Analysis and simulation of nonlinear option pricing problems

Abstract

We present the Black-Scholes Merton partial differential equation (BSMPDE) and its analytical solution. We present the Black-Scholes option pricing model and list some limitations of this model. We also present a nonlinear model (the Frey-Patie model) that may improve on one of these limitations. We apply various numerical methods on the BSMPDE and run simulations to compare which method performs best in approximating the value of a European put option based on the maximum errors each method produces when we vary some parameters like the interest rate and the volatility. We re-apply the same finite difference methods on the nonlinear model.