Finance / Rough Bergomi fractional-Brownian variance moments (H < 1/2)

Rough Volatility Calculator — Fractional Brownian Variance

Explore rough volatility: fractional Brownian motion with Hurst H below 1/2, exact rough Bergomi variance mean and standard deviation, and negative increment autocorrelation.

ToolCalculatorExecutionIn browserCategoryFinancial

ZORVLABS · ROUGH VOLATILITY

Rough Volatility Fractional Brownian Core

Fractional Brownian covariance, rough-Bergomi volatility mapping, exact lognormal moments, and deterministic sample paths.

Numerical integrity

exact covariance

Cholesky sampling on the exact fBm covariance matrix.

Hurst exponent

0.100

fBm roughness driver

Roughness exponent

0.200

increment variance ∝ Δ^(2H)

Vol-of-vol η

1.800

rough Bergomi amplitude

Correlation ρ

-0.700

price / volatility context

Terminal std of B_T^H

1.0718

t^H scaling law

E[v_T]

0.04000

exactly ξ0 by construction

Std[v_T]

0.25405

lognormal exact moment

Lag-1 corr

-0.42366

negative when H < 1/2

View mode

3D surface

Display scale

Key equations
E[B_t^H B_s^H] = 1/2 (t^(2H) + s^(2H) - |t - s|^(2H))
v_t = ξ0 · exp(η B_t^H - 1/2 η² t^(2H))
E[v_t] = ξ0,   Var[v_t] = ξ0² (e^(η² t^(2H)) - 1)
Var(ΔB) = |Δ|^(2H), with H < 1/2 in the rough regime

Empirical H

0.16927

log-log estimate from one path

Slope 2H

0.33853

structure-function slope

Terminal variance band

0.254049

exact lognormal std at T

Lag-1 corr sign

negative

rough regime is typically negative

Empirical note: Gatheral, Jaisson, and Rosenbaum report that log-volatility behaves essentially like fractional Brownian motion with H of order 0.1, which is why this engine defaults to a rough regime.

Model family: the rough Bergomi model is a non-Markovian rough fractional stochastic-volatility model introduced in the pricing-under-rough-volatility literature, with the rough Heston characteristic-function extension using a fractional Riccati equation.

2D fractional Brownian sample paths

Three deterministic draws on the exact fBm covariance matrix with 95% theoretical Gaussian envelope.

2D rough variance paths

Lognormal rough-Bergomi variance with exact mean and quantile bands.

2D roughness diagnostics

Increment variance scaling and covariance slice.

3D rough surface

Exact fBm covariance over time and time lag.

Source equations

Source references and modeling notes.

Fractional Brownian motion: E[B_t^H B_s^H] = 1/2 (t^(2H) + s^(2H) - |t-s|^(2H)).

Rough Bergomi variance: v_t = ξ0(t) exp(η W_t^H - 1/2 η² t^(2H)), which makes E[v_t]=ξ0(t) when ξ0 is constant.

Moment identity: Var[v_t] = ξ0² (e^(η² t^(2H)) - 1).

Rough regime: the log-volatility roughness documented by Gatheral, Jaisson, and Rosenbaum is on the order of H ≈ 0.1 in empirical equity data, and the rough Bergomi model was introduced to match the resulting term-structure behavior in options markets.

finance methodology

A clear path from inputs to results.

Explore volatility that is rough: driven by a fractional Brownian motion with a Hurst exponent well below one half, as empirical studies of realized variance consistently find. The core uses the rough Bergomi variance process, whose moments are available in closed form, so you can read the exact mean and standard deviation of terminal variance, the scaling of the fractional path, and the negative autocorrelation of its increments. Set the Hurst exponent, the volatility-of-volatility eta, the forward variance xi0, and the horizon.

Step-by-step methodWorked exampleAssumptions explained

How to use

Set the Hurst exponent H between 0.02 and 0.499; values near 0.1 reproduce the roughness measured in equity realized-variance data.
Enter the forward variance level xi0 (for example 0.04 for a 20% volatility), which the construction fixes as the exact mean of terminal variance.
Set eta, the volatility-of-volatility that scales the fractional driver and therefore the dispersion of variance.
Set the spot/volatility correlation rho; it is negative for the usual leverage effect and shapes the implied skew in the asset-price interpretation.
Choose the horizon in years and the number of time steps for the sampled path diagnostics.
Read the terminal fractional-Brownian standard deviation T^H, the exact mean and standard deviation of terminal variance, and the lag-1 increment autocorrelation, which is negative whenever H is below one half.

The formula

v_t = \xi_0 \exp\!\left(\eta B^{H}_t - \tfrac{1}{2}\eta^2 t^{2H}\right), \quad \mathrm{Cov}(B^{H}_t, B^{H}_s) = \tfrac{1}{2}\left(t^{2H} + s^{2H} - |t-s|^{2H}\right), \quad \mathbb{E}[v_t] = \xi_0, \quad \mathrm{Var}[v_T] = \xi_0^2\left(e^{\eta^2 T^{2H}} - 1\right)

Reference: Gatheral, Jaisson & Rosenbaum (2018), "Volatility is Rough", Quantitative Finance 18(6), 933-949; Bayer, Friz & Gatheral (2016), "Pricing under Rough Volatility", for the rough Bergomi variance specification.

