Half-Life (Mean Reversion)

Half-life quantifies the speed of mean reversion by measuring how long it takes for a spread or price deviation to close by half. The concept is borrowed from physics — radioactive decay follows the same mathematical pattern. In quantitative trading, the Ornstein-Uhlenbeck (OU) process describes mean-reverting price series, and the half-life is derived from its mean-reversion rate parameter theta. A half-life of 10 calendar days means that a spread trading 20% above its mean is expected to narrow to 10% above the mean within 10 days. This gives traders a probabilistic timeline for position convergence, which is essential for sizing, risk management, and capital efficiency. Without half-life analysis, pairs traders are operating blind on the most important dimension of their trade.

The standard approach fits an Ornstein-Uhlenbeck process to the spread series and extracts the mean-reversion rate. Mathematically, run an ordinary least squares (OLS) regression of the spread change (delta_S) against the lagged spread (S_t-1): delta_S = a + b * S_t-1 + epsilon. The half-life is then computed as: half_life = -log(2) / b, where b is the regression coefficient on the lagged spread (b must be negative for mean reversion to exist). In Python, this is straightforward with NumPy and pandas. Before estimating half-life, verify the spread is stationary using the Augmented Dickey-Fuller test — if the ADF test cannot reject the unit root hypothesis, half-life estimates are unreliable. Half-life should be recalculated periodically (monthly or quarterly) because cointegration relationships can break down over time.

Half-life values create a practical tradability spectrum for pairs trading strategies. Half-lives below 5 days indicate extremely rapid mean reversion — theoretically attractive but difficult to trade in practice because bid-ask spreads, commissions, and market impact erode returns on positions held only a few days. Half-lives between 5 and 30 days represent the sweet spot for active pairs trading: fast enough to generate multiple trades per year with reasonable capital turnover, slow enough to enter and exit without excessive friction. Half-lives between 30 and 90 days are viable for longer-horizon investors and hedge funds with patient capital but generate fewer annual trade opportunities. Half-lives above 100 days suggest mean reversion so slow that the strategy capital is tied up for extended periods, and the risk of the cointegration relationship breaking down before reversion occurs is too high for most systematic traders.

Half-life directly informs position sizing through its relationship to holding period and profit opportunity. A shorter half-life means faster capital recycling but potentially smaller profit-per-trade; a longer half-life suggests larger expected profit but slower turnover. Practitioners combine half-life with entry z-score to estimate expected holding time and annualized return. For a spread currently 2 standard deviations from its mean with a 15-day half-life, the expected time to reach 1 standard deviation is roughly 15 days. This guides decisions on position size relative to overall portfolio capital — you want enough concentration to generate meaningful returns but not so much that you cannot hold through the natural volatility before convergence occurs. Kelly Criterion variants can incorporate half-life as the effective holding period when calculating optimal position fractions.

While half-life is most commonly discussed in equity pairs trading, the concept extends to any mean-reverting spread across asset classes. In fixed income, the yield spread between investment-grade and high-yield bonds shows mean-reverting behavior with half-lives often in the 20-60 day range. In forex, currency pairs tied to purchasing power parity tend toward very long half-lives of months or years, making them unsuitable for active pairs trading. In crypto markets, mean-reverting relationships exist between correlated tokens (BTC and ETH, or tokens in the same protocol ecosystem), though these relationships tend to be less stable than equity pairs and require more frequent re-estimation. Tradewink applies half-life filtering across both equity and crypto pairs, using tighter half-life windows for crypto given the higher volatility and faster-changing correlations in digital asset markets.