Exploring Fractional-Order Models in Computational Finance via an Efficient Hybrid Approach

Abstract

In this study, a hybrid numerical method is applied to solve the time-fractional Black-Scholes model for various options, including traditional (European and American) as well as non-standard options (such as butterfly spread, double barrier, and digital options). The method combines the fractional Liouville-Caputo scheme for time derivatives with the Strang splitting algorithm, while a meshless approach based on Lucas and Fibonacci polynomials is used for spatial derivatives. Numerical experiments are conducted using the $L_{\infty}$ error norm to evaluate the accuracy and effectiveness of the method, with the double mesh technique employed for validation when exact solutions are unavailable. In addition, the model performance is further evaluated through the computation of key sensitivity measures, specifically the Greeks (delta and gamma). The accuracy and robustness of the proposed solution are validated by benchmarking its results against those obtained using alternative methods reported in the literature.