At n = 5, this test has almost no power where equity estimates of H tend to fall. Wu, Muzy and Bacry (2022) supplied the parameterisation that places rough and multifractal volatility in one family. Zarhali, Bacry and Muzy derive the scaling relations, deviation inequalities, a test of H = 0 and closed-form moment formulas for small intermittency. Yet with cΔ = 1, the test rejects only 7.97% of the time at H = 0.10. At H = 0.05, the rejection rate is 5.05% against a nominal 5%.
We did not run any of this on market data. Latent log-volatility cannot be observed from OHLCV, so an implementation would use a realized-volatility proxy constructed from 1-minute returns.
One family for roughness and multifractality
The Log S-fBM model treats log-volatility as a stationary Gaussian process. On |τ| < T, its covariance is (ν²/2)(1 - (|τ|/T)^{2H}), where ν² = λ²/(H(1 - 2H)). Three parameters carry the model. Local roughness comes from H, while the intermittency λ controls amplitude. T sets a hard decorrelation horizon. Beyond T, covariance is zero.
The Log S-fBM random measure is M_{H,T}(dt) = exp(ω_{H,T}(t))dt, normalised to have mean dt. Subordinating Brownian motion to that clock gives the process X_t = B_{M_{H,T}(t)}.
The reconciliation is the paper's strongest idea.
Rough-volatility estimates place H near 0.1. Multifractal cascades put log-volatility in the logarithmically correlated class, effectively H = 0. Wu et al. showed that M_{H,T} converges weakly to the multifractal random measure as H approaches zero, making both descriptions limiting cases of the same family. Across a range of financial assets, their calibration produced λ² of roughly 0.02. Estimates of H were around 0.1 for broad indices and closer to 0.01 for individual stocks. The paper's asymptotics and its test need to work at those values.
Time rescaling enters through the decorrelation horizon rather than amplitude. The processes (ω_{H,αT}(αt)) and (ω_{H,T}(t)) have the same law, an invariance distinct from fBM self-similarity. Independent S-fBM processes sharing H and T aggregate exactly into another S-fBM, whose intermittency is sqrt(Σλ_i²). Every numerical exercise uses synthetic paths generated by the model. For the moment plots, the settings are L = 2^12, T = 2^12, λ² = 0.02 and H = 0.01.
Can the test detect H = 0.10?
The statistic sums the log measure: Σ_n = Σ_j ln(M_{H,T,Δ}((j-1)Δ)/Δ). Its definition standardises this quantity as Z_n(H) = Σ_n/(λ sqrt(V_n(H))). Here V_n(H) = L_n²/(2H(1-2H)) × (T^{2H} - 2L_n^{2H}/((2H+1)(2H+2))) and L_n = nΔ ≤ T.
The implemented test instead standardises with V_n(0), rejecting when |Z_n| > z_{1-α/2}. Although the expression contains a 1/H prefactor, its numerator vanishes at the same rate, leaving a removable singularity. The limit is V_n(0) = (L_n²/2)[2 ln(T/L_n) + 3]. This limit makes the two-sided z-test at H = 0 expressible.
The test distinguishes alternatives entirely through R, the square root of V_n(H)/V_n(0). With n = 5 and Δ = 1, R equals 1.0000 at H = 0.05, 1.1116 at H = 0.10, 1.6583 at H = 0.25 and 4.8091 at H = 0.45. Empirical power at those values is 0.0505, 0.0797, 0.2407 and 0.6884. Across all nine H values tested, those figures remain within 0.005 of the theoretical π(H).
At n = 5, the H = 0.05 variance agrees with its H = 0 value to four decimals, leaving power at 0.0505. With n = 5 and Δ = 1, the test distinguishes almost nothing across the range of H associated with equities.
The paper acknowledges this result in its discussion of Table 1. Power climbs from the nominal level α ≈ 0.05 to almost 0.69 at H = 0.45, while H = 0.05 remains at the nominal level. Figure 3 supplies the authors' answer by plotting empirical power against the number of observations for H = 0.01 and H = 0.10. Power grows slowly in the near-null case. At H = 0.10, it approaches unit power much faster.
Larger samples solve the problem, but only n = 5 is tabulated. We did not find the number of observations required to reach, say, 80% power at H = 0.10. A trader needs exactly that figure before applying the test to five years of daily data.
Z_n also depends on λ. The paper says λ must be estimated separately because, at this order, λ and H are confounded through λ²V_n(H). A consistent λ̂ retains the stated asymptotic distribution. One example is the generalised method of moments fit in Wu et al. The reported rejection rates, however, use known λ, and we did not find any treatment of how λ̂ error carries into them.
R minus one is 0.1116 at H = 0.10. A ten percent error in λ̂ can therefore move the statistic by as much as the whole H = 0.10 signal. There is another unresolved issue: H = 0 lies on the boundary of the model's domain H ∈ (0, 1/2). We did not find a discussion of how a two-sided z-test at that boundary affects size.
Concentration weakens near single-stock estimates
I would keep the deviation inequalities. For the log measure on a compact D, the upper bound on its limsup x^{-2} ln P is -H(1 - 2H)/λ². For deviation from the Gaussian small-intermittency proxy λΩ_{H,T}(D)/|D|, the corresponding bound is -H(1 - 2H)/(2λ²) × (T/diam(D))^{2H}.
The guaranteed rate H(1 - 2H)/λ² collapses as H approaches zero. The paper gives a structural reason. Lower H makes the covariance increasingly singular near the origin, while ν² = λ²/(H(1 - 2H)) explodes. Wu et al. estimated H near 0.01 for individual stocks. Under this model, roughness and tail weight therefore move together when sizing tail risk. The troublesome parameter is the one estimated for single names.
Observation-window length enters the second rate through (T/diam(D))^{2H}, so smaller sets concentrate more strongly. Both results are upper bounds on limsup x^{-2} ln P. In the multi-component case, the S-fBM supremum bound includes a 2md prefactor from applying the union bound across d independent processes. Its quality deteriorates linearly with the number of components.
Synthetic evidence only
The paper validates the results entirely on synthetic data. Every path comes from the tested model, making the specification correct by construction. Agreement of 0.005 between empirical and theoretical power confirms the algebra rather than a market fit.
We did not implement the test. A practical version would operate on an estimated discrete-time realized-volatility proxy, whereas the paper analyses a continuously observed integrated measure. Any portfolio result or volatility-forecast result would require a new empirical exercise.
The scale-invariance evidence is visual. Figure 4 overlays increment log densities at τ = 1, 20, 50 for H = 0.005, 0.05 and 0.1. Figure 5 shows q-th order moments at q = 0.5, 1, 1.5, using 150 points in each panel. These plots are compared with the closed-form small-intermittency formulas of Theorem 3. The S-fBM panel uses the closed form of Eq. (57). A regression fitted to the plotted points appears only in the log-volatility panel.
The paper states that scale invariance of the increment log density persists across a continuum of H values and announces a numerical evaluation of goodness of fit. We did not find a reported statistic or confidence band for that evaluation. Theory lines use the same parameters that generated the paths. Agreement is assessed by eye.
The model earns its place as a parameterisation for volatility risk. Its aggregation property and explicit tail rates provide quantities for sizing exposure when H is small, and the H(1 - 2H)/λ² rate identifies the parameter that deserves attention. As a classifier, though, the reported test is too weak in the relevant range. Figure 3 already gives λ-known power curves at H = 0.01 and H = 0.10. The missing evidence is the same set of curves with a plug-in λ̂, run on samples as long as a real equity history and producing rejection rates clearly above 5%.