3.12.1 Rough Volatility from Zero

Basic Prerequisites: Black–Scholes–Merton (or none - this page is written to stand alone).

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1. Intuition & Practical Objective

Stochastic-volatility models made volatility random but kept it smooth: a diffusion’s increments scale like $\sqrt{\Delta t}$. The market disagrees. Measure daily log-volatility $\ln {\sigma }_t$ and look at how its increments scale with the lag $\Delta$:

$$\mathbb{E}[(\ln {\sigma }_{t+\Delta }-\ln {\sigma }_t{)}^2]={\nu }^2{\Delta }^{2H}.$$

If $\ln \sigma$ were a diffusion, $H=\frac{1}{2}$. The data say $H\approx 0.1$ (Gatheral–Jaisson–Rosenbaum 2018). A process with $H<\frac{1}{2}$ is rougher than Brownian motion: its increments are anti-correlated (a step up is more likely followed by a step down), and its paths have fractal dimension $2-H>1.5$ - visibly wiggly, not smooth.

Three “aha”s:

1. Rough ≠ jumpy, rough = anti-persistent. A fractional Brownian motion with $H=0.14$ is a continuous Gaussian process (no jumps) whose increments have negative autocorrelation. This is the opposite of “long memory” (which is $H>\frac{1}{2}$). Rough vol is anti-persistent noise with a power-law covariance kernel $(t-s{)}^{H-1/2}$.
2. H is a single knob that sets every short-dated property. The same exponent controls the increment variance (${\Delta }^{2H}$), the autocorrelation sign, the fractal dimension ($2-H$), and the ATMF skew term structure $\psi (T)\propto T^{H-\frac{1}{2}}$. One number, and the whole short end follows.
3. The skew is the forensic evidence. Markovian SV models (Heston, SABR, $n$-factor Bergomi) generate an ATMF skew that caps as $T\to 0$. The market’s SPX skew grows like $T^{-0.44}$. The only way to get a blowing-up short skew is a power-law kernel with $H<1/2$.

The practical objective: internalise that “rough” is a precise, measurable statement about the scaling exponent of log-vol increments - not a vague notion of turbulence - and that the single number $H\approx 0.1$ is both an empirical fact and the design parameter that fixes the short-dated skew.

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2. Mathematical Ground Truth & Derivations

2.1 Fractional Brownian motion in one line

fBm $W_t^H$ is the Gaussian process with autocovariance (Mandelbrot–Van Ness 1968)

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

For $H=\frac{1}{2}$ this collapses to $\mathbb{E}[W_t{W}_s]=\min (t,s)$ - ordinary Brownian motion. For $H\ne \frac{1}{2}$ it is neither a martingale nor a Markov process, but it has stationary increments: $W_{t+\Delta }^H-W_t^H$ has the same law as $W_{\Delta }^H$, and

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

(stationary increments give this exactly, as verified in §3). It is this exact ${\Delta }^{2H}$ law that GJR estimate as ${\nu }^2{\Delta }^{2H}$ on log-volatility.

2.2 Why H<1/2 means anti-persistence

The lag-1 autocorrelation of fBm increments $X_i=W_{i+1}^H-W_i^H$ is

$${\rho }_1=\frac{\mathbb{E}[X_i{X}_{i+1}]}{\mathbb{E}[X_i^2]}=\frac{1}{2}(2^{2H}-2),$$

which is negative for $H<\frac{1}{2}$, zero at $H=\frac{1}{2}$, positive for $H>\frac{1}{2}$. So $H=0.14\Rightarrow {\rho }_1=\frac{1}{2}(2^{0.28}-2)\approx -0.393$: after a volatility spike the next increment is more likely to reverse it - vol clusters because it reverts. This is the single cleanest number distinguishing rough ($H\ll \frac{1}{2}$) from smooth ($H>\frac{1}{2}$) from Brownian ($H=\frac{1}{2}$).

