FRACTIONAL DERIVATIVES AND FRACTAL DYNAMICS IN OPTION PRICING: EXTENDING THE BLACK-SCHOLES EQUATION.

Abstract

This study develops a fractional extension of the classical Black-Scholes model to incorporate memory effects and anomalous diffusion in financial markets. By introducing a Caputo fractional derivative of order α∈(0,1], the model captures non-local temporal dynamics that better reflect empirical features such as volatility clustering and heavy-tailed returns. A spline collocation method is applied for spatial discretization, while finite difference approximations address the memory kernel inherent in fractional systems. Numerical experiments demonstrate that smaller fractional orders enhance the model’s ability to reproduce long-range dependence observed in market data. The findings confirm that fractional PDE formulations provide a mathematically rigorous alternative to the classical Black-Scholes framework, bridging theoretical advances in fractional calculus with practical improvements in option pricing accuracy. This work highlights the potential of fractional models to overcome restrictive assumptions of Gaussian returns and constant volatility, offering a robust foundation for future financial applications.