High-order compact finite difference schemes for the time-fractional Black-Scholes model governing European options

Abstract

Based on the price fluctuations of the underlying fractal transmission system, the α-order time-fractional Black-Scholes model is obtained. In this paper, we introduce two compact finite difference schemes for the solution of the time-fractional Black-Scholes equation governing European option pricing. In proposed schemes, in order to gain sixth-order and eighth-order accuracy in space, we first use exponential transformation to eliminate the convection term of the Black-Scholes equation. Then, the time-fractional derivative discretizes by a 3−αth order numerical formula (called the L1–2 formula here) which is constructed by a linear interpolating polynomial on the first subinterval and the quadratic interpolating polynomials on the other subintervals. We investigate stability and convergence of proposed schemes by Fourier method. Finally, some numerical examples perform to demonstrate the theoretical order of accuracy and illustrate the effectiveness of proposed methods. We also discuss the influence of different parameters on the option price in the time-fractional Black-Scholes model.