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

作者：Lesmana Donny C; Wang Song^*

来源：Applied Mathematics and Computation, 2013, 219(16): 8811-8828.

DOI：10.1016/j.amc.2012.12.077

• Nonlinear Black-Scholes equation
• Option pricing
• Upwind finite difference method
• Convergence
• Stability
• Newton method

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.

• 出版日期2013-4-15