The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 645 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 645

A natural, and simple, discretization scheme is given by the Euler

approximation. The Euler approximation replaces the differentials with

finite differences. If we divide the unit interval in n subintervals, the

Euler approximation replaces the SDE with the following recursive

scheme:

 k 1  k 1

Xk + 1 – Xk = μXk, --- ---+ σXk, --- -------εk + 1

 n n  n n

where εk + 1 are independent random draws from a standard normal,

N(0,1). A computer implementation of this scheme would start from

some initial value and compute the solution recursively using a random

number generator to generate the εk + 1. Repeating the process many

times over, one obtains many paths and many final points from which

quantities such as averages can be easily computed. More complex

schemes can be used in order to obtain a smaller approximation error.

As an illustration of the above, Exhibit 20.5 presents random paths

generated using the Euler approximation to approximate several one-

factor interest rate models described earlier in this chapter.

EXHIBIT 20.5 Ten Paths Generated from Different One-Factor Interest Rate

Models

Ho-Lee model: Vasicek model:

μ = 0.005, σ = 0.1 L = 1, T = 200, σ = 0.1