Numerical Approximation to a Variable-Order Time-Fractional Black–Scholes Model with Applications in Option Pricing

We propose and analyze a fully-discrete finite element method to a variable-order time-fractional Black–Scholes model, which provides adequate descriptions for the option pricing, and the variable fractional order may accommodate the effects of the uncertainties or fluctuations in the financial market on the memory of the fractional operator. Due to the impact of the variable order, the temporal discretization coefficients of the fractional operator lose the monotonicity that is critical in error estimates of the time-fractional problems. Furthermore, the Riemann–Liouville fractional derivative is usually adopted in time-fractional Black–Scholes equations, while rigorous numerical analysis for Riemann–Liouville variable-order fractional problems are rarely founded in the literature. Thus the main contributions of this work lie in developing novel techniques to resolve the aforementioned issues and proving the stability and optimal-order convergence estimate of the fully-discrete finite element scheme. Numerical experiments are carried out to substantiate the numerical analysis and to demonstrate the potential applications in option pricing.