Roughness in VIX Index and in Realized Volatility: Rolling Window Estimation by Randomized Kolmogorov-Smirnov Distribution

The modeling and forecasting of financial market volatility constitute fundamental components of effective risk management and optimal asset allocation. Traditional models like GARCH and SV often fail to capture the long memory and roughness empirically observed in volatility, prompting the adoption of fractional processes. Accurate esti-mation of the log-volatility roughness parameter is thus key to validating rough volatility models, with several methodologies proposed, includ-ing spectral, wavelet, and machine learning techniques. In contrast to approaches focused on moment behavior, we adopt a novel method based on the selfsimilarity of fractional processes, examining how the entire log-volatility distribution scales across time horizons. We deduce the variance of the estimator and study the roughness of both CBOE VIX and realized volatility.