Analytical Investigation of Time-Fractional Black-Scholes Equations with Caputo Derivatives using the Yang Perturbation Transform Method

The Black-Scholes model plays a significant role in the analysis of financial derivatives and option pricing. However, the classical framework may not fully capture the memory and hereditary effects observed in financial markets. This study investigates a time-fractional Black-Scholes equation involving the Caputo fractional derivative and develops a semi-analytical approach for its solution. The proposed scheme transforms the governing fractional differential equation into a recursive algebraic formulation without requiring temporal or spatial discretization, linearization, or restrictive assumptions. The effectiveness of the proposed approach is demonstrated through several illustrative examples, and the resulting solutions are compared with the corresponding classical solutions. Furthermore, the numerical results and graphical analyses demonstrate the accuracy and effciency of the proposed method, while the rapid convergence of the resulting series confirms its consistency and computational effectiveness.