Evaluation of Derivative Security Prices in the Heath-Jarrow-Morton Framework as Path Integrals Using Fast Fourier Transform Techniques

Abstract

This paper considers the evaluation of derivative security prices within the Heath-Jarrow-Morton framework of stochastic interest rates, such as bond options. Within this framework, the stochastic dynamics driving prices are in general non-Markovian. Hence, in principle the partial differential equations governing prices require an infinite dimensinal state space. We discuss a class of forward rate volatility functions which allow the stochastic dynamics to be expressed in Markovian form and hence obtain a finite dimensional state space for the partial differential equations governing prices. By applying to the Markovian form, the transformed suggested by Eydeland (1994), the pricing problem can be set up as a path integral in function space. These integrals are evaluated using fast fourier transform techniques. We apply the technique to the pricing of American bond options and compare the computational time with other methods currently employed such as the method of lines and more traditional partial differential equation solution techniques.