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This paper measures how jagged price volatility is across many traded assets and finds the same rough pattern everywhere. The authors use the Hurst parameter H, a number between 0 and 1 that describes smoothness: smaller H means more irregular, jagged volatility. Across their sample H tends to be far below the Brownian value 0.5, with class medians from about 0.05 up to 0.20, so volatility looks rough in almost every market they study.
To reach this conclusion the researchers built a single data pipeline covering 3,926 U.S. equities, 34 futures roots on the CME (including equity indices, rates, FX, energy, metals, agriculture and livestock), and options on 44 underlyings over roughly 2010–2025. Their main estimator assumes a single scaling law (monofractality) and fits an ordinary least squares regression on the log of the second moment of increments of log realized variance. They tested alternative estimators, applied quality checks, used higher-frequency (one-second) data for the most liquid names, and validated the approach with simulations.
Key numerical results are concrete. Median H by asset class runs from about 0.05 for livestock, through roughly 0.07–0.10 for rates, foreign exchange, agriculture, energy and metals, to about 0.13 for single stocks and about 0.20 for equity indices. For the most liquid equities the median H rises a bit and saturates near 0.15. Regression fits for the realized‑volatility scaling are strong (near R^2 = 0.98), indicating the simple scaling model fits the realized data well.
The paper also compares two ways to estimate H. One uses realized volatility time series. The other uses option prices and the short‑maturity at‑the‑money skew (an option term called the ATM skew). The option-based route identifies H only when a market shows a clear leverage/skew term structure. For equity indices the implied H estimates are about 0.21–0.28, slightly above the realized estimates. For rates and FX the ATM skew regression fails—the R^2 is near zero—so options do not give a usable H there even though realized volatility remains rough. Implied estimates for seasonal commodities are fragile too, partly because futures maturity effects complicate the option signals.
The authors highlight important caveats. Mean reversion in volatility biases the second‑moment estimator; they give a contamination formula for the stationary fractional Ornstein–Uhlenbeck process showing the local slope of the second moment departs from 2H by (1−H)Γ(2H+1)(κΔ)^{2−2H}. Using lags up to ten days avoids much of that bias. Measurement noise in log realized volatility also biases H; applying a correction raised H estimates slightly but never near the Brownian diffusion benchmark. Finally, they present a failure taxonomy that explains where rough‑volatility methods apply and where they break down, and they suggest alternative approaches for the problem cases. The overall message is cautious: volatility looks very rough in many markets, but option‑based identification and simple estimators can fail or be biased in important places.