Rough Bergomi Model

Trace a coastline on a map and it looks smooth from an airplane; zoom in on the same coastline from a boat and it becomes a jagged, irregular mess of inlets and rocks — and zooming in further just reveals more jaggedness at every scale, not a smooth curve underneath. Ordinary Brownian motion, the standard driver in most volatility models, is not like a coastline — it's "rough" at only one specific, moderate degree, the same at every zoom level, but real historical volatility, when researchers actually measured it at high frequency, turned out to be rougher than that: jumpier at short horizons than standard models predict.

What "rough" means, precisely

Roughness here is measured by the Hurst exponent $H$, a number between 0 and 1 that describes how a path's typical size of movement scales with the length of the time window you look at it over.

$$\mathbb{E}\big[(\log \sigma_{t+h} - \log \sigma_t)^2\big] \propto h^{2H}$$

In words: look at how much log-volatility typically changes over a short window of length $h$, squared and averaged; that quantity scales with $h$ raised to the power $2H$. Standard Brownian motion has $H = 0.5$. Empirically, fitting this relationship to real high-frequency realized volatility data across many assets consistently finds $H$ around 0.1, far below 0.5 — meaning volatility's short-term changes are larger and more frequent, relative to how they scale with the observation window, than a standard Brownian driver would produce. The rough Bergomi model plugs a fractional Brownian motion with this low $H$ into the forward variance dynamics in place of standard Brownian motion.

Worked example 1 — how scaling with $H$ differs numerically

Suppose the typical size of a log-vol move over a 1-day window is 0.05 (in log terms). Under standard Brownian scaling ($H = 0.5$), the typical move over a 4-day window scales as $4^{0.5} = 2$ times larger, so about 0.10. Under rough scaling with $H = 0.1$, the same 4-day window scales as $4^{0.1} \approx 1.15$ times larger, so only about 0.0575 — barely bigger than the 1-day move. In words: a rough process's volatility-of-volatility barely grows as you widen the observation window, which is another way of saying almost all the "action" is concentrated at the shortest time scales — exactly the jagged-coastline picture, where zooming out doesn't smooth things out the way it would for an ordinary random walk.

Worked example 2 — why this fixes a known smile-fitting problem

Classical stochastic volatility models (Heston, standard Bergomi) struggle to simultaneously fit the volatility smile's steep short-dated skew and its shallower long-dated skew with the same parameters — short-dated skew in real markets is empirically much steeper (falls off faster as $T \to 0$) than these models predict unless forced with an unrealistic vol-of-vol parameter. The rough Bergomi model, using $H \approx 0.1$, produces a short-dated at-the-money skew that behaves like $T^{H - 1/2} = T^{-0.4}$, which blows up much faster as maturity shrinks than the $T^{-0.5}$-ish behavior classical models are stuck with — matching the steep, empirically observed short-term skew shape far more naturally, without needing an unrealistically large vol-of-vol input.

What this means in practice

Rough volatility models are increasingly used on volatility trading desks specifically because they fit the whole implied volatility surface — especially the short-dated skew — with fewer, more stable parameters than classical models, and those parameters (roughly, $H$, vol-of-vol, and correlation) tend to stay steadier over time, reducing recalibration noise. The tradeoff is computational: because the driver isn't a standard Markov process, pricing and simulation are more involved than in Heston or standard Bergomi.

Practice

1. If a log-vol move over a 1-day window is typically 0.04, what is the typical move over a 9-day window under standard Brownian scaling ($H=0.5$) versus rough scaling with $H=0.1$?
2. Why does a lower Hurst exponent make the short-dated at-the-money skew steeper as maturity shrinks toward zero?