An Adaptive Mesh Based Computational Approach to the Option Price and Their Greeks in Time Fractional Black-Scholes Framework

This article deals with an efficient numerical method for solving the time fractional Black-Scholes equation governing the European option pricing model and their Greeks. The Caputo fractional derivative involved in time results a mild singularity and forms a layer near the initial time. For discretization, a graded mesh is introduced in the temporal direction, and in space, a uniform mesh is constructed. The L1 scheme is used to discretize the time fractional derivative, while the second-order finite difference approximations are used for the spatial derivatives. The proposed approach effectively resolves the initial layer with a graded mesh in time, achieving higher temporal accuracy of O(N-(2-gamma))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(N<^>{-(2-\gamma )})$$\end{document}. It provides valuable insights into the error bounds through stability and convergence analysis and captures the behavior of option Greeks, highlighting the impact of fractional derivatives. Compared to uniform mesh-based methods and other existing approaches, it demonstrates superior accuracy and efficiency for time-fractional Black-Scholes equations, ensuring space-time higher-order accuracy. Some numerical results on the solution and their Greeks prove the theoretical analysis. The proposed scheme is applied to European option pricing models governed by the time fractional Black-Scholes equation to examine the impact of the fractional derivative on option pricing.