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A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics. (English) Zbl 1498.91497

Summary: 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 \(\mathcal{O}(k^2+h^2)\). 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.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
60G22 Fractional processes, including fractional Brownian motion
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
91G20 Derivative securities (option pricing, hedging, etc.)
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