Finite element solutions of the nonlinear RAPM Black-Scholes model

Abstract

The main purpose of our Capstone project is to study the Risk-Adjusted Pricing Methodology (RAMP) Black-Scholes model and to find the finite element solutions of the nonlinear Black-Scholes equation. The RAPM is one of the many nonlinear models in option pricing considering factors which affect the volatility in the original Black-Scholes equation. This model can be simplifed to a nonlinear parabolic equation in a new vari- able which equals the product of Gamma and the price of the underlying asset. Galerkin nite element method is applied to the parabolic equation. Two types of solutions will be presented: one using the linear elements and the other using quadratic elements. Local finite element equations for the linear and quadratic elements are derived with some specifc inter- polations of the nonlinear terms. Numerical solutions are obtained and compared to the results in literature. The explanation of the discrepancies will be given together with the future goals of this study.