3.12 Rough Volatility & Fractional Models

Basic Prerequisites: Volatility Surfaces & Smiles and Heston, SABR & Stochastic-Vol Dynamics. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Black–Scholes gives you a constant volatility. Stochastic-volatility models upgrade that to a diffusion - but the market is telling you something sharper still. Gatheral, Jaisson & Rosenbaum (2018) measured the daily log-volatility of 21 global indices and found its increments scale with lag as $(\Delta {)}^{2H}$ with $H\approx 0.1$, not $H=\frac{1}{2}$. A diffusion’s increments grow like ${\Delta }^{1/2}$; a process with $H<1/2$ has anti-correlated increments and is rougher than Brownian motion. That single empirical fact rewrites the whole vol-pricing paradigm.

The one-sentence essence:

Log-volatility is well-modelled as a fractional Brownian motion with Hurst $H\approx 0.1$: its increments are stationary, near-Gaussian, and scale as ${\Delta }^H$ with $H\ll \frac{1}{2}$, which produces an ATMF skew that decays as $T^{H-\frac{1}{2}}$ - blowing up far faster at short maturities than any Markovian stochastic-volatility model can generate.

Why this matters on a desk: the short-dated skew is where Markovian SV models (Heston, SABR, $n$-factor Bergomi) all fail structurally. A two-factor OU model needs its slowest factor to fit a power-law skew, and even then the vol-of-vol term structure is exponential-kernel while the data are power-law (${\nu }_T\propto T^{-0.4}$). Rough vol gets both right with three parameters ($H,\eta ,\rho$). This folder is the topic-hub: intuition, the fractional-Brownian ground truth, the rBergomi pricing model, Hurst estimation, the failure modes, and the frontier (rough Heston, Markovian lifts).

2. Mathematical Ground Truth & Derivations

Notation. $S$ spot, $F_T=S_0{e}^{(r-q)T}$ forward, $K$ strike, $k=\ln (K/F_T)$ log-moneyness, $T$ maturity, $v_t$ instantaneous variance, ${\xi }_t(u)={\mathbb{E}}_t[v_u]$ the forward-variance curve, ${\sigma }_{BS}$ Black–Scholes implied vol, $W^H$ fractional Brownian motion with Hurst $H$, $\psi (T)=|{\partial }_k{\sigma }_{BS}{|}_{k=0}$ the ATMF skew.

Quick-Reference Lookup (job #1). Every formula is transcribed from the cited corpus (Gatheral–Jaisson–Rosenbaum 2018; Bayer–Friz–Gatheral 2016; Bennedsen–Lunde–Pakkanen 2017; Bergomi 2016; Mandelbrot–Van Ness 1968) and every number in the check column was re-executed (§3 and the sub-pages).

Critical caveat (flagged in the corpus). The $H\approx 0.1$ figure and the power-law ${\nu }_T\propto T^{-0.4}$ are empirical estimates with statistical error bars (GJR bootstrap CI’s, §3.4). And the RFSV scaling is not classical long memory: GJR (§4) show that naive long-memory estimators applied to a well-calibrated RFSV path produce the same “long-memory” verdict as on real data - the apparent long memory of volatility is spurious, an artefact of anti-persistence at short scales, not a true $k^{-2H-1}$ tail.

3. Computational Implementation - the fractional-engine (hybrid scheme + Hurst variogram)

Stdlib only (math, random) - no numpy/scipy. The whole folder runs on two primitives: the hybrid scheme for the Volterra process $W^{\alpha }$ (the fractional driving noise of rBergomi) and the variogram OLS for Hurst estimation. Every number in §2 and the sub-pages was produced by these.

$\mathrm{Var}[W_t^{\alpha }]=t^{2H}$ is the defining property (the exponent of the Volterra kernel sets it), and $\mathbb{E}[v_t]={\xi }_0(t)$ is the forward-variance martingale that anchors the whole pricing construction - if the drift term ${\eta }^2{t}^{2H}/2$ had the wrong sign, the martingale would fail. This is the first thing to check on any rBergomi implementation, exactly as $\phi (-i)=1$ is the first check for Heston.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Rough vol is not long memory. H<1/2 means anti-persistent increments and a decaying-but-not-classical autocorrelation; classical long-memory estimators misread it as true long memory (GJR §4).
2. The singular kernel is the cost. $(t-u{)}^{\alpha }$ with $\alpha <0$ blows up at $u\to t$: naive Euler simulation of rBergomi is divergent and biased, which is exactly why the hybrid scheme exists.
3. Markovian models are structurally wrong at the short end. Any finite-factor OU/vol model produces an ATMF skew that caps as $T\to 0$; the data require $\psi (T)\propto T^{-0.4}$ to blow up. Calibrating harder cannot fix this.
4. Non-Markovianity kills the PDE and Fourier machinery. No finite-dimensional Markov representation, so pricing is Monte Carlo or an approximating Markovian lift - slower and harder to calibrate jointly to SPX and VIX.
5. H and η are confounded in the short-time skew. $\psi (T)=\rho \eta \sqrt{2H}{T}^{H-\frac{1}{2}}/(\cdots )$: from one skew slice you see the combination $\eta \sqrt{2H}$, not each separately - a classical identification degeneracy.
6. The martingale + positivity checks are mandatory. Get the drift sign or the $\alpha =H-\frac{1}{2}$ mapping wrong and $\mathbb{E}[v_t]\ne {\xi }_0(t)$ silently - every option price is then wrong.

5. References

• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough, Quantitative Finance 18(6), 933–949
• Bayer, Friz & Gatheral (2016), Pricing under rough volatility, Quantitative Finance 16(6), 887–904
• Bennedsen, Lunde & Pakkanen (2017), Hybrid scheme for Brownian semistationary processes, Finance and Stochastics 21(4), 931–965
• Bergomi (2016), Stochastic Volatility Modeling, CRC Press
• Mandelbrot & Van Ness (1968), Fractional Brownian motions, fractional noises and applications, SIAM Review 10(4), 422–437
• Fukasawa (2017), Short-time at-the-money skew and rough fractional volatility
• Gatheral (2006), The Volatility Surface, ch 7 (short-expiration skew asymptotics that rough vol generalises).

6. Connected Graph Bridges

• Foundational base: Stochastic Calculus & Itô’s Lemma · Probability & Measure Theory · Numerical Methods (MC) · Econometrics & Time Series (Hurst estimation lives here)
• Sibling topics: Heston, SABR & Stochastic-Vol Dynamics (the Markovian baseline this folder generalises) · Volatility Surfaces & Smiles (the object whose short-end skew drives the need for roughness)
• Related flat notes: SV · 06 Advanced Extensions (the launchpad that flags rough vol as the frontier) · VS · 04 Advanced Dynamics (skew stickiness, SSR) · ES) (fat tails, the daily Student-$t$ one-day return)
• Sub-pages (in-folder): 01 From Zero · 02 The rBergomi Model · 03 fBm & Derivations · 04 Hurst Estimation & Simulation · 05 Failure Modes & Practice · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05