Pseudospectral Symbolic Computation for Financial Models

The modeling of financial markets as continuous stochastic processes provides the means to analyze the implications of models and to compute prices for a host of financial instruments. We code, as a symbolic computing program, the analysis, initiated by Black, Scholes and Merton, of the formation of a partial differential equation whose solution is the value of a derivative security, from the specification of an underlying security's process. The Pseudospectral method is a high-order solution method for partial differential equations that approximates the solution by global basis functions. 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 parameterized with this information to create instrument and model-specific pricing code. The Black-Scholes model and the Cox-Ingersoll-Ross term-structure model are used as examples.