A sixth order numerical method and its convergence for generalized Black–Scholes PDE

Abstract

The main aim of this paper is to construct a new computational approach for the numerical solution of generalized Black–Scholes equation. In this approach, the temporal variable is discretized using Crank–Nicolson scheme and spatial variable is discretized using sextic B-spline collocation method. Convergence analysis of the method is discussed. The proposed method is proved to be stable and has second-order convergence with respect to time variable and sixth order convergence with respect to space variable. To illustrate the applicability and efficiency of the proposed method, we consider some benchmark problems describing European call options. Numerical results verify the orders of convergence predicted by the analysis. Numerical results reveal that the present method provides better results as compared to some existing numerical methods based on B-spline collocation approach.