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.

翻译：我们提出一种基于全新分布估计量的对数波动率赫斯特参数估计方法，该方法利用柯尔莫哥洛夫-斯米尔诺夫统计量评估完整分布的标度行为，而非仅针对个别矩。为应对金融波动率的时间依赖性，我们设计了一种随机置换程序，该程序在保持边际分布的同时有效消除序列相关性，从而使KS框架能够严密适用于相依数据。我们推导了该估计量的渐近方差，可用于统计推断与置信区间构建。从计算角度而言，我们证明无导数优化方法——特别是布伦特法和Nelder-Mead单纯形法——在保持估计精度的前提下，相比网格搜索可显著提升计算效率。对CBOE VIX指数与标普500指数5分钟已实现波动率的实证分析揭示了统计显著的粗糙度层级结构：隐含波动率较已实现波动率更为平滑。然而，两类波动率的赫斯特指数均显著低于0.5，这不仅强化了粗糙波动率范式，更凸显出分数阶模型中局部粗糙度与长记忆效应分离这一开放性挑战。