An upwind numerical approach for an American and European option pricing model

The numerical solution of several mathematical models arising in financial economics for the valuation of both European and American call options on different types of assets is considered. All the models are based on the Black-Scholes partial differential equation. In the case of European options a numerical upwind scheme for solving the boundary backward parabolic partial differential equation problem is presented. When treating with American options an additional inequality constraint leads to a discretized linear complementarity problem. In each case, the numerical approximation of option values is computed by means of optimization algorithms. In particular, Uzawa's method allows to compute the optimal exercise boundary which corresponds to the classical concept of moving boundary in continuum mechanics.