Alternating Direction Implicit Finite Element Method for Multi-Dimensional Black-Scholes Models

Abstract

A new numerical method is proposed and investigated for solving two- dimensional Black-Scholes option pricing model. This model is represented  by Dirichlet initial-boundary value problem in a rectangular domain for a parabolic equation with advection-diffusion operator containing mixed derivatives. It is approximated by using a finite element method in spatial variables and alternating direction implicit (ADI) method in time variable. The ADI scheme is based on the semi-implicit approximation. The stability and convergence of the constructed scheme is proved rigorously. The provided computational results demonstrate the efficiency and high accuracy of the proposed method.