0

curve forward of slope follow

in changes

)1(2

)),0((),0(

)( ofdrift

at

,tFtrr

ea

rtFat

tFartr

adjustmentconvexity

22

)0( return to toit tends ),0(

from deviates If

curve forward of slope follow

in changes

)1(2

)),0((),0(

)( ofdrift at

,tF,tF

rr

ea

rtFat

tFartr

Aside: Why Is There a “Convexity Adjustment”?

Consider the probabilistic characterization of the bond price

We also have the characterization in terms of forward rates:

If = 0 then r(t) = F(0,t). But if > 0 then we cannot have E[r(t) | r(0)] = F(0,t), because due to the convexity of the function ex we have


)0(d)(exp),0(

0

ruurETPT

Q

T

uuFTP0

d),0(exp),0(

TQ

TQ ruurEruurE

00

)0(|d)(exp)0(d)(exp

Bond Prices in the Hull-White Model

Bond prices in the Hull-White model are given by

where

and

If we know bond prices at time t, then we can compute all zero-coupon rates, forward rates, swap rates, and any quantity (e.g., the payoff of an option on a bond) that depends on bond prices or interest rates.


)(),(),(),( trTtBeTtATtP

ae

TtBtTa )(1

),(

)1)((41

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

ln),(ln 223 atataT eee

aTFTtB

tPTP

TtA

Remark: Key Limitation of 1-Factor Models

Bond prices in the Hull-White model are given by

where

and

All bond prices and yields, regardless of maturity, depend only on the short-rate r, which driven by a single Brownian motion. An implication of this is that changes in the prices/yields of all bonds are (locally) perfectly correlated.


)(),(),(),( trTtBeTtATtP

ae

TtBtTa )(1

),(

)1)((41

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

ln),(ln 223 atataT eee

aTFTtB

tPTP

TtA

Implementing the Hull-White Model Using Simulation

Given the interest rate process

dr(t) = ((t) – ar(t))dt + dBQ(t)

and the function

we can implement the Hull-White model using simulation. If we consider a discrete time-step t (e.g., t = 1/12 year), then we can approximate the interest rate process as

r(t+t) r(t) = ((t) – ar(t))t + tt+t

where t+t ~ N(0,1) is a standard Normal r.v.


)1(2

),0(),0(

)( 22

atea

taFt

tFt


There is a slightly better approximatio:

r(t+t) r(t) = ((t) – ar(t))t + tt+t

Due to the mean reversion, the variance of the short-rate process is not var[r(t+t) r(t)] = 2t. Instead, it is

which for small t is approximately equal to but smaller than 2t. This leads us to use


)1(2

)]()(var[ 22

taea

trttr

ttta aettarttrttr

)2/()1()]()([)()( 2

Implementing the Hull-White Model: Estimate We will simulate:

where

An open issue is how to estimate the forward curve slope F(0,t)/t. Should we estimate it as a forward difference F(0,t)/t (F(0,t+t) F(0,t))/t? Or a backward difference (F(0,t) F(0,tt))/t? Or a central difference?

Your first thought might be that it doesn’t matter; but, if we want to use a long time-step, say t = 1/12, then this issue can be important


ttta aettarttrttr

)2/()1()]()([)()( 2

)1(2

),0(),0(

)( 22

this?estimate toHow

atea

taFt

tFt

“Reprice” BondsWe want the simulation to reprice the bonds, that is if we simulate N interest rate paths and let rn denote an interest rate on the nth path, we want

We want this to be true for all parameters a and .

All values of a and include the choice a, = 0, and we will focus on this choice.


),0(d)(exp1

lim

paths on based price bond of estimate

1 0

TPuurN

N

N

n

T

nN

“Reprice” Bonds: a, = 0If a, = 0, we have

that is the interest rate r follows the forward curve.

With a discrete time-step dt, we need to reinterpret the interest rates. Now r(t) means a rate that is quoted at t, and covers the period from t to t + dt, and r(t+dt) is quoted at t+dt, and covers the period from t+dt to t+2dt.

Similarly, F(0,t) is the forward rate, quoted at 0, covering the period from t to t + dt, and F(0,t+dt) covers the period from t+dt to t+2dt.


ttF

ttr

tt

tFtr

),0(

d)(d

ord),0(

)(d

“Reprice” Bonds: a, = 0

The continuous-time expression

suggests

which in turn suggests using a forward difference

which is what we want.


tt

tFtr d

),0()(d

tt

tFtrttr

),0(

)()(

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

)()( tFttFtt

tFttFtrttr

“Reprice” Bonds: a, = 0To recap, we want to estimate the derivative of the forward curve using a forward difference over the same period that we are simulating the interest rates:

If we do this then the simulated interest rates will be consistent with the forward curve.

In the limit dt 0 the choice of how to estimate the derivative doesn’t matter, but we want to be able to work with relatively large dt. For large dt and rapidly changing yield curves, this issue can be important.*

tt

tFttFtrttr

dt t t tofrom period over the differenceforward a using derivative theestimate

dt t t tofromr simulate

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

*It is not unusual for the short end of the yield curve to display rapid changes in rates.© 2011 Neil D. Pearson


To summarize, we will simulate interest rates using

where (t) is given by


ttta aettarttrttr

)2/()1()]()([)()( 2

)1(2

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

)( 22

difference forward usingestimated derivative curve Forward

atea

taFt

tFttFt

Spreadsheet ImplementationSpreadsheet PRDC_Simulation.xlsm implements 1-factor Hull-White models of the USD and JPY term structures (it also simulates the USD/JPY exchange rate). We focus on the USD model

Column J of the tab InitialTermStructure contains zero-coupon rates, estimated from swap rates. Columns K, L, and M contain forward rates, the forward derivative F(0,t)/t (F(0,t+t) F(0,t))/t, and (t).

Column L of the tab Simulation_Crystal” simulates the short-rate process.

The simulated sample paths (up to 5,000) are stored in the tab r_USD (and also r_JPY and FX). These simulated paths can be used to estimate the values of securities.


© 2011 Neil D. Pearson A Simulation Implementation of the Hull- White Model Neil D. Pearson.

Documents

Transcript of © 2011 Neil D. Pearson A Simulation Implementation of the Hull- White Model Neil D. Pearson.