1.1.3 Pairs Selection & Hedging

Basic Prerequisites: 02 · Cointegration & the Spread.

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1. Intuition & Practical Objective

A trading rule is only as good as the pair it is applied to. This page answers the two questions that come before any signal: which two assets? and in what ratio? There are two schools:

• Distance method (Gatev, Goetzmann & Rouwenhorst, 2006) - a nonparametric screen. Normalise each price to a cumulative-return index, compute the sum of squared deviations between every pair, and trade the closest pairs. It assumes $y$ and $x$ move together in levels (implicitly $\beta =1$ on the normalised index). Cheap, robust, but it ignores the statistical structure of the spread.
• Cointegration method (Vidyamurthy 2004; Avellaneda & Lee 2010) - estimate the equilibrium relationship formally, test the residual for stationarity, and trade the residual of a regression (or of a factor decomposition). More sound econometrically, but more parameters to over-fit.

The hedge ratio is the bridge: the distance method’s “buy one, sell one” is the special case $\beta =1$ of the regression hedge. Getting $\beta$ right is what makes the book market-neutral - zero exposure to the systematic factors.

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2. Mathematical Ground Truth & Derivations

2.1 The distance method (GGR 2006, §2.1)

Let $P_t^i$ be the normalised cumulative total-return index of stock $i$ over the formation window (start $=1$). For every candidate pair $(i,j)$,

$$D_{ij}=\sum _{t=1}^M(P_t^i-P_t^j{)}^2.$$

Rank all pairs by $D_{ij}$ and trade the top $n$ (GGR study the top 5 and top 20, plus pairs 101–120 as a control). Matching in normalised price space is equivalent to assuming a cointegrating vector with two nonzero coordinates and unit scale - “the sum or difference of scaled prices will be reverting to zero” (GGR §1.4). The danger flagged by GGR itself: normalisation is not a test, and spuriously correlated prices will pass.

2.2 The regression hedge ratio

If we accept the cointegrating model $y_t=\mu +\beta x_t+z_t$, the OLS estimate is

$$\hat{\beta }=\frac{\widehat{\mathrm{Cov}}(y,x)}{\widehat{\mathrm{Var}}(x)},\hat{\mu }=\bar{y}-\hat{\beta }\bar{x}.$$

The market-neutral portfolio (long $1 of $y$, short $\hat{\beta }$ of $x$) has return equal to the idio residual:

$$r_{p,t+1}=r_{t+1}^y-\hat{\beta }{r}_{t+1}^x\approx \Delta z_{t+1},$$

so its exposure to any factor $F$ loading ${\beta }_F^y,{\beta }_F^x$ cancels when $\hat{\beta }={\beta }_F^y/{\beta }_F^x$. In log-price space, $\Delta z_{t+1}=\Delta \ln y_{t+1}-\hat{\beta }\Delta \ln x_{t+1}$ is the portfolio return of the dollar-neutral book.

2.3 Factor-neutral generalisation (Avellaneda–Lee 2010)

Pairs trading is the two-asset case of a broader statistical arbitrage: decompose each stock’s return against systematic factors,

$$R_i=\sum _{j=1}^m{\beta }_{ij}{F}_j+{\tilde{R}}_i,$$

and trade the idiosyncratic residual ${\tilde{R}}_i$. The factors $F_j$ can be sector ETFs (each stock regressed on its peers’ ETF, ${\beta }_{ij}=\mathrm{Cov}(R_i,R_{I_j})/\mathrm{Var}(R_{I_j})$) or PCA eigenportfolios (eigenvectors of the return correlation matrix; weights $Q_i^{(j)}=v_i^{(j)}/{\sigma }_i$). A portfolio $\{Q_i\}$ is market-neutral iff ${\sum }_i{\beta }_{ij}{Q}_i=0$ for all $j$; the first eigenportfolio is the market, higher eigenportfolios are interpretable long–short sector bets (“coherence”).

2.4 Distance vs cointegration - when they agree

On normalised prices, the distance metric $D_{ij}$ is minimised by the pair whose level spread is tightest. If the true relationship is $y=\mu +\beta x+z$ with $\beta \approx 1$ and small $\mathrm{Var}(z)$, distance and cointegration agree. When $\beta \ne 1$ (different volatilities/leverage), distance systematically mis-hedges and the regression hedge is required.

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3. Computational Implementation - the distance screen and the hedge ratio

Stdlib only. Build a small synthetic universe: four stocks on factor A (with stationary idio), three on factor B, and one unrelated loner. The distance screen should recover the within-sector pairs, and the top pair’s OLS hedge ratio should be close to $1$ on the normalised index.

Every top-5 pair is within sector A - the screen correctly identifies the comoving block and never touches the loner Z. The OLS hedge ratio on the normalised indices is $\hat{\beta }=0.9985\approx 1$, confirming that for this pair the distance method’s implicit unit hedge is essentially correct.

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4. Failure Modes & First-Principles Breakdowns

1. Selection is a multiple test. Screening $N$ stocks gives $N(N-1)/2$ pairs; the “best” distance is an extreme order statistic and is biased toward spurious fits. GGR mitigate this by fixing the formation/trading split ex ante; modern practice uses the Deflated Sharpe Ratio (Backtesting Hygiene).
2. Distance ignores $\beta$. Normalised-price matching forces $\beta =1$; a pair with different volatilities is mis-hedged, leaving residual factor exposure that surfaces as a directional loss in a market move.
3. Economic link ≠ statistical link. Same-sector stocks can have different regulatory, leverage or commodity exposure. The distance screen finds optical comovement; a structural break in the link (Ch 5) is invisible to it.
4. Factor model choice drives the residuals (Avellaneda–Lee). PCA vs ETF factors give different residuals and hence different P&L; ETF weights are biased to large caps, PCA factors are not, and the number of significant eigenvalues is itself time-varying.
5. Look-ahead in normalisation. Using the entire sample to normalise prices leaks future information into the formation window. Normalise strictly on the formation period.

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5. References

• Gatev, Goetzmann & Rouwenhorst, RFS 19(3), 2006
• Avellaneda, M. & Lee, J.-H., Quantitative Finance 10(7), 2010
• Vidyamurthy, G., Pairs Trading: Quantitative Methods and Analysis, Wiley, 2004
• Krauss, C., Journal of Economic Surveys 31(2), 2017
• Tsay, Analysis of Financial Time Series

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6. Connected Graph Bridges

• Back: 02 · Cointegration & the Spread · Index Hub
• Forward: 04 · Trading Rules & Backtest
• Sibling: Fundamental Multi-Factor Models (factor definitions that drive the residual) · Signal Processing & Kalman Filtering (time-varying $\beta$)
• Portfolio: Transaction Costs & Turnover