A Robust Semi-Analytical Approach to Study Time-Fractional Black-Scholes Equation with Non-Local Derivative

Abstract

The time-fractional Black-Scholes equation has a significant impact on market anomalies and irregularities, which offer long-range dependence and heavy-tailed distributions, which led to a more precise depiction of financial markets, particularly in predicting extreme events and in the valuation of derivatives. The primary goal of this work is to examine Black-Scholes equations of arbitrary order with the assistance of the Caputo fractional derivative. Here, we apply an effective semi-analytical technique called an approximate analytical method. We briefly introduce the Black-Scholes equation, its history, and its applications in the field of economics. The aforementioned equation in financial problems is addressed by employing the analytical method, and this concept is used to assess the value of the option (buy or sell an asset) without a transaction cost. Solutions from the proposed method are obtained in series form, which converges swiftly and also carries out numerical simulations by comparing to different methods. The obtained outcomes are discussed through the 3D plots and graphs with the minimum error that expresses the physical representation of the considered equation. The preferred method to examine fractional Black-Scholes equations is efficient, reliable, and robust.