Option Pricing Under Fractional Brownian Motion: A Caputo–Black–Scholes Approach

We investigate the fractional extension of the classical Black–Scholes model for option pricing, introduced by a time-fractional derivative, in the sense of Caputo, and replacing the standard Brownian motion with a fractional Brownian motion (fBm) where the Hurst parameter H>0. 5. The use of fBm captures long-range dependence and memory effects often observed in financial markets, while the Caputo derivative generalizes the temporal dynamics beyond the classical Markovian framework. We first derive the fractional Black–Scholes equation, by incorporating memory effects through the Caputo fractional derivative and the fractional Brownian motion. Then, we establish a theorem that guarantees the existence and uniqueness of the solution. Using Laplace transform techniques and the Mittag-Leffler function, we derive a closed-form representation of the solution. Furthermore, we discuss numerical methods for evaluating the solution, and illustrate the impact of fractional parameters on option prices. Our results highlight how memory effects induced by fractional Brownian motion significantly modify option valuation, potentially offering a more realistic modeling framework compared to the classical Black–Scholes model.