The realized-volatility half of this paper is the result. The option-implied half is the warning label: it documents where the skew-based estimator produces numbers you should refuse to use.

Mouti concedes that second point in the abstract. "The option-implied measure identifies $H$ only where the leverage effect produces a clean skew term structure." And "for rates and FX the ATM skew regression fails with an R-squared near zero even though realized volatility remains rough." Section 7.2 then draws the conclusion I would draw: "The cross-asset conclusions therefore rest on the realized estimates, and the implied channel is used where it is identified." Section 9 goes further. It asks for a replacement measure in rates, either a curvature term structure or variance-swap replication. So the live argument is on the realized side. Does the second-moment regression identify H there?

We could not reproduce this work. We have no continuous futures, no FX spot and no futures-option data, so the cross-asset and futures-option parts are out of reach. Our equity data is one-minute bars, so the paper's one-second liquid-equity check cannot be run. Minute OHLCV carries no raw trade and quote detail, so any microstructure treatment we attempted would rest on bar returns. Every figure below is the paper's own.

Start with what Mouti built. One pipeline, one set of estimator choices, applied to everything. Daily realized variance from intraday bars, using seven different estimators. Then the Gatheral-Jaisson-Rosenbaum move: regress the log second moment of log-realized-variance increments on log lag, and read off half the slope as the Hurst exponent H. Below 0.5 means the volatility path is locally rougher than Brownian, which is the whole rough-volatility claim. The primary specification uses lags of one to ten trading days.

The second channel comes from options. Under rough volatility the at-the-money implied skew decays as a power law in maturity, with exponent H minus one half. Regress log absolute skew on log maturity and you recover H without touching a single realized-variance number. Mouti measures the ATM skew for each date and expiry as an OLS slope: implied vol on log-moneyness, inside a moneyness band of 0.08. He then fits the maturity power law over expiries from 14 days to one year.

The data. Equity data is one-minute OHLC prices of Nasdaq-listed stocks, May 2018 to December 2025. The universe is 3,926 Nasdaq common stocks and ETFs with at least 120 usable one-minute data points. Futures come from the CME Globex MDP 3.0 feed from June 2010: 34 roots at one minute, front-month continuous with a volume-based roll. Implied volatility covers 44 underlyings, about 190,000 asset-days converted into implied volatilities. Overnight returns are excluded from realized variance throughout.

Every root, every class, far below one half

Median realized H across all 3,926 equities is 0.119, interquartile range 0.100 to 0.137. Restrict to the quality subset and it rises to 0.131. That subset is 1,409 names with at least 500 days and at least 80% minute fill on the median day. Median regression R-squared there is 0.988. On futures the class medians run 0.048 for livestock, 0.074 rates, 0.081 FX, 0.085 agriculture, 0.088 energy, 0.091 metals, and 0.195 for equity indices. Every one of the 34 roots sits far below 0.5.

That cross-section contains something a desk should notice. Index futures are the smoothest equity object at a median of 0.195, while single stocks come in at 0.131. The paper's literature review cites Zarhali et al. (2025) for single stocks being much rougher than indices, and this cross-section points the same way. Calibrating an index vol model and a single-name vol model to the same H is a choice that gap does not support. That last reading is mine, not the paper's.

Three refinements in the paper push the equity estimate up, and then it stops. Quality conditioning takes the median from 0.119 to 0.131. The offset correction moves the quality-subset realized-kernel median from 0.125 to 0.140 at lag ten, and from 0.111 to 0.123 at lag 40. One-second prices on the 40 most liquid names give medians of 0.186 for the realized kernel, 0.15 for two-scale realized variance and 0.170 for pre-averaging. Weekly and monthly aggregation give 0.149 and 0.148. Nothing approaches 0.5.

The skew regression fails where there is no leverage effect

One statistic separates the two halves of the market. Skew-regression R-squared runs 0.21 to 0.82 for equity indices and 0.02 to 0.03 for rates.

Pooled implied estimates for the indices are ES 0.277, XSP 0.253, SPY 0.243, SPX 0.230, YM 0.225 and NQ 0.207. Clustered standard errors are below 0.008. Realized estimates for the same underlyings run 0.186 to 0.201, so implied sits roughly 0.02 to 0.08 higher.

The paper's failure taxonomy reports pooled implied numbers of 0.70 (ZN), 0.35 (ZB), 0.32 (ZT) and 0.31 (ZF), with R-squared of 0.02 to 0.03. Section 7.2 prints 0.69 for the same ZN estimate. FX runs R-squared from 0.004 (6A) to 0.16 (6M), with estimates of 0.29 to 0.60. Realized H for the same contracts is 0.06 to 0.09 in rates and 0.06 to 0.12 in FX, with regression R-squared above 0.86. Treasury and FX smiles are near-symmetric. The measured skew flips sign across dates and maturities, and the log absolute skew is noise.

