Z-Score Normalization for Statistical Arbitrage

Understanding Z-Score normalization

The Z-Score transforms a raw value into units of standard deviations from the mean using the formula:

$Z = \frac{x - \mu}{\sigma}$

Where:

• $x$ is the current value (price ratio or spread)
• $\mu$ is the historical mean
• $\sigma$ is the historical standard deviation

For pairs trading, the price ratio between two correlated securities is typically normalized:

$Z_{ratio} = \frac{(P_A/P_B) - \mu_{ratio}}{\sigma_{ratio}}$

Applications in statistical arbitrage

Pairs trading signals

Z-Score normalization helps identify trading opportunities when pairs deviate from their historical relationship:

Common signal thresholds:

• |Z| > 2: Potential trading opportunity
• |Z| > 3: Strong divergence signal
• Z returning to 0: Mean reversion target

Statistical properties and assumptions

Mean reversion testing

Before applying Z-Score normalization, the price ratio should be tested for mean-reverting properties:

$\Delta(P_A/P_B) = \theta(\mu - P_A/P_B) + \epsilon$

Where:

• $\theta$ is the mean reversion speed
• $\epsilon$ is random noise

Stationarity requirements

The price ratio must exhibit:

• Constant mean
• Constant variance
• Time-independent autocorrelation

Risk considerations

Dynamic volatility adjustment

Standard Z-Scores assume constant volatility. For more robust signals, consider using:

1. GARCH models for volatility estimation
2. Exponentially weighted standard deviation
3. Regime-dependent normalization

Position sizing

Position sizes can be scaled inversely to the Z-Score magnitude:

$Position Size = \frac{Risk Budget}{|Z|}$

This approach increases position sizes as pairs move closer to equilibrium.

Implementation challenges

Look-ahead bias prevention

Rolling statistics must use only historical data:

Parameter selection

Critical parameters include:

• Historical window length
• Signal thresholds
• Position sizing limits
• Rebalancing frequency

These should be optimized through:

• Walk-forward analysis
• Cross-validation
• Sensitivity testing

Market applications

Z-Score normalization is particularly effective in:

1. Equity pairs trading
2. Fixed income relative value
3. Currency carry trades
4. Commodity spreads
5. ETF arbitrage

The technique helps identify temporary mispricings while providing a standardized framework for risk management and position sizing across different market contexts.