Option Pricing with Orthogonal Polynomial Expansions

TL;DR

This paper introduces a series expansion method using orthogonal polynomials for pricing European options in various polynomial stochastic volatility models, offering an efficient alternative to Fourier methods and extending to Greeks and exotic options.

Contribution

It develops analytic polynomial series representations for option prices and Greeks in multiple stochastic volatility models, including extensions to exotic options.

Findings

Series expansion matches Fourier method accuracy in affine cases

Numerical validation confirms efficiency of polynomial approach

Extension to exotic options demonstrated successfully

Abstract

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier transform based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.