Design and analysis of a high order computational technique for time-fractional Black-Scholes model describing option pricing

This work deals with the construction and analysis of a high-order computational scheme for a time-fractional Black-Scholes model that governs the European option pricing. The time-fractional derivative is considered in the sense of Caputo and the L1 - 2 formula is employed to approximate the Caputo temporal-fractional derivative of order alpha, where alpha is an element of (0, 1). A compact difference scheme is designed for discretization of space variable. The convergence of the method is discussed in detail. It is shown that the proposed method has fourth order accuracy in space and (3-alpha)-th order accuracy in time. One numerical example with the known exact solution is considered to demonstrate the applicability and accuracy of present numerical scheme. Moreover, the suggested numerical scheme is employed for pricing three European option problems, namely European call option, European double barrier knock-out option and European put option. The effect of fractional order derivative on option price profile is investigated. Furthermore, the effects of three relevant parameters, namely volatility, strike price and interest rate on the price of European double barrier knock-out option are investigated. The computational time for the proposed method is provided.