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：本文提出了一种基于分布的新型估计器，用于估计对数波动率的赫斯特参数。该估计器利用Kolmogorov-Smirnov统计量来评估整个分布而非单个矩的标度行为。为应对金融波动率的时间依赖性，我们提出了一种随机置换程序，该程序能有效消除序列相关性，同时保留边缘分布，从而使KS框架能够严格应用于依赖数据。我们建立了估计量的渐近方差，该方差可用于推断和置信区间构建。从计算角度，我们证明了无导数优化方法（特别是Brent方法和Nelder-Mead单纯形法）相较于网格搜索能实现显著的效率提升，同时保持估计精度。对CBOE VIX指数和标普500指数5分钟实现波动率的实证分析揭示了一个统计上显著的粗糙度层级：隐含波动率比实现波动率更为平滑。然而，两种测度均表现出远低于二分之一的赫斯特指数，这强化了粗糙波动率范式，并凸显了在分形建模中将局部粗糙度与长记忆效应分离这一开放挑战。