A New Homotopy Transformation Method for Solving the Fuzzy Fractional Black–Scholes European Option Pricing Equations under the Concept of Granular Differentiability

Abstract

The Black–Scholes option pricing model is one of the most significant achievements in modern investment science. However, many factors are constantly fluctuating in the actual financial market option pricing, such as risk-free interest rate, stock price, option underlying price, and security price volatility may be inaccurate in the real world. Therefore, it is of great practical significance to study the fractional fuzzy option pricing model. In this paper, we proposed a reliable approximation method, the Elzaki transform homotopy perturbation method (ETHPM) based on granular differentiability, to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Firstly, the fuzzy function is converted to a real number function based on the horizontal membership function (HMF). Secondly, the specific steps of the ETHPM are given to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Finally, some examples demonstrate that the new approach is simple, efficient, and accurate. In addition, the fuzzy approximation solutions have been visualized at the end of this paper.

Keywords

• homotopy perturbation method
• Elaki transform
• fractional Black–Scholes equation
• granular differentiability