Half-life is how long a pair stays interesting

If you trade pairs and you do not know the half-life of your spread, you do not know when to exit.

Half-life is one of the most useful numbers in pairs trading and one of the least discussed. It tells you the expected time for a shocked spread to revert halfway back to equilibrium. If the half-life of your BTC/ETH spread is twelve hours and you are holding a position for four days, the trade is no longer about the mean-reversion you entered on.

This post explains what half-life is, how to compute it from your own data and how to use it to set holding-period rules.

A mean-reverting spread is often modelled as an Ornstein-Uhlenbeck process:

dZ(t) = θ × (μ − Z(t)) × dt + σ × dW(t)

In plain English: the spread moves around a long-run mean μ. The farther it gets from that mean, the harder it is pulled back. The strength of the pull is θ. Random noise σ keeps the series rough rather than perfectly smooth.

The single number that determines how fast the spread reverts is θ. Large θ means fast reversion. Small θ means slow reversion. From θ you compute the half-life:

τ ≈ ln(2) / θ

After one half-life the shock is half gone. After two half-lives it is three-quarters gone. The decay is geometric.

You probably do not have continuous-time data. You have bars. Fit a discrete-time AR(1) on the empirical spread:

ΔZ(t) = κ + φ × Z(t−1) + η(t)

The discrete-time half-life is:

τ = −ln(2) / ln(1 + φ)

One AR(1) fit gives you a number in the same units as your bars. If you use 5-minute bars and get 24, the spread reverts halfway in two hours. If φ is close to zero, the spread is close to a random walk and the half-life becomes enormous. That is a sign the pair is not actually tradeable.

The standard advice from the literature is that holding periods should be on the order of one to several half-lives. A common practical range is between two and four half-lives.

If the half-life is 12 hours and you use a multiplier of 3, you close any position still open after 36 hours regardless of where the z-score sits. This keeps capital from being trapped in trades where the assumed mean-reversion never materialised.

Half-life timeouts pair naturally with z-score thresholds and with optimal-stopping rules. Under the OU model, even the theoretically optimal entry threshold depends on how fast the spread reverts and how much it costs to trade in and out.

Equity pair half-lives are often days to weeks. Crypto pair half-lives are shorter: hours to days for major pairs and sometimes far shorter for active altcoin pairs. That means holding-period rules borrowed from equity papers do not transfer directly.

It also means the half-life itself is non-stationary. The number fitted last week may not be the number that applies this week. A platform that keeps a stale half-life in its rules is trading with a stale time horizon.

If you read one number off your spread, read the half-life. It tells you how long the trade has before it stops being the trade you thought you were taking.

The full formal treatment is in the technical guide. The implementation paper covers how half-life pairs with optimal stopping and risk overlays.

Read the technical guide →