Statistical arbitrage

Pairs Trading

Learn statistical pairs trading with price spreads, regression, cointegration tests, z-scores and half-life estimates, including costs and structural-break risks.

Mechanics & mathematicsHypothetical examples

What Is It?

Pairs trading is statistical arbitrage: it trades a historical relationship between two assets in the expectation that an unusual spread will revert. It is not a matched-payoff arbitrage, and neither cointegration nor a large z-score guarantees convergence or profit.

Correlation measures co-movement, often of returns. Cointegration asks whether nonstationary price series admit a stationary linear combination. High return correlation alone does not establish a stable price spread. Economic linkage, data adjustments and sample selection matter alongside test statistics.

How It Works

A price-level regression models asset A as an intercept plus β times asset B plus a residual. For β > 0, a high residual can motivate shorting one A share and buying β B shares; a low residual reverses those legs. This is a hedge for the modeled price spread, not automatically a dollar-neutral or beta-neutral book.

A log-price regression has a different interpretation: β is an elasticity. Approximate log-return hedging uses β units of B notional per unit of A notional, so the number of B shares depends on both prices. Reusing β as a share count from a log regression is a sizing error.

Normalize the residual using a trailing mean and standard deviation, with parameters fitted using information available before the decision. An illustrative rule might enter above |z| = 2, exit near zero and enforce a separate loss or time limit. Threshold selection and repeated pair testing can overfit; use held-out and walk-forward evaluation.

Mathematical Formula

P_{A,t}=\alpha+\beta P_{B,t}+\varepsilon_t,\qquad z_t=\frac{\varepsilon_t-\mu_t}{\sigma_t}

For a price-level model, residuals are in A-price units. μ and σ are estimated from an explicitly chosen trailing window; σ must be positive. The intercept matters for fair-value projection but is not a separate traded share.

\Delta\varepsilon_t=a+b\varepsilon_{t-1}+u_t,\qquad h=\frac{\ln(0.5)}{\ln(1+b)}

For a discrete AR(1) approximation, the exact half-life in sampling intervals requires 0 < 1+b < 1. The small-|b| approximation is −ln(2)/b. A fitted non-mean-reverting or oscillatory coefficient does not justify this formula unchanged.

\Pi_{\mathrm{gross}}=n_A(P_{A,0}-P_{A,1})+n_B(P_{B,1}-P_{B,0})

This cash P&L is for short A and long B with fixed positive share counts. Deduct both legs' round-trip trading costs, financing, stock-borrow costs and short dividend compensation. Account separately for distributions received.

Worked Example

Illustrative prices · not market data

Use a hypothetical price-level model A = 10 + 2B + residual. At entry A = 115 and B = 50. The trailing residual mean is zero and its standard deviation is 2.

The entry residual is 115 − 10 − 2 × 50 = 5, giving z = 2.5. An illustrative high-spread trade shorts 100 A shares and buys 200 B shares.
Suppose A later becomes 112 and B becomes 51; the residual is now zero. The A short gains 100 × (115 − 112) = 300 and the B long gains 200 × (51 − 50) = 200.
Gross P&L is 500. With aggregate round-trip, borrow and financing costs of 80, net P&L is 420.
The initial short notional is 11,500 while the long notional is 10,000. A β hedge is visibly not dollar-neutral. If b = −0.10 in a separate AR(1) fit, the exact estimated half-life is about 6.58 sampling intervals.

This favorable outcome is conditional, not expected guaranteed income. A spread that widens or breaks structurally can lose on both the signal and the hedge.

Execution Considerations

Align trading calendars, splits, dividends and currencies. Test nonstationarity and residual stationarity using a suitable Engle–Granger procedure and residual-based critical values, rather than interpreting ordinary ADF p-values as exact cointegration significance. Account for multiple comparisons and parameter uncertainty.

The pairs analytics use price-level regression, a simplified approximate ADF screen and an approximate half-life. Treat these outputs as diagnostics. Check simulated-data flags before interpreting signals. Re-estimate on a defined schedule, reserve margin, confirm stock borrow and apply position, loss and holding-time limits.

Risks & Limitations

Structural breaks, mergers, business changes and crowded positioning can stop mean reversion permanently. A historically stationary residual is not a contractual convergence mechanism.

Borrow recalls, dividend payments, liquidity gaps, beta exposure and changing hedge ratios undermine neutrality. Frequent rebalancing can consume the statistical edge. Half-life estimates describe a fitted model, not a deadline for profitable exit.

Frequently Asked Questions

Is high correlation enough to trade a pair?

No. Correlated returns can coexist with a price spread that drifts indefinitely. Check the economic hypothesis, spread definition, stationarity and out-of-sample behavior.

Does β make a pair market-neutral?

Not automatically. Price-level share hedging, dollar neutrality and market-beta neutrality are distinct objectives and generally require different position sizes.

Does a z-score of 2 imply a known win probability?

No. Financial residuals need not be independent, normally distributed or stable. A z-score measures standardized distance under the chosen window, not a calibrated profit probability.

Explore the Existing Tool

Analytics available · simulated fallback

The existing pairs tool calculates historical statistics from Yahoo price series and uses clearly flagged simulated series when that source fails. Its simplified ADF p-values and half-life estimates are screening approximations, not proof of cointegration or trade profitability.

Open Pairs Trading Analytics

References & Further Reading

Background sources for the concepts above. Apply the assumptions stated in this guide to each formula.

Educational material only; not investment advice or an offer to trade. Examples are hypothetical. Trading involves execution, financing and capital-loss risks; no return is guaranteed.