An upwind finite difference method for a nonlinear Black–Scholes equation governing European option valuation under transaction costs

Abstract

In this paper we develop a numerical method for a nonlinear parabolic partial differential equation arising from pricing European options under transaction costs. The method is based on an upwind finite difference scheme for the spatial discretization and a fully implicit time-stepping scheme. We prove that the system matrix from this scheme is an M-matrix and that the approximate solution converges unconditionally to the viscosity solution to the equation by showing that the scheme is consistent, monotone and unconditionally stable. A Newton iterative algorithm is proposed for solving the discretized nonlinear system of which the Jacobian matrix is shown to be also an M-matrix. Numerical experiments are performed to demonstrate the accuracy and robustness of the method.