The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 636 Wednesday, February 4, 2004 1:33 PM

636 The Mathematics of Financial Modeling and Investment Management

The Vasicek Model

In 1977, Oldrich Vasicek proposed the Ornstein-Uhlenbeck process as a

model of interest rates to produce a one-factor equilibrium model.^13 In

the Vasicek model α= 0,

μ(it, )= (Li– )

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

T

Li– ˆ

di = -----------dt + σdB

T

where L and T are constants.

The Vasicek model is a mean-reverting process as interest rates are

pulled back to the value L. Interest rates exhibit mean reversion proper-

ties, a fact that the Vasicek models correctly address. However, having

only three free parameters, the Vasicek model is difficult to fit to the ini-

tial term structure.

The Hull-White Model

In 1990 Hull and White proposed a mean-reverting model that generalizes

the Vasicek model.^14 The Hull-White model is given by the choice α= 0,

(Lt()– i)

μ(it, )= -----------------------

Tt()

with time-variable volatility

Lt()– i

di = ------------------dt + σ()tdBˆ

Tt()

The Hull-White model has enough parameters to be fitted to any

initial term structure.

(^13) Oldrich Vasicek, “An Equilibrium Characterization of the Term Structure,” Jour-
nal of Financial Economics (1977), pp. 177–188.
(^14) J. Hull and A. White, “Pricing Interest Rate Derivative Securities,” Review of Fi-
nancial Studies 3 (1990), pp. 573–592, and, “One Factor Interest Rate Models and
the Valuation of Interest Rate Derivative Securities,” Journal of Financial and Quan-
titative Analysis (1993), pp. 235–254.