SpreadHurst

The Hurst exponent of the spread a − b over a rolling window — a regime detector for pairs trading.

Quick reference

Field	Value
Family	Price Statistics
Input type	(f64, f64) (a pair of prices a, b)
Output type	f64 (Hurst exponent H)
Output range	~[0, 1]; 0.5 is the random-walk midpoint
Default parameters	period is required (>= 8)
Warmup period	period
Interpretation	H < 0.5 mean-reverting; H ≈ 0.5 random walk; H > 0.5 trending.

Formula

Each update forms the spread sₜ = aₜ − bₜ and estimates the Hurst exponent H from how the variance of τ-lagged differences grows with the lag τ:

V(τ) = mean_t (s_{t+τ} − s_t)²   ∝   τ^(2H)
H    = slope of log V(τ) on log τ, divided by two

H classifies the spread's regime:

• H < 0.5 — mean-reverting (anti-persistent): the spread snaps back, the regime pairs traders want.
• H ≈ 0.5 — a random walk: no exploitable structure.
• H > 0.5 — trending (persistent): the spread keeps diverging.

The fit uses lags 1..=period/4 (at least two). When the spread is flat — every lagged difference is zero, so the log-regression has too few valid points — the indicator returns the neutral midpoint 0.5.

Source: crates/wickra-core/src/indicators/spread_hurst.rs.

Parameters

Name	Type	Default	Valid range	Source	Description
period	usize	none	>= 8	spread_hurst.rs:62	Look-back window of spreads. < 8 errors with Error::InvalidPeriod (the fit needs at least two lags).

Inputs / Outputs

From crates/wickra-core/src/indicators/spread_hurst.rs:

use wickra::{Indicator, SpreadHurst};
// SpreadHurst: Input = (f64, f64), Output = f64
const _: fn(&mut SpreadHurst, (f64, f64)) -> Option<f64> =
    <SpreadHurst as Indicator>::update;

Python streams as update(a, b) -> float | None and batches over two equal-length arrays. Node streams as update(a, b) and batches over a[], b[].

Warmup

warmup_period() == period. The unit test accessors_and_metadata pins warmup_period() == 40 for period = 40; warmup_returns_none pins the first Some at the period-th pair.

Edge cases

• Oscillating spread. A mean-reverting (anti-persistent) spread has H < 0.5; pinned by oscillating_spread_is_anti_persistent.
• Trending spread. A linearly diverging spread has H ≈ 1; pinned by linear_trend_spread_is_persistent.
• Flat spread. A constant spread yields the neutral midpoint 0.5; pinned by flat_spread_returns_midpoint.
• Reset. reset() clears the spread window.

Examples

Rust

use wickra::{BatchExt, Indicator, SpreadHurst};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut h = SpreadHurst::new(60)?;
    // Mixed-frequency oscillating spread -> anti-persistent (H < 0.5).
    let pairs: Vec<(f64, f64)> = (0..200)
        .map(|t| {
            let b = 100.0 + f64::from(t);
            let tt = f64::from(t);
            (b + 3.0 * (tt * 0.8).sin() + 1.5 * (tt * 0.17).sin(), b)
        })
        .collect();
    let last = h.batch(&pairs).into_iter().flatten().last().unwrap();
    println!("{last:.6}");
    Ok(())
}

Output:

0.114399

Python

import numpy as np
import wickra as ta

t = np.arange(200)
b = 100.0 + t
a = b + 3.0 * np.sin(t * 0.8) + 1.5 * np.sin(t * 0.17)
print(round(ta.SpreadHurst(60).batch(a, b)[-1], 6))

Output:

0.114399

Node

const ta = require('wickra');
const b = Array.from({ length: 200 }, (_, t) => 100 + t);
const a = b.map((bv, t) => bv + 3 * Math.sin(t * 0.8) + 1.5 * Math.sin(t * 0.17));
console.log(new ta.SpreadHurst(60).batch(a, b).at(-1).toFixed(6));

Output:

0.114399

Interpretation

SpreadHurst answers the prerequisite question of pairs trading: is this spread the kind that reverts? An H well below 0.5 is a green light for a mean-reversion entry; an H near or above 0.5 says the spread wanders or trends and a fade will bleed. Use it as a regime gate ahead of a timing signal: only arm PairSpreadZScore or SpreadBollingerBands when H confirms anti-persistence, and size the holding period with OuHalfLife.

Common pitfalls

• Short windows are noisy. The log–log slope is fit from only period/4 lags; small period gives a high-variance estimate. Prefer windows of several dozen bars or more.
• 0.5 is the null, not a value. A flat or degenerate spread returns exactly 0.5. Treat values right at the midpoint with suspicion.

References

Hurst, H. E. (1951), the rescaled-range analysis; the variance-of-differences estimator follows the structure-function approach to H.

See also

• Indicator-OuHalfLife — reversion time scale once H < 0.5.
• Indicator-VarianceRatio — a complementary mean-reversion test.
• Indicators-Overview — the full taxonomy.