The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 637 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 637

The Cox-Ingersoll-Ross Model

In 1985 John Cox, Jonathan Ingersoll, and Stephen Ross (CIR)^15 pro-

posed an equilibrium model with

α=^1

---

2

μ(it, )= (Li– )

----------------

T

Li–

di = -----------dt + σ idBˆ

T

where L and T are constants. The CIR model is mean reverting but has

only three free parameters to fit the initial term structure. It can be

shown that in this model interest rates always remain non-negative.

Kalotay, Williams, and Fabozzi

In 1993 Andrew Kalotay, George Williams, and Frank Fabozzi (KWF)^16

proposed a model with α= 1, μ= θ(t)i described by the following SDE:

di = θ()tidt + σidBt

For θ= constant the model becomes a geometric random walk. As the

model is lognormal, interest rates never become negative.

Black-Karasinski

In 1991 Fisher Black and Piotr Karasinski^17 proposed a model with α=

1 described by the following SDE:

d ln i = [θ(t) – φ(t)ln i]dt + σ(t)dBt

(^15) John Cox, Jonathan Ingersoll, and Stephen. Ross, “A Theory of the Term Struc-
ture of Interest Rates,” Econometrica (1985), pp. 385–408.
(^16) Andrew J. Kalotay, George Williams, and Frank J. Fabozzi, “A Model for the Val-
uation of Bonds and Embedded Options,” Financial Analyst Journal (May–June
1993), pp. 35–46.
(^17) Fischer Black and Piotr Karasinski, “Bond and Option Pricing when Short Rates
are Lognormal,” Financial Analysts Journal (July–August 1991), pp. 2–59.