A ZN implied H of 0.70 and an RB implied H near minus 0.4 sit outside (0,1). Treat an implied H of 0.70 as a diagnostic. Reading it as evidence against rough realized volatility in rates reads a broken instrument as data. Single stocks fail the same test: all seven names have skew R-squared below 0.3, with AAPL at minus 0.016 and NVDA at minus 0.013, against realized H of 0.11 to 0.14.

One non-equity-index asset clears the identification bar. Wheat, at R-squared 0.34, implied 0.080 against realized 0.088 (0.079 against 0.104 corrected). XSP just misses at 0.27.

Why ten days?

The estimation window is where the result lives, and the paper says so out loud. Figure 1 reports SPY at 0.182 for lags up to ten, 0.160 up to 40, and 0.071 on lags 40 to 250. ES on the same three windows: 0.191, 0.156 and 0.075. The prose gives slightly different fits: 0.186 (SPY) and 0.201 (ES) at lag ten, 0.161 and 0.162 at 40, 0.076 and 0.092 on 40 to 250. The prose pair at lag ten is what produces the 0.186 to 0.201 range quoted above.

Proposition 1 is the justification for stopping at ten. Take log-variance to follow a stationary fractional Ornstein-Uhlenbeck process. The local log-log slope then deviates from 2H by a term of order (kappa times lag) to the power 2 minus 2H. At a maximum lag of ten that bias is at most 0.005 in H, for mean reversion up to 0.02. It grows about fivefold at 40 lags and twentyfold at 100. Mean reversion contaminates rough processes less than diffusive ones, because the exponent exceeds one when H is below a half.

The simulation is where the reader should be uncomfortable, and again the paper puts it on the table.

With a true H of 0.5 and no mean reversion, the lag-ten estimator on five-minute realized variance returns 0.128.

At lag 40 it returns 0.233. Measurement noise alone manufactures roughness on a smooth truth. Mouti's answer is that the two regimes leave different fingerprints. Under a rough truth, raw and corrected estimates bracket the truth, and the two corrected refits agree. Under a smooth truth they diverge, 0.29 at lag ten against 0.38 at lag 40. The empirical panel shows tight brackets and window stability. The defence holds for the equity panel and nowhere else in the sample. The correction itself overshoots in the rough regime, returning 0.146 at a truth of 0.05. So the paper reports raw and corrected as a bracket rather than a point estimate.

The conclusion concedes the sharper version. The joint fit of H, mean reversion and noise is not identified in realistic samples, and it inflates H in rough regimes.

Do not carry a crude calibration into gas

The within-class spread in commodities is as wide as the spread between classes. Energy: CL 0.161 and HO 0.131 against NG 0.065, RB 0.088, BZ 0.076. Metals: HG 0.155 and SI 0.111 against GC 0.091, PA 0.073, PL 0.063. Mouti's own line is that a single commodity H is not a well-defined object, and that models calibrated on CL should not be transferred to NG or LE. Anyone pricing a rough forward-variance model on one contract and reusing the exponent on another is asserting something the data denies.

Seasonality is innocent on the realized side. Deseasonalizing log realized variance changes H by at most 0.002 across NG, ZC, ZS, ZW, HO and CL. The implied side breaks. With monthly natural gas options the pooled skew regression gives 0.26. Adding weeklies triples the R-squared and moves the estimate to 0.080. A listing-calendar choice worth 0.18 in H, on a contract where it should be worth nothing.

What we could not check

We have no continuous futures, no FX spot and no futures options. The cross-asset core of the paper is 34 roots and 33 CME option roots, all of it out of reach. Our equity bars are one minute, so the one-second liquid-name check that lifts the realized-kernel median to 0.186 cannot be run either. Minute OHLCV also lacks the trade and quote detail behind the noise treatment, so any microstructure work we did would rest on bar returns rather than raw transactions.

Judged as measurement

The paper reports no trading strategy, no P&L and no forecasting test. It measures a scaling exponent. Judge it on that and it does the job: one pipeline across the exchange-traded universe, seven realized estimators, and an explicit bias formula for the window choice. The taxonomy then names three places the option channel fails: rates and FX with no leverage skew, seasonal natural gas, and commodity heterogeneity.

Table 6 already runs the smooth truth. At a true H of 0.5 with kappa 0.010 it reads raw 0.261, corrected lag-ten 0.656 and corrected lag-40 0.534. At kappa 0.035, the same three are 0.363, 0.677 and 0.486. The refits disagree and the bracket is wide, which is exactly the fingerprint Mouti says separates the regimes. The equity quality subset behaves differently: realized kernel 0.125 raw against 0.140 corrected at lag ten, 0.111 against 0.123 at lag 40. What is missing is the calibration. "The parameters are calibrated to SPY", and that single grid is then used to validate estimators applied to livestock, FX and rates. A smooth-truth grid calibrated to those classes, with their own vol-of-vol and mean reversion, would change my mind on the headline. Until it exists, the realized side stands.

We have written before about how model rankings flip with the loss function in a GARCH setting. Here the analogue is the option chain. The implied estimate moves 0.18 when weekly natural gas options are added, a listing-calendar effect the same size as the whole cross-asset spread.