A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics

Empirical evidence suggest that fractional stochastic based models are well suited for modelling systems and phenomenons exhibiting memory and hereditary properties. Assuming that the stock market exhibits some unexplained memory structures, described by a non-random fractional stochastic process governed under a standard Brownian motion, we derive a time-fractional Black-Scholes (tfBS) partial differential equation for pricing option contracts on such stocks. We further propose a corresponding robust numerical method which is based on the extension of a Crank Nicholson finite difference method for solving tfBS-PDEs. Through rigorous theoretical analysis, we established that the method is unconditionally stable and convergent up to order $O(k^2+h^2)$. Two numerical examples are presented using realistic market parameters. Our results confirm theoretical observations and general consensus in literature that, stock market dynamics are of a power law nature and follow heavy tailed distributions with memory.