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

•Using efficient market hypothesis and assuming that the stock market exhibits some unexplained memory structures, we derive a time fractional Black-Scholes (tfBS) partial differential equation PDE for pricing stock options.•We present a robust numerical scheme which is based on the extension of the Crank Nicholson difference method in solving the tfBS-PDE obtained under the assumption that the stock market does exhibit some unexplained hereditary/memory features.•We present rigorous theoretical analysis, and prove that our method is both convergent and unconditionally stable.•We provide some numerical results to illustrate the robustness of the method. 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(k2+h2). 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.