Removing non-smoothness in solving Black-Scholes equation using a perturbation method

TL;DR

This paper introduces a variable transformation technique to improve the homotopy perturbation method for solving Black-Scholes equations, effectively removing non-smoothness issues and providing new analytical solutions for complex options.

Contribution

The paper presents a novel variable transformation approach that enhances the homotopy perturbation method for Black-Scholes equations, including new solutions for quanto options.

Findings

High accuracy with the exact solution for the Black-Scholes equation

Effective extension to multi-asset basket and quanto options

New simple analytical solution for single-asset quanto options

Abstract

Black-Scholes equation as one of the most celebrated mathematical models has an explicit analytical solution known as the Black-Scholes formula. Later variations of the equation, such as fractional or nonlinear Black-Scholes equations, do not have a closed form expression for the corresponding formula. In that case, one will need asymptotic expansions, including homotopy perturbation method, to give an approximate analytical solution. However, the solution is non-smooth at a special point. We modify the method by {first} performing variable transformations that push the point to infinity. As a test bed, we apply the method to the solvable Black-Scholes equation, where excellent agreement with the exact solution is obtained. We also extend our study to multi-asset basket and quanto options by reducing the cases to single-asset ones. Additionally we provide a novel analytical solution of the single-asset quanto option that is simple and different from the existing expression.