The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 639 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 639

dXs= μ(Xs ,t)dt + σ(Xs,t)dBˆ

s

where dBˆ s is a standard Brownian motion under an equivalent mar-

tingale measure Q. In the single factor case, the short rate process it

follows an Itô process

dis= μ(is ,t)dt + σ(is,t)dBˆ

s

■ Step 2. Compute the arbitrage-free price of a zero-coupon bond using

the theory of arbitrage-free pricing under an equivalent martingale

measure according to which the price Λu t at time t of a zero-coupon

bond with face-value 1 maturing at time u is

uis() sd

u = EQ

t e

Λ ∫t

t

■ Step 3. Use the Feynman-Kac formula to show that Λut = Fi( t,t),

which solves the following PDE:

∂Fx t( , ) 1 ∂ ( , ( ,

2

Fx t) ∂Fx t)

--------------------+ ---σ , , ( ,

2

(xt)----------------------+ μ(xt)--------------------– xF x t) = 0

∂t^2 ∂x^2 ∂x

with boundary conditions F(x,T) = 1.

The above methodology can be immediately extended to cover the

pricing of a class of interest-rate derivatives whose payoff can be

expressed as a function of short-term interest rates or, alternatively, as a

function of bond prices. Consider, first, the case of a derivative security

whose payoff is given by two functions h(it,t) and g(iτ,τ), which specify,

respectively, the continuous payoff rate and the final payoff at a speci-

fied date τ≤T. This specification covers a rather broad class of deriva-

tive securities and bond optionality, including European options on

zero-coupon bonds, swaps, caps and floors.

The general arbitrage pricing theory (see Chapter 15) can be imme-

diately applied. The price at time t of a derivative security defined as

above is the following extension of the bond pricing formula:

τ uis() sd is() sd

Q ∫t

( ∫t

τ

Fi(t,t) = Et ∫e his,s) sd + e gi(τ, τ)

t