Exact solution of a generalized version of the Black-Scholes equation

TL;DR

This paper presents an exact solution to a generalized Black-Scholes equation parameterized by a, extending the classic model and providing insights into its behavior through Hermite polynomial solutions and numerical comparisons.

Contribution

The paper introduces an exactly solvable generalization of the Black-Scholes equation using Hermite polynomials, expanding analytical tools for option pricing models.

Findings

Exact solution expressed via Hermite polynomials

The generalized model satisfies the martingale condition

Numerical comparison with classical Black-Scholes solution

Abstract

We analyze a generalized version of the Black-Scholes equation depending on a parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We show that the generalized equation is exactly solvable in terms of Hermite polynomials and numerically compare its solution with the solution of the Black-Scholes equation.