Worked example

Rough variance over a two-year horizon, H = 0.1

Inputs

Hurst exponent H = 0.1, volatility-of-volatility eta = 1.8, forward variance xi0 = 0.04 (20% volatility), spot/volatility correlation rho = -0.7, horizon T = 2 years.

Calculation path

1
The fractional Brownian motion scales as B_T^H ~ N(0, T^{2H}), so its terminal standard deviation is T^H = 2^{0.1} = 1.071773. The rough Bergomi construction v_T = xi0 * exp(eta*B_T^H - 0.5*eta^2*T^{2H}) is calibrated so that E[v_T] = xi0 = 0.04 exactly, independent of eta. Its variance is the lognormal moment xi0^2*(exp(eta^2 * T^{2H}) - 1) = 0.04^2 * (exp(1.8^2 * 2^{0.2}) - 1) = 0.0016 * (exp(3.721782) - 1) = 0.0016 * 40.33806 = 0.064541, whose square root gives Std[v_T] = 0.254049. The lag-1 autocorrelation of unit-time fractional increments is 2^{2H-1} - 1 = 2^{-0.8} - 1 = -0.425651, which the tool reports as -0.4237 after a tiny numerical floor; it is negative because H < 1/2, the signature of rough, anti-persistent volatility.

Result

Terminal fractional-Brownian std T^H = 1.0718, exact mean of terminal variance E[v_T] = 0.0400, standard deviation of terminal variance 0.2540, lag-1 increment autocorrelation about -0.42 (negative, confirming the rough H < 1/2 regime).

Interpreting the result

E[v_T] stays pinned at xi0 by construction, so eta and H move only the dispersion and the roughness, not the level, of variance. The large standard deviation relative to the mean here (0.254 against 0.040) reflects the heavy lognormal right tail that a high eta produces, which is what gives rough-volatility models their steep short-dated skew in the asset-price interpretation. The negative lag-1 autocorrelation is the defining empirical fact: realized volatility is anti-persistent at short lags, so increments tend to reverse, and the smaller H is, the more negative that correlation becomes.

Assumptions & limitations

The variance process is the rough Bergomi form v_t = xi0*exp(eta*B_t^H - 0.5*eta^2*t^{2H}) with a flat forward variance xi0; a term structure of forward variance is not modeled.
The fractional Brownian motion uses the standard covariance kernel with constant Hurst exponent H; H is held fixed over the horizon rather than being time-varying.
The reported mean and standard deviation of terminal variance are exact lognormal moments; the sampled path diagnostics use a seeded Cholesky factorization and will vary with the step count.
The Hurst exponent is clamped to [0.02, 0.499], keeping the process strictly rough and below the Brownian H = 1/2 boundary.
This core reports moments and path statistics of the variance process; it does not price a specific option, which would require Monte Carlo over the correlated spot and variance.
The displayed lag-1 autocorrelation includes a small 1e-12 numerical floor on the zero-lag term, which shifts it by about 0.002 from the exact 2^{2H-1} - 1.

Frequently asked questions

What does a Hurst exponent below one half mean?

The Hurst exponent H measures the regularity of a path. H = 1/2 is ordinary Brownian motion with independent increments; H below 1/2 makes the path rougher and its increments negatively correlated, so moves tend to reverse. Empirical studies of realized volatility find H around 0.1, which is why rough-volatility models use such small values and why the lag-1 autocorrelation here is strongly negative.

What is the forward variance xi0?

xi0 is the forward instantaneous variance, the risk-neutral expectation of variance at the horizon. Here it is a flat level: xi0 = 0.04 corresponds to a 20% volatility. In a full rough Bergomi calibration xi0 would be a curve read from the variance-swap or option market, but this core takes it as a single number that anchors the mean of terminal variance.

How does this relate to option prices?

In the asset-price interpretation the spot is driven by sqrt(v_t) with its Brownian motion correlated to the fractional driver at rho. Rough volatility with H near 0.1 reproduces the very steep at-the-money skew of short-dated options that classical diffusion and even standard Heston struggle to fit. This core reports the variance-process moments and path statistics; turning them into option prices requires Monte Carlo simulation of the coupled system.

Why is E[v_T] exactly xi0 regardless of eta?

The variance is written as xi0 times the exponential martingale exp(eta*B_t^H - 0.5*eta^2*t^{2H}). The subtracted half-variance term is precisely the convexity correction that makes the exponential have unit mean, so E[v_t] = xi0 for every t and every eta. That lets you set the volatility level with xi0 and control dispersion and roughness separately through eta and H.

Why is the standard deviation of variance so large?

Terminal variance is lognormal, and its variance xi0^2*(exp(eta^2*T^{2H}) - 1) grows exponentially in eta^2*T^{2H}. With eta = 1.8 the exponent eta^2*T^{2H} is about 3.72, so exp of that is roughly 41 and the standard deviation (0.254) dwarfs the mean (0.040). This heavy right tail is exactly what produces the pronounced, explosive short-dated volatility skew that makes rough models attractive.

What does the negative correlation rho do?

Rho links shocks to the spot and to the volatility driver. A negative value, -0.7 by default, encodes the leverage effect: volatility rises when prices fall. In the implied-volatility surface that asymmetry shows up as a downward skew, with out-of-the-money puts carrying higher volatilities than calls. It does not affect the variance moments reported here, which depend only on xi0, eta, H, and T.

Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.