Rough Volatility

Classical stochastic volatility models have one embarrassment they cannot argue their way out of. Ask Heston, or SABR, or any model where volatility is driven by an ordinary Brownian motion, what the at-the-money skew should look like for options expiring next week. They answer: about the same as for options expiring next year. The market answers: several times steeper. Traders have papered over this for decades by recalibrating the model separately at each maturity, which works but means the "model" is really a different model at every expiry. Rough volatility fixes it by changing one thing, and the thing it changes is not a parameter of the volatility process. It is the texture of the volatility path itself.

Coastlines and rolling hills

Walk along a rolling hill and zoom in. The closer you look, the smoother the ground gets: over a metre it is nearly flat, over a centimetre it is flat. Now look at a coastline on a map, then look at a photograph of one bay, then at one rock pool. It never smooths out. At every zoom level you see the same kind of jaggedness. That property has a number attached to it, and it is the reason coastlines have no well-defined length.

A Brownian motion is somewhere between the two: zoom in and a wiggle over a time gap $\Delta$ has typical size proportional to $\sqrt{\Delta}$. Halve the interval and the typical move shrinks by about 30 percent. Volatility, when you actually measure it, does not behave like that. Its path over an hour is nearly as large as its path over a day, and its path over a day is nearly as large as its path over a month. It is much closer to the coastline than to the hill, and "rough volatility" is nothing more elaborate than taking that measurement seriously.

The roughness parameter

Give roughness a name. For a process $X$, the Hurst exponent $H$ is the number that makes

$$\mathbb{E}\,\bigl| X_{t+\Delta} - X_t \bigr| \;\propto\; \Delta^{H} .$$

In plain English: if you look at the process over a gap $\Delta$, the typical size of its move grows like $\Delta$ raised to the power $H$. Here $X_t$ is the quantity you are watching — for volatility work it is the logarithm of volatility — $\Delta$ is the time gap, and $H$ is a number between 0 and 1 that measures how quickly moves grow as you widen the window.

Three cases to hold on to. $H = 0.5$ is ordinary Brownian motion: widen the window by a factor of 100 and moves get 10 times bigger. $H$ close to 1 is a very smooth, trending path: widen by 100 and moves get almost 100 times bigger. $H$ close to 0 is the coastline: widening the window barely changes anything, because almost all of the movement is already happening at the finest scale.

Measured on realized volatility from essentially every liquid asset ever tested, $H$ comes out around 0.1.

Substitute that into any model and something breaks in a useful direction: the process is so jagged that it is not differentiable in any ordinary sense and is no longer a Markov process, so the standard PDE machinery does not apply. That is the price. What you get back is the skew.

Worked example 1: measuring H from realized volatility

The estimate is a one-line calculation, which is part of why the finding held up so fast.

Take a series of daily realized volatilities, take logs, and compute the average absolute change over different lags. Suppose you find:

lag	average absolute change in log volatility
1 day	0.2000
10 days	0.2518
100 days	0.3170

Now read off the exponent. Between lag 1 and lag 100 the gap widened by a factor of 100 while the typical move grew by a factor of $0.3170 / 0.2000 = 1.585$. So

$$100^{H} = 1.585 \quad \Longrightarrow \quad H = \frac{\log_{10} 1.585}{\log_{10} 100} = \frac{0.2000}{2} = 0.10 .$$

Cross-check on the shorter leg: from lag 1 to lag 10 the gap widened tenfold and moves grew by $0.2518 / 0.2000 = 1.259$, so $H = \log_{10}(1.259)/1 = 0.100$. Same answer. On log-log paper the three points sit on a straight line of slope 0.1.

Compare that with what Brownian motion predicts. At $H = 0.5$, a 100-day change should be $\sqrt{100} = 10$ times a one-day change, so 2.00, not 0.317. The data is off by a factor of six. That is not a subtlety at the edge of statistical significance; it is the difference between a hill and a coastline.

Read it the other way and it says something you can feel: volatility jumps around a great deal day to day, and almost none of that jumping accumulates into long-horizon drift. Almost everything reverses.

Worked example 2: why the skew explodes

The commercially important consequence. Define the at-the-money skew $\psi(T)$ as how fast implied volatility changes as you move the strike away from the money, at expiry $T$, measured per unit of log-moneyness. Rough volatility predicts

$$\psi(T) \;\propto\; T^{\,H - 1/2} .$$

In plain English: the skew grows as expiry shrinks, at a rate set entirely by the roughness. With $H = 0.1$ the exponent is $0.1 - 0.5 = -0.4$, so the skew scales like $T^{-0.4}$.

