Closed-form solutions for VIX derivatives in a Legendre empirical model

TL;DR

This paper presents a novel single-parameter Markov diffusion model for VIX derivatives using Legendre polynomials, providing analytical solutions that outperform traditional models in accuracy and flexibility.

Contribution

It introduces a new data-driven, Legendre polynomial-based model for VIX, deriving closed-form solutions for derivatives and demonstrating improved performance over existing models.

Findings

Analytical series solutions for VIX futures and options.

Model achieves equal or better accuracy than the 3/2 model.

Provides an efficient and robust alternative for VIX pricing.

Abstract

In this paper, we introduce a data-driven, single-parameter Markov diffusion model for the VIX. The volatility factor evolves in $(-1,1)$ with a uniform invariant distribution ensured by Legendre polynomials, mapped to the empirical distribution. We derive analytical series solutions for VIX futures and options using separation of variables to solve the Feynman-Kac PDE. Compared to the 3/2 model, our approach offers equal or superior accuracy and flexibility, providing an efficient, robust alternative for VIX pricing and risk management. Code and data are available at github.com/gagawjbytw/empirical-VIX.