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

Paper: arXiv 2309.08175

Authors: Ying-Li Wang, Cheng-Long Xu, Ping He

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.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a mathematically sophisticated approach using spectral decomposition and Legendre polynomials to achieve analytical series solutions for VIX derivatives. It provides a solid data-driven framework with explicit code and data provided for replication, effectively bridging advanced PDE techniques with practical derivative pricing.

Research Flowchart

  flowchart TD
    A[Goal: Efficient Closed-Form VIX Futures & Options Pricing] --> B[Input: Empirical VIX Data & Volatility Factor in -1,1]
    B --> C[Model Formulation: Single-Parameter Markov Diffusion via Legendre Polynomials]
    C --> D[PIDE/PDE Derivation: Apply Feynman-Kac Formula]
    D --> E[Computation: Analytical Series Solutions via Separation of Variables]
    E --> F[Outcomes: Superior/Equal Accuracy to 3/2 Model & Robust Risk Management]