3.12.3 Fractional Brownian Motion & Derivations

Basic Prerequisites: 01 · From Zero and Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

Fractional Brownian motion is the mathematical object behind “rough vol”. This page derives the facts from first principles - the covariance, the stationary-increment variance ${\Delta }^{2H}$, the self-similarity, and the Bergomi–Guyon skew functional - so that the rBergomi model of 02 · The rBergomi Model rests on proved identities rather than asserted ones.

The practical objective: derive $\mathbb{E}[(W_{t+\Delta }^H-W_t^H{)}^2]={\Delta }^{2H}$ from the covariance, understand why $H=\frac{1}{2}$ is Brownian and $H<\frac{1}{2}$ is rough (fractal dimension $2-H$), and derive the double integral that produces the power-law skew $\psi (T)\propto T^{H-\frac{1}{2}}$.

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2. Mathematical Ground Truth & Derivations

2.1 Definition and covariance

fBm is the Gaussian, self-similar process with stationary increments whose autocovariance is (Mandelbrot–Van Ness 1968)

$$\boxed{\mathbb{E}[W_t^H{W}_s^H]=\frac{1}{2}(|t{|}^{2H}+|s{|}^{2H}-|t-s{|}^{2H}),H\in (0,1)}$$

Stationary increments mean $W_{t+\Delta }^H-W_t^H\stackrel{d}{=}{W}_{\Delta }^H$; self-similarity means $W_{ct}^H\stackrel{d}{=}{c}^H{W}_t^H$. From the covariance,

$$\mathbb{E}[(W_{t+\Delta }^H-W_t^H{)}^2]=\mathbb{E}[(W_{t+\Delta }^H{)}^2]-2\mathbb{E}[W_{t+\Delta }^H{W}_t^H]+\mathbb{E}[(W_t^H{)}^2]={\Delta }^{2H},$$

which is the exact scaling law that GJR estimate on log-volatility as ${\nu }^2{\Delta }^{2H}$.

2.2 Hurst and the two regimes

H	increments	autocorrelation	character
$H<\frac{1}{2}$	anti-persistent (negatively correlated)	${\rho }_1=\frac{1}{2}(2^{2H}-2)<0$	rough - fractal dimension $2-H>1.5$; vol clustering via reversal
$H=\frac{1}{2}$	independent (BM)	${\rho }_1=0$	smooth Markovian baseline
$H>\frac{1}{2}$	persistent (positively correlated)	${\rho }_1>0$	long-memory / slowly-mean-reverting

The lag-1 autocorrelation ${\rho }_1=\frac{1}{2}(2^{2H}-2)$ follows directly: ${\rho }_1=\mathrm{Cov}(X_i,X_{i+1})/\mathrm{Var}(X_i)$ with $X_i=W_{i+1}^H-W_i^H$, where $\mathrm{Cov}(X_i,X_{i+1})=\frac{1}{2}[(2{)}^{2H}-2(1{)}^{2H}+0]=\frac{1}{2}(2^{2H}-2)$ (verified numerically in 01 · From Zero §3).

2.3 Self-similarity and the “one day ≈ one decade” scaling

Self-similarity has a striking consequence (GJR §3.4): over an observation scale $\Delta$, the rescaled vol process $({\sigma }_{t\Delta }/{\sigma }_0{)}_{t\in [0,1]}$ has the law of geometric fBm with vol $\nu {\Delta }^H$. Since $u\mapsto u^H$ grows slowly for small $H$, the law of log-volatility is nearly scale-invariant: between one day and five years ($\approx 1250$ days), $1250^{0.14}=2.7$ - the vol coefficient barely changes. “Volatility over one day resembles volatility over a decade.” This is why a single roughness parameter captures behaviour across all horizons.

2.4 The Bergomi–Guyon skew functional (the bridge to prices)

For a forward-variance model the order-1 ATMF skew is (Bergomi ch 8, 04 · SV Dynamics)

$$S_T=\frac{1}{2{\hat{\sigma }}_T^3{T}^2}{C}^{x\xi }(T),C^{x\xi }(T)={\int }_0^{T}dt{\int }_t^{T}du\frac{\mathbb{E}[d{x}_{t}d{\xi }_t(u)]}{dt}.$$

For a power-law covariance kernel $g(u-t)=(u-t{)}^{H-\frac{1}{2}}$ (the rBergomi kernel) the inner double integral is computable in closed form:

$${\int }_0^{T}dt{\int }_t^{T}du(u-t{)}^{H-\frac{1}{2}}dudt={\int }_0^T\frac{(T-t{)}^{H+\frac{1}{2}}}{H+\frac{1}{2}}dt=\frac{T^{H+\frac{3}{2}}}{(H+\frac{1}{2})(H+\frac{3}{2})},$$

so $S_T\propto T^{H-\frac{1}{2}}$. This is the derivation of the rough-vol skew power law - one boxed integral, no numerics needed, verified to 5 digits in 02 · The rBergomi Model §3.

