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

**Authors:** Sergio Bianchi, Daniele Angelini  
**arXiv:** [2509.20015](https://arxiv.org/abs/2509.20015)  
**Date:** 2025-09-24  
**Summary page:** https://www.summarizepaper.com/en/paper/2509.20015v1/

## Key points

- Authors Sergio Bianchi and Daniele Angelini introduce a novel distribution-based estimator for the Hurst parameter of log-volatility
- They leverage the Kolmogorov-Smirnov statistic to assess scaling behavior of entire distributions rather than individual moments
- The authors propose a random permutation procedure to address temporal dependence of financial volatility, effectively removing serial correlation while preserving marginal distributions
- The study establishes the asymptotic variance of the estimator, valuable for inference and confidence interval construction
- Bianchi and Angelini demonstrate that derivative-free optimization methods such as Brent's method and Nelder-Mead simplex offer efficiency gains compared to grid search while maintaining estimation accuracy
- Empirical analysis on CBOE VIX index and 5-minute realized volatility of S&P 500 reveals statistically significant hierarchy of roughness, with implied volatility smoother than realized volatility
- Both measures exhibit Hurst exponents well below one-half, reinforcing the rough volatility paradigm
- Challenges are highlighted in disentangling local roughness from long-memory effects in fractional modeling

## Summary

In their paper titled "Roughness Analysis of Realized Volatility and VIX through Randomized Kolmogorov-Smirnov Distribution," authors Sergio Bianchi and Daniele Angelini introduce a novel distribution-based estimator for the Hurst parameter of log-volatility. By leveraging the Kolmogorov-Smirnov statistic, they assess the scaling behavior of entire distributions rather than individual moments. To address the temporal dependence of financial volatility, the authors propose a random permutation procedure that effectively removes serial correlation while preserving marginal distributions. This enables the rigorous application of the KS framework to dependent data. The study establishes the asymptotic variance of the estimator, which is valuable for inference and confidence interval construction. From a computational perspective, Bianchi and Angelini demonstrate that derivative-free optimization methods, specifically Brent's method and the Nelder-Mead simplex, offer substantial efficiency gains compared to grid search while maintaining estimation accuracy. Through empirical analysis of the CBOE VIX index and the 5-minute realized volatility of the S&P 500, the authors reveal a statistically significant hierarchy of roughness. They find that implied volatility is smoother than realized volatility. However, both measures exhibit Hurst exponents well below one-half, reinforcing the rough volatility paradigm. The study also highlights challenges in disentangling local roughness from long-memory effects in fractional modeling. Overall, Bianchi and Angelini's research provides valuable insights into understanding and estimating financial volatility dynamics using innovative statistical methods and computational techniques.