EQUATION-BASED MODELING OF EUROPEAN OPTIONS UNDER THE BROWNIAN AND CGMY PROCESSES BY USING THE COMSOL MULTIPHYSICS® SOFTWARE

Abstract

The Brownian process-based Black-Scholes (BS) PDE model and the nonlinear PDE developed by H.E. Leland, which includes transaction costs are first evaluated numerically by the COMSOL Multiphysics® software. Then, the corresponding partial integro-differential equation (PIDE) models based on Levy process are modeled numerically. The numerical solutions of all four models are compared favorably to those in the literature obtained by the finite difference methods. Unlike the Brownian motion, the Lévy motion allows for jumps and discontinuous paths that might better reflect the observed jump behavior for some assets. Some technical modifications and transformations are introduced to the PDEs and the PIDEs so that the resulting systems of equations are suitable for application for the equation-based modeling framework in the COMSOL Multiphysics® software. The method developed in this work allows us to take the advantage of the prowess of the software and it provides a novel and an efficient procedure for evaluation of the option pricing models.