Numerical Valuation of Time Fractional Black–Scholes Equation in Financial Markets

Abstract

The time-fractional Black--Scholes equation (T-FBSE) is formulated to analyze option price dynamics within a coupled fractal diffusion system. The T-FBSE provides a mathematical basis for valuing both European and American call and put options on stocks that do not pay dividends. In this study, the T-FBSE is numerically investigated for the pricing of European and American options by employing a local meshless interpolation strategy. The temporal fractional derivative is discretized by means of a finite difference approach with accuracy of order \( 2-\alpha \) for \( 0<\alpha<1 \), while the spatial operator is approximated through the local radial point interpolation method (LRPIM). Rigorous theoretical results establish the unconditional stability and convergence of the resulting semi-discretized approach in the \( H^1 \)-norm. Numerical examples are provided to verify the theoretical analysis and to demonstrate the accuracy, stability, and practical applicability of the proposed LRPIM-based method for fractional option pricing problems.