Pathwise Roughness of Bitcoin Realized Volatility: Stability Across Time, Sampling, and Volatility Measures

Abstract

This paper examines whether Bitcoin realized volatility admits a measurable pathwise roughness index and how stable that estimate is across time and measurement designs. Using one-minute BTC/USD close prices from Bitstamp between 2017 and 2024, realized-volatility paths are constructed at 1-, 5-, 10-, and 15-minute frequencies and evaluated with the model-free normalized p-variation estimator of Cont and Das (2024). A unique root is obtained in 341 of 380 rolling 90-day window-frequency configurations, or 89.7 percent, and in 113 of 128 non-overlapping configurations, or 88.3 percent. Conditional rolling medians of the roughness estimate are 0.054, 0.065, 0.086, and 0.080 at the four respective frequencies, and all finite estimates from the temporal, window-length, and jump-robust analyses are below 1/2. Root availability and estimate magnitude nevertheless vary across periods and measurement procedures. Truncation affects root availability primarily at one minute, while bipower variation yields a unique root in 20 of 24 eligible fixed-window configurations. In the eight five-minute fixed-window samples, comparisons with iterative amplitude-adjusted Fourier transform surrogates identify excess MF-DFA width in three samples and excess log-moment curvature in one. Bitcoin realized volatility therefore generally admits a low pathwise roughness estimate, but that estimate is not invariant to time or measurement design. The results concern observed realized volatility and do not directly identify the roughness of latent spot volatility.