A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options

Abstract

This paper is concerned with the design of a high order numerical approach based on a uniform mesh for efficient numerical solution of time-fractional Black-Scholes equation, governing European options. The time-fractional derivative is defined in the Caputo sense. A collocation method based on quintic B-spline basis functions is used for space discretization and time-stepping is done using a backward Euler method. The stability and convergence of the method are analyzed. The method is shown to be unconditionally stable and fourth order accurate in space and (2−β) order accurate in time, where β is the order of the time fractional derivative (0<β<1). Two numerical examples with the known exact solutions are considered to validate theoretical results and demonstrate the accuracy of the method. Moreover, this scheme is used to price three different European options governed by a time-fractional Black-Scholes model; (i) European double barrier knock-out call option (ii) European call option and (iii) European put option. The effect of the order of fractional derivative on the option price is studied.