Anchor it on a number you can observe. Suppose the one-year at-the-money skew is 4 volatility points per unit of log-moneyness, i.e. $\psi(1) = 0.04$.

One month, $T = 1/12$: $\psi = 0.04 \times 12^{0.4}$. Now $12^{0.4} = e^{0.4 \ln 12} = e^{0.4 \times 2.4849} = e^{0.9940} = 2.702$. So $\psi = 0.04 \times 2.702 = 0.1081$, or 10.8 points.

One week, $T = 1/52$: $52^{0.4} = e^{0.4 \times 3.9512} = e^{1.5805} = 4.857$. So $\psi = 0.04 \times 4.857 = 0.1943$, or 19.4 points.

So the model says the one-week skew should be nearly five times the one-year skew. Now the comparison. In Heston, SABR and every other classical stochastic volatility model, the at-the-money skew tends to a finite constant as expiry goes to zero: the prediction is roughly 4 points at one week too. Desks do observe skews of that order at very short dates, which is why they were forced to recalibrate vol-of-vol upward for every short expiry — a fudge that rough volatility replaces with a parameter that was measured, not fitted.

Two sanity checks on the exponent. If volatility were Brownian, $H = 0.5$ and the exponent would be zero — flat skew across maturities, exactly the classical result, recovered as a special case. And if volatility were maximally rough, $H = 0$, the exponent would be $-0.5$ and the one-month skew would be $0.04 \times \sqrt{12} = 0.139$, or 13.9 points. Real markets sit between, closer to the rough end.

Seeing it, and not seeing it

The explorer below runs a mean-reverting path, which is how classical models picture volatility. Turn the volatility up and resample a few times. The path is busy, but look at any short stretch and it is locally smooth — over a small window it has a direction. Rough volatility is what you get when that local smoothness never appears at any zoom level.

What this means in practice

The direct product of the research is the rough Bergomi model, in which the log of instantaneous variance is driven by a fractional Brownian motion with $H \approx 0.1$. It fits the entire equity index surface, all maturities at once, with about three parameters — roughness $H$, volatility-of-volatility $\eta$, and spot-volatility correlation $\rho$ — where classical models need a fresh set per maturity. That parsimony is the practical case for it: fewer parameters that move less means more stable Greeks and less recalibration noise in the risk report.

The cost is computational and structural. Fractional Brownian motion is not Markov, so there is no low-dimensional state to run a PDE on and no easy tree. Pricing is Monte Carlo with a correlated-noise construction, American exercise is genuinely awkward, and simulation is an order of magnitude slower than Heston. Rough Heston, which keeps an affine structure through a fractional Riccati equation, exists partly to claw some of that back.

Where it changes decisions: anything priced off very short-dated skew. Weekly options market making, variance swap replication near expiry, gap-risk on barriers close to knock-out, and the pricing of very short-dated digitals. Over a year to expiry the difference between $H = 0.1$ and a well-recalibrated Heston is small.

Practice

1. A process has $H = 0.25$. A one-day move is typically 0.4. How large is a typical 16-day move?
2. The one-year at-the-money skew is 3 volatility points and $H = 0.1$. What does rough volatility predict for the three-month skew?
3. You estimate $H = 0.08$ from five-minute realized volatility and $H = 0.14$ from daily realized volatility on the same asset. Which do you trust more, and why?

Answers. (1) $0.4 \times 16^{0.25} = 0.4 \times 2 = 0.8$. (2) $T = 0.25$, so $\psi = 0.03 \times 0.25^{-0.4} = 0.03 \times 4^{0.4} = 0.03 \times 1.741 = 0.052$, about 5.2 points. (3) The daily estimate. Five-minute realized volatility is contaminated by microstructure noise, which adds spurious high-frequency jitter and pulls the estimated roughness down.

Key terms

• Hurst exponent $H$ — the power law linking time gap to typical move size; 0.5 is Brownian, lower is rougher.
• Fractional Brownian motion — the Gaussian process with a tunable $H$; its increments are correlated, so it is not a Markov process.
• At-the-money skew $\psi(T)$ — the slope of implied volatility in log-moneyness at the money, for expiry $T$.
• Rough Bergomi — the standard rough volatility model: forward variance driven by fractional noise.
• Power-law skew — the $T^{H-1/2}$ term structure, which is the model's headline empirical success.