Fast numerical simulation of a new time-space fractional option pricing model governing European call option

When the fluctuation of option price is regarded as a fractal transmission system and the stock price follows a Lévy distribution, a time-space fractional option pricing model (TSFOPM) is obtained. Then we discuss the numerical simulation of the TSFOPM. A discrete implicit numerical scheme with a second-order accuracy in space and a 2−γ order accuracy in time is constructed, where γ is a transmission exponent. The stability and convergence of the obtained numerical scheme are analyzed. Moreover, a fast bi-conjugate gradient stabilized method is proposed to solve the numerical scheme in order to reduce the storage space and computational cost. Then a numerical example with exact solution is presented to demonstrate the accuracy and effectiveness of the proposed numerical method. Finally, the TSFOPM and the above numerical technique are applied to price European call option. The characteristics of the fractional option pricing model are analyzed in comparison with the classical Black-Scholes (B-S) model.