New Analytical Solutions of a Modified Black-Scholes Equation with the European Put Option

TL;DR

This paper derives exact analytical solutions for a modified Black-Scholes equation related to European put options using Laguerre polynomials, and compares these solutions with numerical results from the standard model.

Contribution

It introduces a new solvable form of the modified Black-Scholes equation for European puts and provides explicit solutions in terms of special functions.

Findings

Analytical solutions are expressed via Laguerre polynomials.

The modified equation is exactly solvable, unlike the standard form.

Numerical experiments validate the analytical solutions.

Abstract

Using Maple, we compute some analytical solutions of a modified Black-Scholes equation, recently proposed, in the case of the European put option. We show that the modified Black-Scholes equation with the European put option is exactly solvable in terms of associated Laguerre polynomials. We make some numerical experiments with the analytical solutions and we compare our results with the results derived from numerical experiments using the standard Black-Scholes equation.