Numerical Investigation of the Time-Fractional Black–Scholes Problem Using the Caputo Fractional Derivative in the Financial Industry

The present study addresses the European option pricing problem based on the Black–Scholes (B-S) model using a hybrid analytical approach known as the Sawi homotopy perturbation transform scheme (SHPTS). We formulate fractional derivatives in the Caputo sense to effectively capture the memory effects inherent in financial models. The competency and reliability of the SHPTS are demonstrated through two illustrative examples. This method produces a closed-form series solution that converges to the precise solution. We perform convergence and visual analyses to demonstrate the competency and reliability of the proposed scheme. The numerical findings further reveal that the strategy is straightforward to apply and very successful in resolving the fractional form of the B-S problem.