Applying the Generalized Laplace Residual Power Series Method to the Time-Fractional Multi-Asset Black-Scholes European Option Pricing Model

Abstract

It is a well-known fact that the Black-Scholes model is used in order to analyse the behavior of the financial market with regard to the pricing of options. An explicit analytical solution to the Black-Scholes equation is known as the Black-Scholes formula. The Black-Scholes equation is modified by mathematicians in the form of fractional Black-Scholes equations. Unfortunately, there are certain cases in which the fractional-order Black-Scholes equation does not have a closed-form formula. This article demonstrates the method for deriving analytical solutions to the fractional multi-asset Black-Scholes equation with the left-side Caputo-type Katugampola fractional derivative. The^tρρ^-Laplace residual power series approach, which blends the residual power series method with the^tρ-Laplace transform, is the ρ methodology used to find analytical solutions to this equation. The suggested method is remarkably precise and efficient for the fractional multi-asset Black-Scholes equation, according to numerical analyses. This confirms that the^tρρ^-Laplace residual power series method is among the most effective techniques for finding analytical solutions to fractional-order differential equations.