Numerical solution of the time fractional Black–Scholes model governing European options

When considering the price change of the underlying fractal transmission system, a fractional Black–Scholes(B-S) model with an αα-order time fractional derivative is derived. In this paper, we discuss the numerical simulation of this time fractional Black–Scholes model (TFBSM) governing European options. A discrete implicit numerical scheme with a spatially second-order accuracy and a temporally 2−α2−α order accuracy is constructed. Then, the stability and convergence of the proposed numerical scheme are analyzed using Fourier analysis. Some numerical examples are chosen in order to demonstrate the accuracy and effectiveness of the proposed method. Finally, as an application, we use the TFBSM and the above numerical technique to price several different European options.