The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 642 Wednesday, February 4, 2004 1:33 PM

642 The Mathematics of Financial Modeling and Investment Management

u  u ˆ

() α*(tu)+ ---[σ*(tu)]

2

dΛt = Λ dt– σ*(tu)ΛdBt

u

t



it– ,

1

, ,

 2 

t

This process determines the bond price process in function of a for-

ward rate process. However, to avoid arbitrage, the forward rate pro-

cess must be constrained. In particular, Heath, Jarrow, and Morton

(HJM) demonstrated the following theorems.

Suppose that the forward rate obeys the following SDE under the

probability measure P:

t t

ftu( , )= f( 0 ,u)+ ∫ α(su, )ds+ ∫σ(su, )dBˆs

0 0

Then Pis an equivalent martingale measure if and only if the coeffi-

cients α(t,u), σ(t,u) obey the following relationship:

1

α*(tu, )= ---[σ*(tu, )]^2

2

that is,

u



2

1 

u

∫ α(ts, )ds = ---∫σ(ts, )ds

t^2 t 

where 0 ≤t≤u≤T.

If Pis not an equivalent martingale measure, then there is no arbi-

trage if and only if there is an adapted process θ(τ) satisfying the follow-

ing relationship:

1

α*(tu)= ---[σ*(tu)]

2

, , + σ*(tu, )θ τ(), 0 ≤t≤u≤T

2

or, equivalently, differentiating both sides with respect to u:

α(tu, ) = σ(tu, )σ*(tu, )+ σ(tu, )θ()t , 0 ≤t≤u≤T

Implementing the HJM methodology takes advantage of the available

degrees of freedom. The initial forward rate curve f(0,u) can be deter-

mined by observing the initial curve