The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 643 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 643

∂(log ΛT 0 )

f( 0 ,T)= – -----------------------

∂u

As only a finite number of bond prices can be observed, it is necessary to

use techniques to convert a number of finite observations into a smooth

curve. One cannot simply fit a high-degree polynomial to the available

observations as this would introduce a lot of noise. On the other hand, fit-

ting a low-degree polynomial would create a curve that does not corre-

spond to the true term structure. Splines is an approach that is often used to

create a smooth initial forward curve. This technique involves fitting pieces

of curves in such a way that the transition between the pieces is smooth.

Suppose that the initial forward rate curve has been fitted to empiri-

cal data. Suppose that two deterministic functions σ*(t,u), θ(t) have

been chosen. Let’s define

α(tu, ) = σ(tu, )σ*(tu, )+ σ(tu, )θ() t

With these definitions, the forward rate process is determined by the fol-

lowing equation in the risk neutral probabilities:

df = σ(tu, )σ*(tu, )dt + σ(tu, )dBˆ

t

Solving this equation yields the forward rate process and the short-term

process. The bond pricing equation then becomes

dΛu

t = it t ,

()Λudt – σ*(tu)Λu ˆ

t dBt

In this equation only the volatility σ*(t,u) appears. This shows that,

in order to implement the HJM model, only the initial term structure

and the volatilities are needed.

THE BRACE-GATAREK-MUSIELA MODEL

The Brace-Gatarek-Musiela (BGM) model is a particular implementa-

tion of the HJM model which corresponds to a specific choice of the

volatility.^21 The BGM model is based on defining a forward LIBOR

(^21) Alan Brace, Dariusz Gatarek, and Marek Musiela, “The Market Model of Interest
Rate Dynamics,” Mathematical Finance 7, no. 2 (April 1997), pp 127–155.