Finite difference/Fourier spectral for a time fractional Black-Scholes model with option pricing (Q2007317)

Summary: We study the fractional Black-Scholes model (FBSM) of option pricing in the fractal transmission system. In this work, we develop a full-discrete numerical scheme to investigate the dynamic behavior of FBSM. The proposed scheme implements a known \(L1\) formula for the \(\alpha \)-order fractional derivative and Fourier-spectral method for the discretization of spatial direction. Energy analysis indicates that the constructed discrete method is unconditionally stable. Error estimate indicates that the \(2-\alpha \)-order formula in time and the spectral approximation in space is convergent with order \(\mathcal{O}\left( \Delta t^{2 - \alpha} + N^{1 - m}\right)\), where \(m\) is the regularity of \(\mathbf{u}\) and \(\Delta t\) and \(N\) are step size of time and degree, respectively. Several numerical results are proposed to confirm the accuracy and stability of the numerical scheme. At last, the present method is used to investigate the dynamic behavior of FBSM as well as the impact of different parameters.