Mean reversion and the Ornstein–Uhlenbeck process

Mean reversion is the opposite bet to momentum: when something has moved far from “normal”, expect it to come back. It works on the right objects — spreads, ratios, short-horizon returns — and fails dangerously on the wrong ones.

The Ornstein–Uhlenbeck process

The continuous-time model of a mean-reverting quantity $X_t$ is

dX_t = \theta(\mu - X_t)\,dt + \sigma\,dW_t

$\mu$ is the long-run mean the process is pulled towards,
$\theta$ is the speed of reversion,
$\sigma$ is the noise.

The further $X$ is from $\mu$ , the stronger the pull. Its stationary distribution is normal with variance $\sigma^2 / (2\theta)$ , and the half-life of a deviation is

h = \frac{\ln 2}{\theta}

In discrete time this is the AR(1) process from stationarity, with $\phi = e^{-\theta\,\Delta t}$ . You estimate $\theta$ by regressing $X_t - X_{t-1}$ on $X_{t-1}$ .

The z-score trade

Standardise the deviation:

z_t = \frac{X_t - \hat\mu_t}{\hat\sigma_t}

using a rolling mean and standard deviation. A classic rule: short when $z > 2$ , long when $z < -2$ , exit when $z$ crosses 0. Bollinger Bands are this exact rule applied to price, and RSI-oversold entries are a cousin of it — which is why they dominate the mean-reversion section of the vault.

The lookback should match the half-life

A rolling window much shorter than the half-life chases noise; much longer, and the mean it uses is stale. Estimating the half-life first and setting the window around it is one of the simplest ways to make a mean-reversion system less arbitrary.

Where mean reversion is real

Spreads between related assets — the basis of pairs trading.
Very short horizons — bid–ask bounce and liquidity provision create reversal over minutes to a day.
Volatility itself — implied volatility reverts much more reliably than price.
Funding rates and basis in crypto perpetuals, pulled back by arbitrageurs (see carry).

Single asset prices over medium horizons are mostly not mean-reverting; they behave closer to random walks with momentum.

The risks

Mean reversion has the negative skew that momentum avoids: many small wins and occasional large losses, when a “deviation” turns out to be a regime change. The spread that was 2σ cheap becomes 5σ cheap because one company is going bankrupt, or a stablecoin is losing its peg.

Controls every mean-reversion system needs

A stop in z-score or price, even though it fights the logic of the trade.
A time stop: if the trade hasn’t reverted within two or three half-lives, the model is probably wrong.
A regime filter: stand aside when volatility spikes or the relationship’s stationarity test fails.
Never average down without a hard limit. That is how mean reversion becomes martingale.

Mean reversion and the Ornstein–Uhlenbeck process

The Ornstein–Uhlenbeck process

The z-score trade

Where mean reversion is real

The risks

Put it to work

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