Rough Volatility & Path Signatures

For most of this institute we have treated volatility as something that drifts slowly and politely. Now I want to break that assumption, because the data has been breaking it for years and the market has finally caught up.

Start with a picture you already have in your head. You have seen a realized-volatility series — volatility measured directly from high-frequency returns over short windows. It does not look like a smooth, lazily wandering line. It looks jagged, spiky, almost electric. For a long time we modeled it as if it were smooth, with classical stochastic-volatility models that assume volatility follows an ordinary Brownian-style diffusion. Those models produce paths that, when you zoom in, are far too tame compared with what you actually see. The real series is rougher. That single word is the whole subject of this chapter.

To make "rougher" precise, you need a number that measures the roughness of a path. That number is the Hurst exponent, written as H, and it lives between zero and one. The cleanest way to think about it: take a path and look at how much it typically moves over a small time gap of length delta. For a process with Hurst exponent H, the typical size of that move scales like delta raised to the power H. When H equals one-half, you are looking at ordinary Brownian motion — the increments scale like the square root of time, which is the familiar law you met in your first stochastic-calculus course. When H is larger than one-half, the path is smoother and trends; increments line up. When H is smaller than one-half, the path is rougher, increments fight each other, and the line jitters violently at fine scales.

Here is the empirical fact that launched the field. When Gatheral, Jaisson, and Rosenbaum measured H on log realized-volatility across essentially every liquid asset, they did not get one-half. They got something close to one-tenth. Volatility is not a smooth diffusion at all; it is extraordinarily rough, far rougher even than Brownian motion. This is the core claim of rough volatility: model the log-volatility as a fractional Brownian motion with a tiny Hurst exponent, on the order of zero-point-one, and suddenly the model reproduces the jagged realized series, the steep short-dated implied-volatility smile, and the term structure of at-the-money skew — all of which the smooth models could only fudge.

Why should you care beyond elegance? Because roughness changes what the recent past is worth as evidence. When H is near one-tenth, volatility has long memory and violent local fluctuation at the same time: a burst of activity says a great deal about the next few minutes and surprisingly much about the next few days, but the path is so erratic that any single tick is a weak signal on its own. Reading rough volatility correctly is exactly the institute's throughline — deciding when a number is genuine evidence an asset is about to move, rather than noise dressed up as information.

Now, estimating one number, H, captures the roughness of a path but throws away almost everything about its shape. The order in which moves happened, whether the asset zig-zagged up then down or down then up, the interplay between price and volume — none of that survives in a single exponent. To keep the shape, we need a richer description of the path itself. That is where the signature enters, and with it the theory of rough paths, developed by Terry Lyons.

Build the intuition slowly. Take a path — say, the joint trajectory of price and traded volume over a session. The simplest summary is its total increment: where it ended minus where it started, in each coordinate. That is level one of the signature. But increments forget order. A path up-then-over and a path over-then-up have the same total increment yet are clearly different trades. To tell them apart you need to know how the coordinates moved relative to each other, and the cleanest measure of that is the signed area swept out between the two coordinates — the iterated integral of one coordinate against the other. That signed area is the celebrated Lévy area, and it is the heart of level two of the signature. It is the first quantity that knows the difference between "price led volume" and "volume led price."

The full signature continues this logic forever: it is the infinite collection of iterated integrals of the path against itself, at every level. Level one gives increments and level two gives signed areas — those two have clean geometric meanings — while the higher levels are higher-order iterated integrals that capture progressively finer information about the order and ordering of the moves, without a tidy geometric name. Lyons' theorem is the remarkable payoff — for a sufficiently well-behaved path, the signature determines the path up to trivial reparametrization. Nothing about the shape is lost. And the practical gift is that these iterated integrals are a universal, model-free feature set: any reasonable functional of a path can be approximated by a linear combination of signature terms. You compute a fixed, low-order signature — usually levels one through three — and hand those numbers to whatever learner you like. No need to guess the right hand-crafted indicators; the signature is the principled basis they were all crude approximations of.

The catch is dimension. The number of terms grows like the number of coordinates raised to the level, so a high-order signature on many channels explodes. The discipline of the field is to use the log-signature, which strips the redundancy, and to keep the order low. As always, a feature is only evidence if it survives out of sample.