Hurst Parameter Estimation

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. ...

On the rate of convergence of estimating the Hurst parameter of rough stochastic volatility models ArXiv ID: 2504.09276 “View on arXiv” Authors: Unknown Abstract In [“Han & Schied, 2023, \textit{“arXiv 2307.02582”}”], an easily computable scale-invariant estimator $\widehat{"\mathscr{R"}}^s_n$ was constructed to estimate the Hurst parameter of the drifted fractional Brownian motion $X$ from its antiderivative. This paper extends this convergence result by proving that $\widehat{"\mathscr{R"}}^s_n$ also consistently estimates the Hurst parameter when applied to the antiderivative of $g \circ X$ for a general nonlinear function $g$. We also establish an almost sure rate of convergence in this general setting. Our result applies, in particular, to the estimation of the Hurst parameter of a wide class of rough stochastic volatility models from discrete observations of the integrated variance, including the rough fractional stochastic volatility model. ...