Evaluation of Point Barrier Options in a Path Integral Framework Using Fourier-Hermite Expansions

The pricing of point barrier or discretely monitored barrier options is a difficult problem. In general, there is no known closed form solution for pricing such options. In this paper we develop a path integral approach to the evaluation of barrier options. This leads to a backward recursion functional equation linking the pricing functions at successive barrier points. We solve this functional equation by expanding the pricing functions in Fourier-Hermite series. The backward recursion functional equation then becomes the backward recurrence relation for the coefficients in the Fourier-Hermite expansion of the pricing functions. We thus obtain a very efficient and accurate method for generating the pricing function at any barrier point. We perform a number of numerical experiments with the method in order to gain some understanding of the nature of convergence. We present results for various volatility values and different numbers of basis functions in the Fourier-Hermite expansion. Comparisons will be given between pricing of point barriers in the path integral framework and by use of finite difference methods.