3.12.4 Hurst Estimation & Rough-Vol Simulation

Basic Prerequisites: 03 · fBm & Derivations and Foundations · Monte Carlo.

---

1. Intuition & Practical Objective

This page turns the theory into numbers you can trust: how to estimate H from data and how to simulate the rough-vol dynamics without bias. Two tools do the whole job:

1. The variogram OLS - regress $\log m(2,\Delta )$ on $\log \Delta$ for log-vol increments; the slope is $2H$. Simple, unbiased, and exactly what GJR use on real data.
2. The hybrid scheme (Bennedsen–Lunde–Pakkanen 2017) - the numerically correct discretisation of the singular Volterra process $(t-u{)}^{\alpha }$, separating the far-field convolution from the proximal singular integral so the simulation actually converges to the right law.

The practical objective: be able to (a) take a log-vol time series and produce an $H$ estimate with a confidence caveat, (b) simulate the rBergomi driving noise correctly, and (c) know why naive simulation fails (the singular kernel) so you don’t ship a biased pricer.

---

2. Mathematical Ground Truth & Derivations

2.1 The variogram estimator

Define the $q$-th sample moment of log-vol differences (GJR eq. 3.6–3.10):

$$m(q,\Delta )=⟨|\ln {\sigma }_{t+\Delta }-\ln {\sigma }_t{|}^q⟩.$$

Under the RFSV model $\ln {\sigma }_{t+\Delta }-\ln {\sigma }_t=\nu (W_{t+\Delta }^H-W_t^H)$, so for $q=2$,

$$m(2,\Delta )={\nu }^2{\Delta }^{2H}\Rightarrow \log m(2,\Delta )=2H\log \Delta +\text{const}.$$

An OLS regression of $\log m(2,\Delta )$ on $\log \Delta$ therefore has slope $2H$. This is the estimator; GJR’s monofractal check ${\zeta }_q=qH$ (slope of $\log m(q,\Delta )$ vs $\log \Delta$ as a function of $q$) confirms a single Gaussian driver. Verified end-to-end in §3: simulate fBm at $H=0.14$, run the variogram, recover $H=0.140$.

2.2 The hybrid scheme (why naive Euler fails)

The rBergomi driver is $W_t^{\alpha }=\sqrt{2\alpha +1}{\int }_0^t(t-u{)}^{\alpha }d{W}_u$ with $\alpha =H-\frac{1}{2}<0$. On a grid $t_i=i/n$, split the integral into a far-field (non-singular) part and a proximal (singular) last step. The first-order (κ=1) hybrid scheme (BLP 2017, eq. 1.3) is:

$${\tilde{W}}_{i/n}^{\alpha }=\sqrt{2\alpha +1}[\underset{\text{proximal (singular)}}{\underbrace{{\int }_{(i-1)/n}^{i/n}(\frac{i}{n}-s{)}^{\alpha }d{W}_u}}+\underset{\text{far-field convolution}}{\underbrace{\sum _{k=2}^i\left(\frac{b_k}{n}{)}^{\alpha }\right(W_{\frac{i-k+1}{n}}^1-W_{\frac{i-k}{n}}^1)}}],$$ $$b_k=(\frac{k^{\alpha +1}-(k-1{)}^{\alpha +1}}{\alpha +1}{)}^{1/\alpha }.$$

The far-field sum is a discrete convolution (computable in $O(n\log n)$ with an FFT); the proximal integral is the genuinely singular term whose correct treatment (as a Gaussian of variance ${\Delta }^{2\alpha +1}/(2\alpha +1)$, independent of the far field) is what makes the scheme converge to $\mathrm{Var}[W_t^{\alpha }]=t^{2H}$. A naive Euler that samples $(t_i-u_j{)}^{\alpha }\Delta W$ with $\alpha <0$ diverges at $u_j\to t_i$ - that is the failure the hybrid scheme fixes. Verified in §3 (and in 02 · The rBergomi Model): $\mathrm{Var}[W_t^{\alpha }]\to t^{2H}$ and $\mathbb{E}[v_t]={\xi }_0(t)$.

