Fast solution method and simulation for the 2D time-space fractional Black-Scholes equation governing European two-asset option pricing

Fractional Black-Scholes (FBS) equation has been widely applied in options pricing problems. However, most of the literatures focus on pricing the single-asset option using the FBS model, and research on time-space FBS with multiple assets has remained vacant. Option pricings are always governed by multiple assets. Under the assumption that fluctuation of asset price is regarded as a fractal transmission system and follows two independent geometric Lévy processes, 2D time-space fractional Black-Scholes equation (TDTSFBSE) is proposed to describe the instantaneous price in the financial market. In this work, we discrete the TDTSFBSE with implicit finite difference and present a fast parallel all-at-once iterative method for the resulting linear system. The proposed method can greatly reduce the storage requirements and computational cost. Theoretically, we discuss the convergence property of the fast iterative method. Numerical examples are given to illustrate the effectiveness and efficiency of our method.