A second order numerical method for the time-fractional Black–Scholes European option pricing model

Abstract

In this paper, we design an efficient and accurate numerical method for solving the time-fractional Black–Scholes equation governing European options. The time-fractional Black–Scholes equation is transformed into an equivalent integro-differential equation. The numerical method for the integro-differential equation is developed by using a numerical integration scheme for time discretization and central difference formulas for space discretization. The stability and convergence of the method are analyzed. The numerical method is proved to be second-order accurate in both space and time. Richardson extrapolation is also introduced to obtain a modified version of the method that exhibits faster convergence for the time-fractional Black–Scholes equation with non-smooth initial data. Finally, two numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.