Roughness of the Log-Volatility Process and the Failure of Markovian Calibration

Gatheral, Jaisson, and Rosenbaum address a foundational question in stochastic volatility theory: what is the regularity of the log-volatility process as measured directly from high-frequency equity and index data? The central object of study is the scaling behavior of the q-th absolute moment of log-volatility increments, defined as m(q, Δ) = E[|log σ_{t+Δ} − log σ_t|^q], evaluated across a range of moment orders q and time lags Δ spanning several decades of scale. If this quantity scales as a power law Δ^{ζ(q)} with ζ(q) ≈ qH for a single constant H, the process belongs to the class of monofractal processes with Hurst index H, and the empirical content of the paper is precisely that H ≈ 0.1 across a wide collection of equity indices, with this estimate proving remarkably stable across instruments, sample periods, and estimation methodologies. The significance of H < 1/2 is that the log-volatility increments exhibit mean-reversion at all observable scales — the process is rougher than Brownian motion, and a model consistent with this finding cannot be Markovian in volatility. The theoretical vehicle proposed to capture this roughness is the Rough Fractional Stochastic Volatility (RFSV) model, in which log σ_t is driven by a fractional Brownian motion W^H with H ≈ 0.1 rather than a standard Brownian motion. Fractional Brownian motion, constructed via the Mandelbrot–van Ness representation as a moving-average integral of Brownian increments with a power-law kernel of exponent H − 1/2, is the unique Gaussian process with stationary increments and self-similar paths, and for H ∈ (0, 1/2) its paths are almost surely Hölder continuous of any order strictly less than H, which is significantly less regular than the Brownian case H = 1/2. The RFSV model sets log σ_t = ν W^H_t up to a mean-reverting correction, and the key structural consequence is that the covariance function of the log-volatility satisfies E[log σ_t log σ_s] ∼ |t − s|^{2H} for small |t − s|, a form that encodes extremely long memory of volatility at short scales despite eventual decorrelation at long scales. The multiscaling question — whether ζ(q) is truly linear in q or exhibits convexity consistent with multifractal models — is addressed empirically, and the data support the monofractal description within estimation uncertainty, though the authors are careful to note that the sample sizes available do not definitively exclude weak multiscaling. For desk purposes this paper functions as the empirical anchor for an entire family of non-Markovian models. It establishes that the standard toolkit of Markovian stochastic volatility — including Heston, SABR, and their calibrations — is structurally incapable of reproducing the empirically observed scaling, because any Markovian diffusion driven by a standard Brownian motion necessarily has H = 1/2 locally. The practical implication is not merely academic: any conditioning scheme that treats the current volatility level as a sufficient statistic for the future evolution of the smile is discarding the majority of the information available in the realized volatility path. This reading therefore motivates the use of path functionals of realized log-volatility as primary conditioning inputs, and it places the desk's entire rough-vol adjacent methodology on a quantitative empirical footing.

The primary estimator is the q-th absolute moment m(q, Δ) = E[|log σ_{t+Δ} − log σ_t|^q] computed from non-overlapping realized variance estimates as proxies for σ_t at lag Δ. Under monofractal scaling one expects m(q, Δ) = C(q) Δ^{qH}, so a log–log regression of m(q, Δ) against Δ for each q yields slope estimates ζ(q); linearity of ζ(q) in q with slope H is the monofractal signature. The Hurst index H ∈ (0, 1) parameterizes fractional Brownian motion B^H, defined via the Mandelbrot–van Ness kernel as B^H_t = ∫_{−∞}^t [(t−s)^{H−1/2} − (−s)^{H−1/2}_{+}] dW_s up to a normalizing constant, so that H = 1/2 recovers standard Brownian motion while H ∈ (0, 1/2) yields anti-persistent increments with negative autocovariance at all lags. The RFSV process sets log σ_t = ν W^H_t + ε_t where ε_t is a mean-zero correction capturing the long-run stationary level; the increments W^H_{t+Δ} − W^H_t are Gaussian with variance Δ^{2H}, which propagates into the moment scaling. The Hölder regularity index α of sample paths satisfies α = H − ε for all ε > 0 almost surely, making path roughness a direct consequence of H. Crucially, fBm for H ≠ 1/2 is neither a semimartingale nor a Markov process, so the classical Itô calculus and filtration-based Markovian arguments do not apply without extension, a point central to the model-theoretic consequences of this paper's empirical findings.

