Financial Applications of Symbolically Gener- ated Compact Finite Difference Formulae Jichao Zhao, Robert M. Corless and Matt Davison Abstract. We introduce the standard fourth order compact finite difference formulae. We show how these formulae apply in the special case of the heat equation. It is well known that the American option pricing problem may be formulated in terms of the Black Scholes partial differential equation (PDE) together with a free boundary condition. Standard methods allow this prob- lem to be transformed into a moving boundary heat equation problem. We use the compact finite difference method to reduce this problem to a system of ordinary differential equations with specified initial conditions. We develop three ways of combining the resulting systems with methods designed to cope with free boundary values. We show that the compact finite difference scheme for the heat equation and for the American options pricing problem are un- conditionally stable. After numerical comparison of these methods with a standard Crank Nicholson projected Successive Over Relaxation method, we conclude that the compact finite difference technique respresents an exciting new method for pricing American options. 1. Introduction In this paper, we first introduce the standard compact finite difference formulae and show how to generate them symbolically. Then we adjust compact finite dif- ference formulae for heat equations. The American option pricing problem, i.e. Black-Scholes equation with free boundary conditions, is converted into ordinary differential equation after we employ compact finite difference method on it. It can be modified to use the built-in ordinary differential equation solvers in many software packages, like Matlab and Maple. We use three different ways (refer to [8], [6]) to deal with free boundary values. Last through comparing with the Crank Nicholson projected Successive Over Relaxation (SOR) method, we know that compact finite difference method converges faster than it under some conditions for American option pricing problems.