1.1.4 Trading Rules, the z-score & a Full Pairs Backtest

Basic Prerequisites: 03 · Pairs Selection & Hedge and 02 · Cointegration & the Spread.

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1. Intuition & Practical Objective

Once a pair and its hedge ratio are fixed, the strategy is a rule on the spread: open when the spread is far from equilibrium, close when it returns. The whole art is in two numbers - the entry threshold and the exit threshold - expressed in standard deviations of the spread so that the same constants work across every pair. This page gives the rule, the exact P&L identity of a dollar-neutral pair, and a closed-form backtest you can run and audit.

The standard deviation is the natural unit because the OU spread is Gaussian in equilibrium: a $2\sigma$ excursion is expected to recur with a known frequency, and the time it takes to revert is governed by the half-life. Entry, exit and stop must all be consistent with ${\tau }_{1/2}$: entering a $2\sigma$ move in a spread with a 250-day half-life is not a trade, it is a position.

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2. Mathematical Ground Truth & Derivations

2.1 The z-score signal

Rolling on the formation window (or an expanding window), estimate ${\mu }_z,{\sigma }_z$ of the spread $z_t=y_t-\hat{\beta }{x}_t$, and define

$$Z_t=\frac{z_t-{\mu }_z}{{\sigma }_z}.$$

Canonical rule (the flat-file convention, made precise):

Condition	Action
$Z_t>+2$	Short the spread: short $y$, long $\hat{\beta }x$
$Z_t<-2$	Long the spread: long $y$, short $\hat{\beta }x$
$|Z_t|<0.5$	Flat (converged)
$|Z_t|\ge 3.5$	Stop out (possible cointegration break)

The stop is not arbitrary: under a correct model, $|Z|\ge 3.5$ should occur with probability $\approx 0.05\%$ per observation, so its repeated occurrence is evidence the model has broken, not a bargain.

2.2 The Avellaneda–Lee s-score (a drift-adjusted z-score)

Avellaneda & Lee define the dimensionless s-score from the OU parameters over a 60-day window,

$$s_i=\frac{X_i(t)-m_i}{{\sigma }_{\text{eq},i}},{\sigma }_{\text{eq},i}=\sqrt{\frac{\mathrm{Var}(\zeta )}{1-b^2}},$$

and trade: open short if $s>+1.25$, open long if $s<-1.25$, close short at $s<+0.75$, close long at $s>-0.50$. Including the drift ${\alpha }_i$ gives the “modified s-score” $s_{\text{mod},i}=s_i-{\alpha }_i{\tau }_i/{\sigma }_{\text{eq},i}$, which just shifts the thresholds by $\approx 0.3$ in practice - the built-in momentum term.

2.3 The P&L identity (why pairs trading is “free” of the market)

Long $1 of $y$, short $\hat{\beta }$ of $x$. In log-price space the one-period portfolio return is

$$r_{p,t+1}=\Delta \ln y_{t+1}-\hat{\beta }\Delta \ln x_{t+1}=\Delta z_{t+1}.$$

So the cumulative P&L is the change in the spread between entry and exit, and the market factor cancels to first order. Over the holding period from entry at level $z_{\text{in}}$ to exit at $z_{\text{out}}$ the gross P&L is $z_{\text{in}}-z_{\text{out}}$ (for a short-spread position), i.e. $\propto$ the number of standard deviations captured.

2.4 Sharpe and the cost drag

With per-period returns $r_{p,t}$, the annualised Sharpe is

$$\text{SR}=\frac{{\bar{r}}_p}{\hat{\sigma }(r_p)}\sqrt{252}.$$

Transaction costs enter as $c\cdot |\Delta \text{position}|$ per change with $c\approx 5$ bp per leg (10 bp round-trip for the two-leg book), and borrow cost on the short leg accrues daily. A pairs strategy’s gross Sharpe is high; the net Sharpe is a cost-and-turnover phenomenon, which is exactly why the half-life matters (fast reversion → shorter holding → more turnover).

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3. Computational Implementation - the backtest

Experiment A - full formation/trading backtest. We simulate a cointegrated pair (common log-price factor + stationary log-spread with $\varphi =0.93$, half-life $\approx 9.5$ days), estimate the hedge ratio and spread moments on a 252-day formation window, then trade the next 252 days with the $2\sigma /0.5\sigma /3.5\sigma$ rule. Stdlib only.

The book earns $+5.21\%$ gross over the year ($+3.86\%$ net of 10 bp round-trips) at a gross-exposure Sharpe of $+1.91$ - a market-neutral profile. Note the cost drag: $27$ position changes (i.e. $54$ leg-trades) cost $1.35\%$, about a quarter of the gross P&L. Turnover is the enemy, which is why the half-life screen matters.

Experiment B - the Avellaneda–Lee s-score. Estimate the OU parameters from a 60-day window and read today’s s-score and half-life.

The estimated half-life is $4.78$ days (fast, well inside the $\kappa >252/30$ filter) and the s-score is $-0.462$ - inside the $\pm 1.25$ entry band, so no trade: most of the time a well-behaved spread sits near equilibrium.

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4. Failure Modes & First-Principles Breakdowns

1. Thresholds fitted to the backtest. Choosing $2\sigma /0.5\sigma /3.5\sigma$ because they maximise historical Sharpe is in-sample optimisation; the Deflated Sharpe Ratio corrects for exactly this (Backtesting Hygiene).
2. The stop is a model-rejection test, not a loss cap. Repeated $|Z|\ge 3.5$ means the cointegration has broken; averaging down (“it must revert”) is the classic way a market-neutral book dies.
3. Costs scale with turnover, turnover scales with $1/{\tau }_{1/2}$. A spread with a 5-day half-life can turn the book over every week; two legs of spread cost then dominate. Always report net-of-cost Sharpe.
4. Non-synchronous legs. If one leg trades thinly, the closing-price spread is stale and the backtest fills at prices you cannot get (GGR’s bid-ask discussion). Use quotes or trade both legs simultaneously.
5. Borrow and short availability. The short leg may be unborrowable or expensive exactly when the signal is strongest (big divergence) - an unmodelled asymmetry.

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5. References

• Avellaneda, M. & Lee, J.-H., Quantitative Finance 10(7), 2010
• Gatev, Goetzmann & Rouwenhorst, RFS 19(3), 2006
• Tsay, Analysis of Financial Time Series
• Krauss, C., J. Economic Surveys 31(2), 2017
• Gatev et al. and Do & Faff for the empirical cost-adjusted Sharpe collapse.

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6. Connected Graph Bridges

• Back: 03 · Pairs Selection & Hedge · Index Hub
• Forward: 05 · Failure Modes & Practice · 06 · Advanced Extensions
• Sibling: Backtesting Hygiene & Deflated Sharpe
• Costs: Transaction Costs & Turnover Constraints