The Hurst Exponent

Stare at a chart and you can usually feel whether it's "trendy" or "choppy," but a feeling isn't a filter you can run on 5,000 tickers overnight. The Hurst exponent turns that intuition into a single number between 0 and 1, estimated purely from how a series' price range grows as you widen the window you measure it over.

Reading the exponent

The idea, called rescaled range analysis, is to split a price series into windows of length $n$, measure the range each window covers (relative to its own volatility), and see how that typical range scales as $n$ grows:

$$E\left[\frac{R_n}{S_n}\right] \propto n^{H}$$

In words: the expected rescaled range over a window of length $n$ grows proportionally to $n$ raised to the power $H$. Fit that relationship across several window lengths and the slope, on a log-log plot, is $H$. A value near 0.5 means the series has no memory — knowing the last move tells you nothing about the next. A value like 0.7 means moves tend to be followed by more moves in the same direction. A value like 0.3 means moves tend to be reversed.

Drag the exponent in the plot above and watch the curve bend: exponents below 1 bow upward and flatten (the mean-reverting shape, range growing slower than proportionally), and above 1 bow the other way (the trending shape). The Hurst exponent is exactly this exponent, just estimated on a security's rescaled price range instead of an abstract $x$.

Worked example

A stat-arb desk computes rescaled range over windows of 10, 20, 40, and 80 days for a commodity spread and gets $R/S$ values of roughly 3.2, 4.4, 6.1, and 8.4. Fitting $\log(R/S)$ against $\log(n)$ gives a slope of about 0.42. Since $0.42 < 0.5$, the spread shows mild mean-reverting behavior — consistent with, though not proof of, a cointegrated relationship worth building a pairs strategy around. The same exercise on a trending commodity future might give a slope near 0.58, arguing against mean-reversion trades on that instrument.

What this means in practice

The Hurst exponent is a screening tool, not a trading signal by itself: a desk with thousands of candidate pairs or spreads can rank them by how far H sits below 0.5 to prioritize which ones are worth the deeper cointegration and stability testing. It says nothing about magnitude, speed, or why a series mean-reverts, only that historically it has.