Statistical Arbitrage and Pairs Trading

Buy the cheap one, sell the expensive one, wait for them to converge. Simple to say, harder to make work — and the math decides whether you have a pair or a trap.

Statistical arbitrage (stat arb) is a family of strategies that profit from temporary mispricings between statistically related assets. The most accessible form for retail traders is pairs trading.

The core idea

Two assets that should move together occasionally diverge. You short the overvalued one and buy the undervalued one, betting the spread reverts to its historical mean.

The trick: the assets must be cointegrated, not just correlated. Correlated assets move together; cointegrated assets have a stationary spread — a relationship that doesn't drift away forever.

Cointegration, not correlation

Two non-stationary price series X and Y are cointegrated if a linear combination is stationary:

Spread = Y − β·X

Where β is the hedge ratio from regressing Y on X. If the spread is stationary (passes the ADF test), the relationship is tradeable: when the spread widens, expect it to revert.

Building a pairs trade

1. Find candidates: economically related assets (e.g., KO and PEP, GLD and GDX, two ADRs of the same firm)
2. Test cointegration: run OLS to get β, then ADF on the spread
3. Compute z-score of the spread:

   Z = (Spread − SMA_n) ÷ σ_n
   

4. Trade rules:
   • Long the spread when Z < −2 (buy Y, short β·X)
   • Short the spread when Z > +2
   • Exit when |Z| < 0.5
5. Size each leg by β so the position is dollar-neutral

Why it can fail

1. Cointegration breaks: the relationship is statistical, not contractual. A merger, regulatory change, or business pivot can break the link permanently
2. Regime change: a spread that was stationary for years can start trending. Stop-loss on the spread is essential
3. Capital efficiency: pairs are dollar-neutral but require margin on both legs, tying up capital
4. Shorting risk: borrow costs, recalls, and short squeezes on the short leg
5. Crowding: popular pairs converge instantly; edge decays as capital chases the same signals

Risk management

• Hard stop on the spread z-score (e.g., exit if |Z| > 4 — the relationship is breaking)
• Time-based exit: if convergence doesn't happen within N days, close the pair
• Volatility-scaled position sizing
• Limit exposure to any single pair (e.g., max 10% of book)

Beyond simple pairs

Stat arb extends to basket trades (long a portfolio, short another), index arbitrage (ETF vs underlying basket), and lead-lag (one asset leads another by a few ticks). All rely on a stationary relationship that will eventually revert — until it doesn't.

Summary

Statistical arbitrage harvests the gap between related assets when they diverge. The math — cointegration, hedge ratios, z-scores — tells you whether you have a real relationship or just two random walks. Build the pair, test the spread's stationarity, size it dollar-neutral, and always plan for the day the relationship breaks. That day will come.