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Hurst exponent

Also called: Hurst coefficient, H exponent, rescaled range exponent

The Hurst exponent measures long-range memory in a time series: about 0.5 for a random walk, above for persistent trends, below for mean reversion.

Definition

The Hurst exponent H is a number between 0 and 1 that describes how the typical size of a series' fluctuations grows with the length of the window you look at. It was introduced by the hydrologist Harold Edwin Hurst, studying Nile river levels, and later formalized through fractional Brownian motion.

For returns that behave like a random walk, H is close to 0.5. Values above 0.5 indicate persistence (moves tend to be followed by moves in the same direction); values below 0.5 indicate anti-persistence or mean reversion.

Intuition

If price increments are independent, the range a price covers grows with the square root of time. If increments are positively correlated over long horizons, the range grows faster, because trends compound; if they are negatively correlated, it grows more slowly, because moves tend to reverse.

Traders use the exponent as a regime descriptor: a market drifting into a persistent phase may suit trend-following logic, while an anti-persistent phase may suit range or reversion logic.

How it's calculated

Classical rescaled-range (R/S) analysis: split the return series into windows of length n. In each window, build the cumulative sum of demeaned returns, take its range R(n), and divide by the window's standard deviation S(n). Then fit

E[ R(n) / S(n) ] ∝ nH ⇒ log E[R/S] = c + H · log n

The slope of the log–log regression across window sizes is the estimate of H.

Detrended fluctuation analysis (DFA) is a more robust alternative: integrate the series, remove a local linear trend in each window, measure the root-mean-square fluctuation F(n), and estimate H from F(n) ∝ nH. DFA is less sensitive to slow trends that would otherwise inflate R/S estimates.

How to read it

Treat values within roughly ±0.05 of 0.5 on short samples as indistinguishable from a random walk; the sampling error of Hurst estimates is large. A rolling estimate that moves persistently above 0.55 or below 0.45 is more informative than a single reading.

Apply the exponent to returns or increments, not to price levels, unless the method explicitly integrates the series first (as R/S and DFA do).

Common mistakes

  • Running R/S on short windows, where small-sample bias pushes estimates above 0.5. Corrections such as the Anis–Lloyd adjustment exist for this reason.
  • Mistaking short-memory autocorrelation or volatility clustering for long memory.
  • Using price levels where the method expects increments, or the reverse.
  • Reading a single estimate without any confidence interval.

In Saqrid

SV's library of 100 built-in proprietary formulas includes Hurst tools alongside momentum, entropy, expected-move and probability-cone formulas.

Open SV.

  • Shannon entropy in marketsShannon entropy measures how uncertain a distribution of outcomes is; applied to returns or price patterns, it gauges how disordered a market is.
  • Market regime detectionMarket regime detection classifies periods into states such as calm, trending or stressed, so analysis and risk can adapt to conditions.
  • Statistical indicatorsA statistical indicator estimates a defined property of returns, such as volatility, memory or a probability, rather than applying a fixed chart rule.
  • Log returnsA log return is the natural log of the price ratio between two dates; log returns add across time, which makes them the default unit in quant work.

Go deeper

Saqrid Institute lesson: Regime Detection: BOCPD, Hurst & Bubble Tests. Lesson text is open to read without an account.

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