The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 638 Wednesday, February 4, 2004 1:33 PM

638 The Mathematics of Financial Modeling and Investment Management

If φ(t) = 0 then the Black-Karasinki model becomes the KWF model.

The Black-Karasinki model is lognormal and therefore interest rates

cannot be negative. The error correction term also prevents rates from

diverging.

The Black-Derman-Toy Model

In 1990 Fischer Black, Emanuel Derman, and William Toy^18 proposed a

lognormal arbitrage-free model with α= 1, μ(i,t) = c(t)i:

di = c t()idt + σ()tidBˆ

Two-Factor Models

A number of two factor models have also been proposed. Brennan and

Schwarz, for example, proposed in 1979 a model based on a short rate i

and a long rate y.^19 This model is written as a set of two equations,

di = μ 1 (i ,, τ y)dt + σ 1 (i ,,τ y)dBˆ

dy = μ 2 (i ,, τy)ydt + σ 2 (i ,,τ y)ydBˆ *

where the two Brownian motions are correlated.

PRICING OF INTEREST-RATE DERIVATIVES

The models of the term structure described thus far are based on deriv-

ing the arbitrage-free prices of zero-coupon bonds from the short-term

rate process. In a nutshell, the methodology involves the following

steps:

■ Step 1. Assume that the short rate process it is a function of an N-

dimensional Itô process Xt (the factors):

it = FX( t,t)

(^18) Fischer Black, Emanuel Derman, and William Toy, “A One Factor Model of In-
terest Rates and Its Application to the Treasury Bond Options,” Financial Analyst
Journal (January–February 1990), pp. 33–39.
(^19) Michael J. Brennan. and Eduardo S. Schwartz, “A Continuous Time Approach to
the Pricing of Bonds,” Journal of Banking and Finance 3 (1979), pp. 133–155.