Wavelet‐Based <i>L</i> 2‐1 _σ Scheme and Its Higher Order Convergence Analysis for Time‐Fractional Option Pricing Model Under Jump‐Diffusion

ABSTRACT This paper introduces a novel numerical methodology employing wavelet‐based ‐ discretization and its higher order convergence analysis to address the time‐fractional Black‐Scholes model within a jump‐diffusion framework, specifically in the context of pricing European options. The Haar wavelet approximation is applied to solve the resulting semi‐discrete problem, and the fractional time operator is effectively approximated through ‐ discretization. The theoretical estimates of the error bounds are provided. A detailed analysis demonstrates that the method yields a second‐order accuracy over the space‐time domain, depending on the solution's regularity as required for theoretical convergence. Moreover, the study explores the impact of fractional operators on option pricing through various test examples and applications based on Merton's as well as Kou's jump‐diffusion model. We rigorously compared our results with the spline‐based finite difference method and an implicit finite difference method, explicitly highlighting the efficacy and accuracy of our proposed method. In addition, the present research contributes not only to a robust numerical solution for time‐fractional Black‐Scholes models but also valuable insights into the influence of fractional operators on option pricing dynamics under jump‐diffusion.