Rough Volatility Across Assets

Abstract: We measure volatility roughness across asset classes using a common data infrastructure and pipeline. Our data covers 3,926 United States equities, 34 CME futures roots, rates, FX, and commodities, and options on 44 underlyings over 2010-2025. Realized volatility is rough everywhere. The class-median Hurst estimate ranges from $0.05$ (livestock) through $0.07-0.10$ (rates, FX, agriculture, energy, metals) to $0.13$ (single stocks) and $0.20$ (equity indices). The option-implied measure identifies $H$ only where the leverage effect produces a clean skew term structure. For the equity indices, implied estimates of $0.21-0.28$ are just above realized volatility $H$, while for rates and FX the ATM skew regression fails with an R-squared near zero even though realized volatility remains rough. We also show a mean-reversion contamination formula for the second-moment estimator of the roughness for the stationary fractional Ornstein-Uhlenbeck process. The local slope of the increment second moment deviates from $2H$ by $(1-H)Γ(2H+1)(κΔ)<sup>{2-2H}$ for all $H\in(0,1)$. A correction framework for the second moment, when the log realized volatility measure has additive noise, raised the $H$ estimate slightly but nowhere near the Brownian diffusion framework. Finally, a failure taxonomy discusses where rough-volatility methods apply and where they fail, suggesting alternative paths to further explore the rough-volatility paradigm.

• The paper develops a unified pipeline using realized-volatility scaling and ATM implied-skew term structures, finding realized Hurst exponents far below 0.5 across all asset classes and validating short-lag estimates against mean-reversion and measurement-error biases.
• The paper finds median realized H values ranging from 0.048 for livestock and 0.074–0.091 for rates, FX, agriculture, energy, and metals to 0.119 for equities and 0.195 for equity indices, showing universal roughness with substantial cross-asset heterogeneity.
• The paper shows that option-implied roughness is reliably identified mainly for equity indices, where H ranges from 0.207 to 0.277, while rates, FX, and many commodities require alternative implied measures because weak or distorted leverage skews undermine identification.

The paper "Rough Volatility Across Assets" (2608.16749) provides the first systematic, single-pipeline measurement of volatility roughness across the exchange-traded universe. Using a common data infrastructure spanning 2010–2025, the author estimates the Hurst exponent $H$ of log-volatility from two independent channels—realized volatility scaling and the at-the-money (ATM) implied skew term structure—for 3,926 U.S. equities, 34 CME futures roots, and options on 44 underlyings. The central findings are that realized volatility is rough in every asset class (class medians from 0.05 to 0.20, all far below $H=1/2$), but that the option-implied identification of $H$ succeeds only where the leverage effect produces a clean skew term structure, namely equity indices.

Data and estimation design

The equity panel consists of 3,926 Nasdaq-listed stocks and ETFs with 1-minute OHLC prices (2018–2025), with a quality subset of 1,409 names requiring at least 500 estimation days and 80% median fill fraction. Futures data come from CME Globex MDP 3.0 (June 2010 onward) as volume-rolled front-month continuous series covering equity indices (ES, NQ, YM), rates (ZT–ZB, GE), FX (6A–6S), and 18 commodity roots. Options are priced from CME cleared settlements and OPRA closes, with forwards and discount factors extracted from put-call parity regressions, Black-76 for futures options near the money, and the Barone-Adesi-Whaley method for American single-stock options. ATM skew is the OLS slope of implied volatility on log-moneyness within $|k| \le 0.08$; maturities below 14 days are excluded because jump-dominated expiry-week skews are unreliable.

On the realized side, the primary estimator is the OLS slope of $\log m(2,\Delta)$ on $\log\Delta$ over lags $\Delta = 1, \dots, 10$ applied to $\log \mathrm{RV}_t$, with TSRV as the headline realized-variance measure and six alternatives (realized kernel, RV5m, pre-averaging, pre-averaged bipower, bipower, corrected threshold) as cross-checks. Empirical fits are tight, with regression $R^2$ around 0.98. On the implied side, $H$ is recovered from $H=1/2$0 via both per-date regressions and a pooled panel regression with date effects, with the pooled $H=1/2$1 serving as an identification diagnostic.

