Pseudospectral Symbolic Computation for Financial Models

F.O. Bunnin, Y. Ren, Y. Guo and J. Darlington

Abstract
The modelling of nancial markets as continuous stochastic processes provides the means to analyse the implications of models and to compute prices for a host of nancial instruments. We code as a symbolic computing program the analysis, initiated by Black, Scholes and Merton, of the formation of a partial di erential equation whose solution is the value of a derivative security, from the speci cation of an undelying security's process. The Pseudospectral method is a high order solution method for partial di erential equations that approximates the solution by global basis funcations. We apply symbolic transformations and approximating rewrite rules to extract essential information for the Pseudospectral Chebyshev solution. We write these programs in Mathematica. Our C++ template implementing general solver code is parameterised with this information to create instrument and model speci c pricing code. The Black- Scholes model and the Cox Ingersoll Ross term structure model are used as examples.