Exactly Solvable Diffusion Equations and Pricing Models Based on Exceptional Hermite Polynomials

Abstract

In 1973, Black-Scholes and Merton developed a partial differential equation that models the price evolution of a European call option, now referred to as the Black-Scholes equation. Because of its importance in options pricing, there has been a lot of research put into developing solvable derivative models. Through a gauge transformation, the classical Black-Scholes equation can be transformed into a Schrodinger equation. From there, we apply supersymmetric methods to construct a family of orthogonal solutions in terms of exceptional Hermite polynomials. We use these techniques to generalize the classical Black-Scholes equation and obtain solvable derivative models.