SIMULATION OF MARKET MEMORY BASED ON BLACK-SCHOLES USING FRACTIONAL CALCULUS

Abstract

This study presents a fractional extension of the classical Black-Scholes model to capture memory effects and anomalous dynamics in financial markets. Traditional option pricing models, such as the Black-Scholes equation, assume Markovian behavior and constant volatility, which often fail to reflect empirical characteristics like long memory, volatility clustering, and non-Gaussian returns. To address these limitations, we incorporate fractional calculus into the Black-Scholes framework by replacing the classical time derivative with a fractional derivative of order α ∈ (0, 1]. This transformation yields the time-fractional Black-Scholes equation, which introduces non-local temporal dynamics and more accurately models the influence of past events on current option values.

To solve the resulting fractional partial differential equation, we employ a spline collocation method, leveraging the smoothness and accuracy of cubic spline basis functions for spatial discretization. The Caputo derivative is approximated using a finite difference scheme that captures the memory effect inherent in fractional systems. Numerical simulations are performed to analyze the effect of varying the fractional order α, with results demonstrating that smaller values of α increase the model’s ability to reflect real-world market behavior.

The integration of fractional calculus and spline methods offers a robust framework for simulating market memory in option pricing. The proposed approach enhances the descriptive power of financial models and provides a more realistic valuation of derivatives under non-Markovian dynamics.