Paper: arXiv 2611.07120

Authors: Tianfeng Yang, Wen Su, Blanka Horvath

Abstract

The VIX term structure provides forward-looking measures of market uncertainty across horizons, while VIX options allow investors to hedge volatility risk. This paper proposes a rough instantaneous squared VIX (RISVIX) model by introducing fractional memory into the conventional Markovian framework. We derive an explicit representation of the VIX term structure governed by Mittag–Leffler decay and a semi-closed-form pricing formula for VIX options based on an affine Fourier–Laplace transform. These results support a two-step calibration procedure that estimates memory, mean reversion, and date-specific initial states from VIX term-structure indices before calibrating the remaining distributional parameters to VIX option prices. Using daily VIX term-structure indices and VIX option quotes from January 2018 to June 2025, we find that RISVIX improves upon the Markovian benchmark in fitting the VIX term structure, with larger gains during elevated-volatility periods and for its slope and curvature. RISVIX also improves out-of-sample option pricing, reducing MSE and MAE by 13.06% and 6.33%, respectively, with larger gains at shorter maturities and in high-VIX states. These findings suggest that the gains from fractional memory are particularly pronounced when volatility is elevated and can be obtained within a tractable direct model of VIX derivatives. Comparisons with additional benchmarks from the recent literature also show improvements to varying degrees in both VIX term-structure fitting and option pricing.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly complex mathematical model incorporating fractional memory and affine Volterra processes for VIX derivatives. It demonstrates strong empirical rigor through extensive backtesting and out-of-sample validation using real market data, showing significant improvements over benchmarks. The novelty lies in directly applying fractional memory to an ISVIX framework, leading to a tractable model with improved performance.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop a model for VIX term structure & option pricing with fractional memory] --> B{Proposed Model: Rough Instantaneous Squared VIX (RISVIX)};
    B --> C[Methodology: Introduce fractional memory into Markovian framework];
    C --> D{Model Derivations: Explicit VIX term structure (Mittag-Leffler decay) & semi-closed VIX option pricing (Affine Fourier-Laplace transform)};
    D --> E[Data/Inputs: Daily VIX term-structure indices & VIX option quotes (Jan 2018 - June 2025)];
    E --> F{Calibration Procedure: 1. Estimate memory, mean reversion, initial states from VIX term structure. 2. Calibrate distributional parameters to VIX option prices};
    F --> G[Key Findings - VIX Term Structure: Improved fit vs. Markovian benchmark, larger gains during elevated volatility, for slope & curvature];
    F --> H[Key Findings - VIX Option Pricing: Improved out-of-sample pricing (13.06% MSE, 6.33% MAE reduction), larger gains at shorter maturities & high-VIX states];
    G & H --> I[Conclusion: Fractional memory significantly improves VIX derivative modeling, especially during elevated volatility, within a tractable direct model];