A bias formula for mean-reversion contamination

A theoretical contribution is a proposition on the second moment of the stationary fractional Ornstein–Uhlenbeck (fOU) process. For $H=1/2$2, the local log-log slope of the increment second moment satisfies

$H=1/2$3

valid for all $H=1/2$4, proved via the spectral density and a Mellin-type integral evaluation. The exponent $H=1/2$5 whenever $H=1/2$6, so contamination is sub-linear precisely in the rough regime, and at $H=1/2$7 the formula recovers the exact Ornstein–Uhlenbeck result. The practical consequence is that restricting the regression to $H=1/2$8 days keeps mean-reversion bias below 0.005 in $H=1/2$9 for $H$0—an order of magnitude smaller than the effects reported—whereas extending to 40 or 100 lags inflates the bias fivefold and twentyfold. This justifies the short-lag window as the primary specification.

Measurement error and the two-estimator correction

Additive log-scale measurement noise attenuates the slope by the factor $H$1, most severely at short lags and small $H$2. The paper develops a two-estimator correction that pairs RV5m with a second realized measure, exploiting the lag-invariant offset $H$3 and a joint fit of the first three autocovariances of the fractional Gaussian noise increments. Simulation validation on 1,000 fOU paths per cell shows that raw and corrected estimates bracket the truth in the rough regime (e.g., raw 0.180 and lag-40 corrected 0.238 against a truth of 0.20 at the benchmark cell), and that the corrected RV5m estimate is stable across pairings (0.199–0.202 at lag 40). Two cautions emerge. First, the correction fails for jump-robust estimators whose error is volatility-dependent rather than serially uncorrelated. Second, and more pointedly, the design does not identify a smooth truth: at $H$4 with $H$5, the pipeline still reports rough estimates (raw 0.128), because measurement noise flattens the short-lag scaling. The empirical estimates, however, display the rough-regime fingerprints—tight brackets, agreeing refits, window stability, and saturation well below one half.

Equity results

Across 3,926 stocks, the lag-10 TSRV distribution is tight and unambiguously rough: median 0.119 (interquartile range [0.100, 0.137]), rising to 0.131 in the quality subset with median regression $H$6 of 0.988. Successive refinements each nudge the estimate upward: the offset correction lifts the quality-subset RK median from 0.125 to 0.140; one-second prices for the 40 most liquid names give medians of 0.186 (RK), 0.15 (TSRV), and 0.170 (PAV); weekly and monthly aggregation give 0.149 and 0.148. The paper concludes that equity $H$7 lies roughly in $H$8, and—importantly—that no refinement moves the estimate toward the diffusive value of 0.5. The most liquid subset median saturates at 0.15, indicating that price-level microstructure noise is not the binding constraint.

For indices, the implied channel works: pooled skew estimates give $H$9 (ES), 0.253 (XSP), 0.243 (SPY), 0.230 (SPX), 0.225 (YM), and 0.207 (NQ), with clustered standard errors below 0.008 and robustness to estimation settings within a 0.06 spread. These sit 0.02–0.08 above the realized estimates for the same underlyings, a gap consistent with the smoothing induced by conditional expectation in option prices. For single stocks the implied channel weakens sharply: MSFT (0.235), GOOGL (0.193), and AMZN (0.151) are moderate, while TSLA (0.060), AMD (0.004), AAPL (−0.016), and NVDA (−0.013) sit at or near zero with $|k| \le 0.08$0 below 0.3, even though their realized $|k| \le 0.08$1 is 0.11–0.14. The daily implied-$|k| \le 0.08$2 series for ES and SPX has medians of 0.270 and 0.229 over fifteen years, ranging from 0.07 to 0.4.

