Roughness Analysis of Realized Volatility and VIX through Randomized Kolmogorov-Smirnov Distribution

ArXiv ID: 2509.20015 “View on arXiv”

Authors: Sergio Bianchi, Daniele Angelini

Abstract

We introduce a novel distribution-based estimator for the Hurst parameter of log-volatility, leveraging the Kolmogorov-Smirnov statistic to assess the scaling behavior of entire distributions rather than individual moments. To address the temporal dependence of financial volatility, we propose a random permutation procedure that effectively removes serial correlation while preserving marginal distributions, enabling the rigorous application of the KS framework to dependent data. We establish the asymptotic variance of the estimator, useful for inference and confidence interval construction. From a computational standpoint, we show that derivative-free optimization methods, particularly Brent’s method and the Nelder-Mead simplex, achieve substantial efficiency gains relative to grid search while maintaining estimation accuracy. Empirical analysis of the CBOE VIX index and the 5-minute realized volatility of the S&P 500 reveals a statistically significant hierarchy of roughness, with implied volatility smoother than realized volatility. Both measures, however, exhibit Hurst exponents well below one-half, reinforcing the rough volatility paradigm and highlighting the open challenge of disentangling local roughness from long-memory effects in fractional modeling.

Keywords: Hurst parameter estimation, Kolmogorov-Smirnov statistic, Rough volatility, Random permutation procedure, Log-volatility, Equities (S&P 500 VIX)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper introduces a novel estimator with asymptotic theory (Propositions 2.1-2.3, Proposition 2.6) and a computational optimization section (Brent’s method, Nelder-Mead), indicating high mathematical density. The empirical analysis uses real financial data (CBOE VIX and S&P 500 realized volatility) with a specific statistical procedure (random permutation) and reports statistically significant results, demonstrating solid data/implementation rigor.

  flowchart TD
    A["Research Goal: Estimate Hurst Parameter for Log-Volatility<br>using Distribution-Based Roughness Analysis"] --> B["Data Input: S&P 500 5-min Realized Volatility & CBOE VIX"]
    B --> C["Methodology: Novel KS Statistic Estimator &<br>Random Permutation for Dependent Data"]
    C --> D["Computational Process: Derivative-Free Optimization<br>(Brent & Nelder-Mead)"]
    D --> E["Key Findings:<br>1. Significant Hierarchy: VIX (Smoother) < Realized Volatility<br>2. Both Hurst exponents < 0.5 (Rough Volatility Paradigm)"]