Problem
Research questions and friction points this paper is trying to address.
Develops a single-parameter Markov model for VIX pricing
Derives analytical solutions for VIX futures and options
Offers accurate and flexible VIX pricing alternative
Closed-form solutions for VIX derivatives in a Legendre empirical model
🤖 AI Summary
This paper addresses the lack of efficient, accurate closed-form solutions for VIX derivatives. We propose a single-parameter Markov diffusion model based on Legendre polynomials: by mapping the VIX process onto (−1,1) and leveraging Legendre orthogonality, we ensure a uniform invariant distribution and construct the first data-driven, analytically tractable empirical modeling framework. We derive explicit series solutions for VIX futures and options—achieving analytical PDE resolution via the Feynman–Kac formula and separation of variables—and calibrate parameters directly to empirical VIX distributions. Empirical results demonstrate pricing accuracy comparable to or exceeding that of the classical 3/2 model, with superior computational efficiency and numerical robustness. An open-source implementation enables plug-and-play risk management. Our core contribution lies in the deep integration of orthogonal polynomial theory with VIX dynamics modeling, unifying theoretical rigor, analytical solvability, and empirical fidelity.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Data-driven single-parameter Markov diffusion model
Legendre polynomials ensure uniform invariant distribution
Analytical series solutions for VIX derivatives
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