On the Deribit BTC options the authors held out, the fractional clock makes pricing worse. Plain Black-Scholes misses exchange marks by 7.30 implied-volatility points; setting the memory parameter to 0.80 raises the error to 13.30.

What the clock changes

Nteumagné, Pindza and Maré retain geometric Brownian motion, then make it run on a random operational clock: the inverse of an α-stable subordinator. Heavy-tailed waiting periods can stop that clock. The authors propose it for illiquid options and stress episodes with pauses between bursts of order flow. They price a call by averaging ordinary Black-Scholes prices across the clock's distribution. In the PDE, a Caputo derivative of order α replaces the first time derivative. At α = 1, the classical model returns.

The other parameter, t0, sets the memory time scale through κα = t0^(1−α). Because the clock cannot be traded, the market is incomplete. The authors choose a product pricing measure that assigns the clock zero risk premium and discounts on that same clock. Expected discounting becomes a Mittag-Leffler function of r t0 θ^α (with θ maturity measured in units of t0).

There is more mathematics around the pricing problem. The Lie symmetry analysis does no pricing work, the authors say, because a fixed-strike payoff does not survive the scalings. They check a graded-mesh finite-difference solver against an independent subordination estimate. On the finest grid, it produces 10.62692 against 10.62804, a difference within one Monte Carlo standard error of 0.00416. They also prove a distinction between time-changed Heston and rough Heston. For the empirical work, they use 17,520 hourly Binance BTC/USDT bars from August 2024 to August 2026 and a Deribit snapshot taken 14 August 2026. Filtering reduces 818 contracts to 245.

The clock can change the shape of prices across maturities. With α = 0.7 and t0 = 1 year, its fractional ATM call is 28.18% above the classical price at three months and 25.55% below it at five years.

Does α help on Deribit?

The authors calibrate α and one constant σ on the four earliest expiries, comprising 144 contracts, using vega-normalised RMSE. They reserve the three later expiries, comprising 101 contracts, for the test. Calibration chooses α = 1. It puts σ at 0.341, with a bootstrap interval of [0.336, 0.347]. Test error is 7.30 at α = 1, then 8.88 at 0.95, 10.40 at 0.90 and 13.30 at 0.80. The boundary-aware lower limit for α is 0.9931. Three tighter filters also choose α = 1: test error is 7.18 with spreads at most 15%, 7.48 with moneyness from 0.85 to 1.15, and 7.18 with open interest at least 5.

Within this family, the added parameter hurts accuracy.

The abstract reports α = 1 for the baseline and all three tighter filters. The paper's error profile also has both errors declining toward α = 1. In the introduction, the authors argue that the clock remains useful as a diagnostic benchmark. They concede that their tests "do not support its general use in liquid markets." I accept that concession and find the defence thin. Their motivation concerned illiquid options and intermittently traded assets, neither of which they tested. On the market they did test, the diagnostic lower limit is 0.9931 and the chosen order is α = 1.

The 7.30 needs its qualifications close by. The search grid ends at α = 1, leaving any preference for an order above 1 unanswered. Even at α = 1, error rises from 4.39 during calibration to 7.30 on the held-out contracts. The authors attribute that gap to using a single constant σ. Those contracts come from later expiries, making term structure the likely culprit.

All expiries belong to the same snapshot, taken at 16:09:53 UTC. The authors say evidence across market regimes would require more dated snapshots, so out-of-time decay remains unmeasured. They convert Deribit mark IVs to prices with zero discounting. The baseline filter permits relative spreads as wide as 25%.

A crossover tied to t0

The 1.095-year fractional-classical price crossover holds only at t0 = 1 year. At α = 0.7, the solver locates the ATM crossover at 1.09480 years; a two-million-draw Monte Carlo puts it at 1.09426. Substitute the clock's mean for the random clock and the crossover shifts to 1.376 years. That gap of about 0.28 years is the paper's sharpest practical warning. Black-Scholes prices are nonlinear in maturity, so averaging the clock before pricing gives a different answer.

The horizon changes with the chosen memory scale. Holding α = 0.7, the crossover moves almost in proportion to t0: 0.2623 years at t0 = 0.25, 0.5365 at 0.5, 1.0943 at 1 and 2.2046 at 2. Across that range, the one-year call price moves from 8.2355 to 12.0904. The authors assume t0 is one year; we found no estimate of it from data. They accordingly reject claims that classical prices on either side of a universal horizon "are therefore not warranted."

Bitcoin's scaling estimates disagree

The scaling exercise supplies another negative result. For non-overlapping 6-hour realised volatility, the semivariogram estimates H at 0.042. R/S gives 0.792 and DFA1 gives 0.815. DFA1 reaches 1.099 on the rolling 24-hour series, which the authors interpret as nonstationarity. Daily realised volatility also yields conflicting routes to α: a direct Mittag-Leffler ACF fit gives 0.990, while 1 − 2H gives 0.873. On that series, the Mittag-Leffler ACF has RMSE 0.1966 against 0.1296 for a plain power law. The authors leave α = 1 − 2H as "an exponent-matching hypothesis, not a theoretical identity."

Their Heston result gives the disagreement a structural counterpart. With a non-degenerate clock and a nonconstant Heston coefficient, the time-changed transform is strictly log-convex in initial variance. The rough Heston transform is log-linear. No mapping β = f(α) can make those transform families identical for all initial variances.

A fit statistic, without a trade

The 7.30-point Deribit error measures pricing against exchange marks. The paper tests no strategy, transaction costs or hedging P&L; it offers no estimate of what a desk would earn. We did not rerun the calibration. The authors provide their Deribit snapshot as supplementary data, and we did not reprocess it.

Dated Deribit snapshots, ideally including a stress episode, could change my view if an α below 1 at a fixed t0 reduced held-out error. On this paper's evidence, the authors built the random clock carefully, and their market test chooses α = 1.