The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 634 Wednesday, February 4, 2004 1:33 PM

634 The Mathematics of Financial Modeling and Investment Management

short-term interest rates are determined. Multiple factors model the

term structure as well as the short-term rate.

In fact, a multifactor term-structure model is a model of the form: it

= F(Xt,t) where it is the short-rate process and Xt is an N-dimensional

Itô process that obeys the following SDE:

dXs= μ(Xs , t)dt + σ(Xs, t)dBˆ

s

where Xs is an N-vector, i is a 1-vector, dBˆ is an N-dimensional

s

Brownian motion under an equivalent martingale measure, μ(Xs,t) is an

N-vector and σ(Xs,t)s is a N×N matrix. The Feynman-Kac formula can

be extended in a multidimensional environment in the sense that the fol-

lowing relationships hold:

• ∫TfX( T, s) sd
  Fx t( , ) = EQt e t Ψ(XT)

and

∂Fx t( , )^1 ∂ ( , ( ,

(^2) Fx t) ∂Fx t)
--------------------+ ---tr σ(xt, )σT(xt, )---------------------- + μ(xt, )--------------------– xF x t( , ) = 0
∂t (^2) ∂x^2 ∂x
Arbitrage-Free Models versus Equilibrium Models
Stochastic differential equations are typically used to model interest
rates. There are two approaches used to implement the same SDE into a
term structure model: equilibrium and no arbitrage. While these two
approaches begin with a given SDE, they differ as to how each approach
applies the SDE to bonds and contingent claims. Equilibrium models
begin with an SDE model and develop pricing mechanisms for bonds
under an equilibrium framework. Arbitrage models, also referred to as
no-arbitrage models, start with the same or similar SDE models as the
equilibrium models. However, no-arbitrage models utilize observed
market prices to generate an interest rate lattice. The lattice represents
the short rate in such a way as to ensure there is a no arbitrage relation-
ship between the observed market price and the model-derived value.
Practitioners prefer arbitrage-free models to value options on bonds
because such models ensure that the prices observed for the underlying
bonds are exact. As a result, bonds and options on those bonds will be
valued in a consistent framework. Equilibrium models, in contrast, will