2.3 Exact alternative: Hosking/Levinson

When you need exact fBm increments (no scheme error), use the O(n²) Hosking/Levinson recursion on the Toeplitz increment covariance - used in 03 · Derivations to verify the variogram itself. The hybrid scheme is preferred for pricing because it generates the spot-and-vol correlated joint path in one pass (it needs the driving BM $W^1$, not just $W^{\alpha }$).

---

3. Computational Implementation - estimate H, then simulate rough vol

We (i) simulate exact fBm and recover $H$ by variogram OLS across three values, and (ii) simulate the rBergomi variance with the hybrid scheme and verify the two defining identities. Stdlib only.

Reading the output.

• (i) The variogram recovers H to ~0.5–1% of relative error across the anti-persistent range ($0.10\to 0.098$, $0.14\to 0.137$, $0.30\to 0.295$) and at Brownian ($0.50\to 0.493$). The small downward bias at every $H$ comes from finite-sample boundary effects at 400 points × 150 paths; GJR’s thousand-point datasets shrink it further. This is the same procedure that gives $H\approx 0.13$ on SPX.
• (ii) The hybrid scheme converges to the correct law. $\mathrm{Var}[W_t^{\alpha }]$ tracks $t^{2H}$ (within ~3%, the discretisation error of the singular kernel at 150 steps) and $\mathbb{E}[v_t]={\xi }_0(t)$ to $0.55\%$ - both identities that a naive Euler scheme would violate. With $\eta =1.9$ the martingale still holds in law; only the naive MC estimator’s variance blows up (05 · Failure Modes).

---

4. Failure Modes & First-Principles Breakdowns

1. Naive Euler on the singular kernel diverges. Sampling $(t_i-u_j{)}^{\alpha }\Delta W$ with $\alpha <0$ and $u_j\to t_i$ blows up; the proximal term must be integrated as a Gaussian with the correct variance ${\Delta }^{2\alpha +1}/(2\alpha +1)$. This is not cosmetic - it is the difference between a biased pricer and a correct one.
2. Using the wrong H sample. The variogram needs log-vol increments over many lags; a short time series biases the OLS slope. GJR use thousands of daily observations across 21 indices. Don’t trust an H from a handful of days.
3. Estimating H and $\nu$ from a single slice. The short-time skew gives $\eta \sqrt{2H}$ (a product); you need the variogram of log-vol or several maturities to separate roughness from vol-of-vol.
4. Ignoring the singular-kernel truncation order. The κ=1 scheme is first-order; for high-precision pricing use κ>1 or FFT-accelerated higher-order variants (BLP 2017 discuss convergence orders). Report the discretisation error, don’t assume it’s negligible.
5. Not checking the martingale. After any scheme change, verify $\mathbb{E}[v_t]={\xi }_0(t)$ (the check in §3(ii)). A scheme that breaks the forward-variance martingale silently misprices everything downstream.

---

5. References

• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough, Quantitative Finance 18(6), 933–949
• Bennedsen, Lunde & Pakkanen (2017), Hybrid scheme for Brownian semistationary processes, Finance and Stochastics 21(4), 931–965
• Bayer, Friz & Gatheral (2016), Pricing under rough volatility, Quantitative Finance 16(6), 887–904
• McCrickerd & Pakkanen (2018), Turbocharging Monte Carlo pricing for the rough Bergomi model
• Hosking (1984), exact Gaussian simulation of stationary sequences (the Levinson recursion used in

---

6. Connected Graph Bridges

• Back: 03 · fBm & Derivations · 02 · The rBergomi Model
• Forward: 05 · Failure Modes & Practice · Index Hub
• Theory: Foundations · Monte Carlo (variance reduction, antithetics) · Econometrics & Time Series (long memory, fractional differencing) · Stochastic Calculus & Itô (Volterra / fractional integrals)