The most striking feature of Gatheral–Jaisson–Rosenbaum's empirical analysis is its cross-sectional robustness: the H ≈ 0.1 estimate replicates across S&P 500, DAX, FTSE, and Nikkei realized variance series, across estimation windows ranging from a few months to several decades of intraday data, and across a range of sampling frequencies from one minute to one day. This universality substantially strengthens the case that roughness is a genuine structural property of equity volatility rather than an artifact of a particular market microstructure or a particular estimation choice, and it correspondingly weakens the defense of any parsimonious Markovian model on universality grounds. The mono-fractal versus multifractal debate is relevant because multifractal models, including the Multifractal Random Walk of Bacry–Delour–Muzy, predict ζ(q) to be a strictly concave function of q rather than linear; the data examined here are broadly consistent with linearity, but the confidence intervals are wide enough that weak multiscaling cannot be excluded. From a desk methodology perspective the distinction matters because multifractal conditioning would require higher-order path statistics beyond what RFSV captures, and the evidence does not compel that extension. The implications for Markovian calibration are precise: any model whose instantaneous variance V_t follows a diffusion dV_t = a(V_t)dt + b(V_t)dW_t is locally Brownian in V and therefore has local H = 1/2. The empirical finding H ≈ 0.1 is therefore not a small deviation from Markovian models but a qualitative incompatibility. The Breuer–Major CLT for functionals of Gaussian processes with long memory is the statistical tool that makes the estimation theory rigorous: it establishes the asymptotic distribution of estimators such as m(q, Δ) when the underlying process has H ∈ (0, 1/2), correcting for the non-standard scaling of variance that arises from the anti-persistence.

The central desk-level consequence of this paper is that realized volatility should be treated as a path statistic rather than a level statistic. Specifically, a single-number summary of current volatility — whether instantaneous or short-window realized — carries a small fraction of the information available in the full trajectory of log-volatility increments over the preceding hours or days. The appropriate conditioning object for any surface-read or signal-generation procedure is therefore a functional of the realized log-volatility path, such as the running value of m(q, Δ) at a calibrated lag, rather than a spot variance estimate. This is a methodological prescription with direct implementation consequences: the desk's path-conditioning signals should be constructed by fitting the scaling exponent H in real time and treating departures from the historical H ≈ 0.1 baseline as a regime indicator. Path-conditioned signals consistently dominate level-based signals at rough-path horizons because the latter discard the anti-persistent increment structure that H < 1/2 encodes. Concretely, a high current realized variance that arrived via a path with strongly negative increment autocorrelations carries different information about near-term dynamics than the same current variance reached via a monotone path, and any signal that reduces the path to a scalar discards this distinction. The desk's multi-scale moment decomposition of realized variance increments is directly motivated by this finding.

The primary estimation confound is market microstructure noise: at sampling frequencies below approximately one to five minutes, bid-ask bounce and discrete tick effects inflate the apparent roughness of the observed price process, biasing H downward toward zero. The standard correction — using a sparse sampling frequency at which the Epps effect is not severe — introduces a bias-variance tradeoff, because the number of usable increments shrinks as the sampling interval widens, inflating the variance of the H estimate. The realized variance estimators used as σ_t proxies are themselves subject to this confound, and the paper's robustness checks across frequencies do not fully resolve whether the empirical H ≈ 0.1 is a true parameter or a residual microstructure artifact at intermediate frequencies. A more pointed challenge comes from Cont and Das, who argue that after applying appropriate finite-sample bias corrections to the moment estimator — corrections that account for the small sample sizes available within any fixed estimation window — the true H may be substantially closer to 1/2 than the raw estimates suggest. The debate turns on whether the persistence of the H ≈ 0.1 finding across windows and instruments is enough to override the finite-sample correction argument; the desk treats this as an open question and applies conservative uncertainty bounds on any H-dependent signal weights accordingly.