2.3 The RFSV model and the monofractal scaling

GJR model log-vol directly as fBm (“Rough Fractional Stochastic Volatility”):

$$\log {\sigma }_{t+\Delta }-\log {\sigma }_t=\nu (W_{t+\Delta }^H-W_t^H),$$

so $\mathbb{E}[(\ln {\sigma }_{t+\Delta }-\ln {\sigma }_t{)}^2]={\nu }^2{\Delta }^{2H}$. They further check the monofractal scaling of the $q$-th sample moments $m(q,\Delta )=⟨|\ln {\sigma }_{t+\Delta }-\ln {\sigma }_t{|}^q⟩\propto {\Delta }^{{\zeta }_q}$ and find ${\zeta }_q=qH$ with $H\approx 0.13$ on SPX - the signature of a single Gaussian (fractional) driver rather than a multifractal cascade. The exponent $\alpha =H-\frac{1}{2}\approx -0.36$ is what the skew will inherit (§03, §02).

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3. Computational Implementation - H’s sign decides persistence

We compute the fBm increment autocorrelation exactly from the covariance formula, for $H$ from anti-persistent ($0.14$) through Brownian ($0.5$) to persistent ($0.9$). Stdlib only - no simulation needed, the answer is a closed-form number.

Read the table:

1. H=0.14 increments are anti-correlated (${\rho }_1=-0.393$), and the autocorrelation decays with lag ($-0.393\to -0.002$): rough vol is not long memory - it is short-range anti-persistence. This is exactly GJR’s §4 point: the data are rough, not long-memory.
2. H=1/2 is pure white noise (all zeros) - ordinary Brownian increments. This is the smooth/diffusive benchmark that Heston and SABR assume.
3. H=0.9 is persistent (${\rho }_1=+0.741$) - the “smooth, slowly-mean-reverting vol” of the pre-rough era, and the direction you must not go if the data say $H\approx 0.1$.

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4. Failure Modes & First-Principles Breakdowns

1. “Rough means turbulent/jumpy.” No - a fractional Brownian motion with $H=0.14$ is continuous and Gaussian. Roughness is about the scaling exponent of increments (and hence fractal dimension $2-H>1.5$), not about discontinuities. Rough ≠ Lévy.
2. “Rough = long memory.” The opposite. $H<1/2$ is anti-persistent; classical long-memory estimators (R/S, fractional differencing) are exactly the tools that falsely certify long memory on rough data (GJR §4). Get the sign of the exponent wrong and you build the wrong model.
3. Estimating H from a too-short sample. The variogram slope needs many lags; with a handful of days the estimator is badly biased (GJR use thousands of daily obs across 21 indices). H≈0.1 is a population claim.
4. Reading H off a single skew slice. The short-time skew gives $\eta \sqrt{2H}$ (product), not H alone. You need the variogram of log-vol or several maturities to separate the roughness from the vol-of-vol.
5. Confusing the physical ($\mathbb{P}$) and risk-neutral ($\mathbb{Q}$) measures. GJR’s H is estimated from realized variance (physical measure). Pricing uses the $\mathbb{Q}$ forward-variance curve; the roughness parameter is assumed to pass through the change of measure (BFG), but the vol-of-vol $\eta$ need not.

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5. References

• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough, Quantitative Finance 18(6), 933–949
• Mandelbrot & Van Ness (1968), Fractional Brownian motions, fractional noises and applications, SIAM Review 10(4), 422–437
• Bennedsen, Lunde & Pakkanen (2017), Hybrid scheme for Brownian semistationary processes, Finance and Stochastics 21(4), 931–965
• Fukasawa (2017), Short-time at-the-money skew and rough fractional volatility
• Hull, Options, Futures, and Other Derivatives, ch 20

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6. Connected Graph Bridges

• Base: Stochastic Calculus & Itô’s Lemma (BM as the H=1/2 case) · Probability & Measure Theory (Gaussian processes, self-similarity) · Black–Scholes–Merton (the constant-σ zero point)
• Continue: 03 · fBm & Derivations · Index Hub
• Sibling: Heston · 01 From Zero (why variance must be a process - the Markovian step that rough vol then out-roughs) · VS · 01 From Zero