Solution of option pricing equations using orthogonal polynomial expansion

TL;DR

This paper develops analytic and numerical methods for solving option pricing equations using orthogonal polynomial expansions, specifically Hermite and Laguerre polynomials, and compares results to existing formulas.

Contribution

It introduces a Galerkin-based approach with orthogonal polynomials for solving Black-Scholes and Heston models, including boundary condition handling.

Findings

Accurate solutions for Black-Scholes and Heston models obtained.

Comparison shows good agreement with semi-closed formulas.

Enhanced understanding of boundary behavior in Heston model.

Abstract

In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of Heston model at the boundary with vanishing volatility.