Mean Reversion and Statistical Arbitrage

Trend-following bets that what moved keeps moving; mean reversion bets that what stretched snaps back. Statistical arbitrage is the industrialization of the second idea, and its core concepts — stationarity, half-life, cointegration — appear regularly in quant-research interviews.

Stationarity is the entry ticket

A series is (weakly) stationary if its mean and variance don't drift over time. Prices are not stationary — they random-walk, and the variance of a random walk grows linearly with time, so "AAPL is below its average price" is not a signal. Mean reversion is only tradable in quantities engineered to be stationary: spreads between related assets, futures baskets vs. their fair value, implied minus realized volatility.

The classic test: in the regression $\Delta y_t = \alpha + \beta\, y_{t-1} + \epsilon_t$, a significantly negative $\beta$ means high levels predict declines — reversion. (Formally the Dickey–Fuller test; the interview version is just "does the change depend negatively on the level?")

The Ornstein–Uhlenbeck process

The canonical model of a mean-reverting quantity:

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

A random walk with a spring: drift pulls toward $\mu$ at speed $\kappa$. The two facts to carry:

Half-life of a deviation. A displacement decays like $e^{-\kappa t}$, so the expected time to close half the gap is

$$t_{1/2} = \frac{\ln 2}{\kappa}.$$

The half-life is your holding period, and it disciplines everything: a spread with a 3-day half-life is tradable; one with a 3-year half-life ties up capital and mostly exposes you to regime change before convergence.

Stationary variance. Long-run, $X$ settles into a distribution with variance $\sigma^2 / 2\kappa$ — the balance point between noise injection ($\sigma$) and the spring ($\kappa$). Entry thresholds ("trade at 2 standard deviations") mean standard deviations of this distribution.

Pairs trading and cointegration

The classic implementation: find two stocks that move together (say two large banks), trade the spread $S_t = A_t - \gamma B_t$ — short the rich one, long the cheap one, exit on convergence. The position is roughly market-neutral; what remains is the relative-value bet.

The load-bearing concept is cointegration: two non-stationary price series whose particular linear combination is stationary. This is much stronger than correlation — two random walks can be highly correlated in returns while drifting arbitrarily far apart in levels (correlated ≠ tethered). Cointegration says a spring connects the levels. Interview one-liner: "correlation is about co-movement of returns; cointegration is about a stationary combination of prices — pairs trading needs the second, and testing it is the Engle–Granger two-step: regress A on B, then test the residual for stationarity."

How it fails — and why that's the real lesson

Mean reversion has a brutal failure signature: it works until the relationship breaks, and the break looks exactly like the best entry you've ever seen. The spread hits 3 standard deviations (you size up), then 5 (the model says "once a century"), then the truth emerges — a merger, a fraud, a regime change — and there is no spring. The stationary model was an approximation with an expiry date.

Structural defenses, not heroics: hard stop-losses expressed in spread units, position limits per pair, diversification across many small uncorrelated spreads (the stat-arb portfolio is the edge — any single pair is weak), and treating deviation beyond historical extremes as evidence against the model rather than opportunity. Bayesian humility, encoded as risk limits.

The interview version

"You find a spread with strong mean reversion in-sample, Sharpe 3 backtest. Concerns?" Expected answer, in order: multiple-testing/selection bias (thousands of pairs screened — some look great by chance; demand out-of-sample confirmation), transaction costs at the implied turnover (half-life sets trade frequency; short half-life = many crossings of the spread), capacity, and the regime-break tail risk above. Naming the statistical failure (data mining) before the market failure (breaks) reads as research maturity.