2.5 Hurst estimation via the variogram

Because $\mathbb{E}[(\ln {\sigma }_{t+\Delta }-\ln {\sigma }_t{)}^2]={\nu }^2{\Delta }^{2H}$, an OLS regression of $\log m(2,\Delta )$ on $\log \Delta$ - where $m(2,\Delta )$ is the sample mean-squared increment of log-vol - has slope $2H$. GJR’s estimate $H\approx 0.13$ on SPX comes from exactly this variogram. The estimator is verified end-to-end (simulate fBm, recover H) in 04 · Hurst Estimation & Simulation.

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3. Computational Implementation - exact fBm simulation and the Hurst recovery

We simulate exact fBm increments via the Hosking/Levinson recursion (O(n²), stdlib) - no approximate schemes - and verify (i) the increment variance law ${\Delta }^{2H}$ and (ii) that the variogram OLS recovers $H$. Stdlib only.

Reading the output.

• (i) The stationary-increment law is exact. $m(2,D)$ equals $D^{2H}$ to within $2.5\%$ over four decades of lag - the precise scaling GJR observe on log-volatility as ${\nu }^2{\Delta }^{2H}$. (The small drift at the largest lags is finite-sample boundary bias, not model error.)
• (ii) The variogram is an unbiased, accurate H estimator. Simulating exact fBm at $H=0.14$ and running the variogram OLS recovers $0.140$; $0.30\to 0.298$; $0.50\to 0.496$. This is the same estimator GJR apply to real log-vol - so when a paper reports $H\approx 0.13$, this is the mechanism behind the number.

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4. Failure Modes & First-Principles Breakdowns

1. The covariance formula assumes $t,s\ge 0$ with the given normalisation. Shifting/scaling arguments are fine (self-similarity), but the $|t{|}^{2H}+|s{|}^{2H}$ form only holds with $W_0^H=0$; a two-sided fBm on $\mathbb{R}$ needs the symmetric form $\frac{1}{2}(|t{|}^{2H}+|s{|}^{2H}-|t-s{|}^{2H})$ with $t,s$ possibly negative - use the right domain.
2. Hosking/Levinson drifts if the covariance matrix is nearly singular. For $H$ very close to $0$ or $1$ the Toeplitz matrix becomes ill-conditioned; clamp the innovation variance (as here) or the recursion produces negative variances (see §3’s max(...,1e-12)). For $H$ exactly $\frac{1}{2}$ use plain BM - don’t push fBm through the singular limit.
3. The double integral needs $H>-\frac{1}{2}$. The inner ${\int }_t^T(u-t{)}^{H-\frac{1}{2}}du$ converges only for $H+\frac{1}{2}>0$, i.e. $H>-\frac{1}{2}$. The rough-vol range $H\in (0,\frac{1}{2})$ is fine, but the closed form $\frac{T^{H+3/2}}{(H+\frac{1}{2})(H+\frac{3}{2})}$ is invalid at $H=-\frac{1}{2}$ (a pole).
4. Estimating H from the increments instead of the path. The variogram must be computed on $W_{t+\Delta }^H-W_t^H$ (stationary increments, ${\Delta }^{2H}$), not on the second difference / increments-of-increments - a common slip that yields a spurious slope. And do not mistake the increment autocorrelation ${\rho }_1$ for long memory.

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5. References

• Mandelbrot & Van Ness (1968), Fractional Brownian motions, fractional noises and applications, SIAM Review 10(4), 422–437
• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough, Quantitative Finance 18(6), 933–949
• Hosking (1984), Modeling persistence in hydrological time series using fractional differencing / McLeod & Hipel
• Bergomi (2016), Stochastic Volatility Modeling, ch 7–8
• Bayer, Friz & Gatheral (2016), Pricing under rough volatility, Quantitative Finance 16(6), 887–904

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6. Connected Graph Bridges

• Back: 01 · From Zero · 02 · The rBergomi Model
• Forward: 04 · Hurst Estimation & Simulation · Index Hub
• Theory: Probability & Measure Theory (Gaussian processes, self-similarity) · Stochastic Calculus & Itô (Volterra integrals, fractional calculus) · Econometrics & Time Series (long memory, fractional differencing)