Numerical Solution of the Time-Fractional Black-Scholes Equation and Its Application to European Option Pricing - Elza Rahma Dihna; Endang Rusyaman; Sukono Sukono (Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang, 2025)
The classical Black-Scholes model is widely used in option pricing but relies on idealized assumptions such as constant volatility and memoryless market dynamics, which limit its accuracy in capturing real-world financial behavior. To overcome these limitations, the time-fractional Black-Scholes model incorporates a fractional-order derivative—specifically the Caputo derivative—which introduces memory effects and accommodates time-varying volatility. This study focuses on numerically solving the time-fractional Black-Scholes equation using the finite difference method (FDM) and applying the results to the pricing of European call options. The model is discretized using an implicit finite difference scheme to ensure stability and accuracy over the time domain. Numerical simulations are conducted for various values of the fractional order α, illustrating that the option price is sensitive to the fractional parameter. Lower values of α tend to increase option prices, highlighting the influence of memory effects on pricing behavior. The results confirm that the finite difference method is an effective numerical tool for solving fractional partial differential equations and demonstrate that the fractional Black-Scholes model offers improved flexibility and realism in option valuation, particularly in markets characterized by irregular volatility and non-Markovian features.