Cross-asset patterns

The headline cross-asset result is heterogeneity within universal roughness. Class medians of the realized $|k| \le 0.08$3 (TSRV, lag 10) are:

Class	Median $|k| \le 0.08$4
Equity index	0.195
Metals	0.091
Energy	0.088
Agriculture	0.085
FX	0.081
Rates	0.074
Livestock	0.048

Every root is far below 0.5, and BPV-based medians move by at most 0.07 relative to TSRV, so price jumps do not drive the result. Within-class spread is as large as between-class spread (metals 0.063–0.155; energy 0.065–0.161), a point with direct modeling consequences discussed below.

The implied-realized comparison across 41 assets splits the market cleanly. Only the equity-index cluster is identified at the $|k| \le 0.08$5 threshold, sitting near the diagonal with implied slightly above realized; the noise-corrected realized estimates close most of the residual gap (NQ and SPY within 0.01, ES from 0.08 to 0.05). One non-equity exception supports the framework: wheat (ZW) passes the threshold and lands near parity (implied 0.080 against realized 0.088).

Failure taxonomy

The paper is explicit about where the implied channel fails, and these failures are informative rather than incidental. Rates: Treasury futures smiles are nearly symmetric because the volatility–return correlation is close to zero; the sign of the skew flips across dates, and pooled regressions return meaningless values ($|k| \le 0.08$6 for ZN, 0.35 for ZB, with $|k| \le 0.08$7 of 0.02–0.03) even though realized estimates of 0.04–0.09 are well fitted. The same failure holds for FX ($|k| \le 0.08$8 from 0.004 to 0.16). This is consistent with unspanned stochastic volatility in fixed income, and it means the rough-volatility paradigm for rates must rest on realized measures or on skew-free implied objects such as curvature term structures. Natural gas: seasonality leaves the realized measure essentially untouched (deseasonalizing changes $|k| \le 0.08$9 by at most 0.002) but corrupts the implied measure through the maturity grid; adding weekly options triples $\log m(2,\Delta)$0 and moves the estimate by 0.18, so NG's implied $\log m(2,\Delta)$1 is a composition artifact. Commodity heterogeneity: a single commodity $\log m(2,\Delta)$2 is not a well-defined object. Within energy, CL (0.16) and HO (0.13) are roughly twice NG, RB, and BZ (0.07–0.09); within metals, HG (0.16) and SI (0.11) exceed GC (0.09) and the platinum group (0.06–0.07); livestock is the roughest class. The spread correlates loosely with storage and delivery mechanics—storable, arbitraged contracts behave like financial assets, while supply-inelastic contracts are roughest—but the paper does not resolve the mechanism.

Limitations

Several limitations bear directly on the results. The non-equity classes rest on a handful of contracts each, so class medians are sensitive to individual roots. The fOU lens is a modeling choice: $\log m(2,\Delta)$3 is a scaling exponent of log-volatility, and its interpretation as fractional-driver memory is model-dependent, a caveat that applies to the entire rough-volatility literature. The implied estimates depend on option-data quality and, for seasonal underlyings, on the listed maturity grid. The simulation further shows that joint estimation of $\log m(2,\Delta)$4, mean reversion, and noise is not identified in realistic samples and inflates $\log m(2,\Delta)$5 in rough regimes, so the correction should be read as bracketing rather than point identification.

Conclusion

The paper delivers a unified cross-asset measurement showing that realized volatility is rough everywhere, with a monotone ordering from livestock (0.05) through rates, FX, agriculture, energy, and metals (0.07–0.10) to single stocks (0.13) and equity indices (0.20), and it proves that the short-lag second-moment regression is protected from mean-reversion contamination. The implied channel identifies $\log m(2,\Delta)$6 only for equity indices, where estimates of 0.21–0.28 sit just above realized values. Two specific open problems follow from the results: rough forward-variance models on commodity futures curves with maturity-dependent roughness and the Samuelson effect, and an implied $\log m(2,\Delta)$7 measure for rates that does not rely on the leverage skew, for example via curvature term structures or variance-swap replication.