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

The paper estimates the smoothness of the log-volatility path from high-frequency realized variance and finds that its increments scale with a Hurst exponent H of order 0.1 — far rougher than diffusive stochastic volatility (H = 1/2) admits. We read this as a structural constraint on any desk model that conditions on a volatility regime: the object being conditioned on is, at the relevant horizon, a non-Markovian fractional process whose recent path carries information that a Markov state cannot encode.

For increments at scale Δ, the q-th absolute moment m(q,Δ) = E[ |log σ_{t+Δ} − log σ_t|^q ] scales as Δ^{ζ_q} with ζ_q ≈ qH and H ≈ 0.1. The proposed RFSV model takes log σ_t as a fractional Ornstein–Uhlenbeck process driven by fractional Brownian motion B^H with H < 1/2.

The empirical scaling is strikingly stable across indices and across estimation windows, which is what makes the result more than a curve fit: H near 0.1 is recovered as a near-universal exponent rather than a fitted parameter. For the desk, the mono-fractal scaling is the more important claim than any specific pricing model built on top of it.

Memos that reference an implied volatility regime should treat the regime as a path statistic, not a level. A single-snapshot read of where vol sits relative to history is weaker than a read of how the recent realized path arrived there.

Estimation conflates the roughness of latent log-volatility with measurement error in realized variance; subsequent work debates how much of the apparent H ≈ 0.1 is microstructure-induced. We treat H as an effective, horizon-dependent quantity rather than a fundamental constant.