The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 635 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 635

not price bonds exactly so they do not provide a consistent framework

for valuing options on bonds and the underlying bonds.

Examples of One-Factor Term Structure Models

A number of one-factor and multifactor term structure models have

been proposed in the literature. We will discuss some of the more popu-

lar one-factor models here:

■ The Ho-Lee model

■ The Vasicek model

■ The Hull-White model

■ The Cox-Ingersoll-Ross model

■ The Kalotay-Williams-Fabozzi model

■ Black-Karasinski model

■ The Black-Derman-Toy model

Our coverage is not intended to be exhaustive.^11

Most of these models are based on a short-term process which satis-

fies an SDE of the following type:

di = μ(it, )dt + σiαdBˆ

The various models differ for the choice of the drift μ(i,t) and of the

exponent α.

The Ho-Lee Model

The first arbitrage-free model was introduced by Thomas Ho and Sang-

Bin Lee in 1986.^12 In the Ho-Lee model α = 0, μ(i,t) = μ = constant.

di = μdt + σdBˆ

This model is quite simple. It has the disadvantage that interest rates

might drift and become negative, which is inconsistent with what is

observed in financial markets. In addition, having only two free param-

eters, it cannot be easily fitted to the initial observed term structure.

(^11) For a more detailed discussion of these models, see Gerald W. Buetow, Jr., Frank
J. Fabozzi, and James Sochacki, “A Review of No Arbitrage Interest Rate Models,”
Chapter 3 in Fabozzi, Interest Rate, Term Structure, and Valuation Modeling.
(^12) Thomas Ho and Sang Bin Lee, “Term Structure Movements and Pricing Interest
Rate Contingent Claims,” Journal of Finance (1986), pp. 1011–1029.