Pair Trading Indicators

1 Statistical Indicators

1.1 Correlation Analysis

Purpose: Measure the strength of linear relationship between two asset price series.

How:

  • Computed using the Pearson correlation coefficient.
  • Inputs: Price series of both assets.
  • Outputs: A value between -1 and 1.

Interpretation:

  • Values close to 1 indicate strong positive correlation.
  • Values close to -1 indicate strong negative correlation.
  • Values close to 0 indicate little or no correlation.
  • For pair trading, correlations above 0.7 are generally desirable.

1.2 Cointegration Test

Purpose: Test for a long-term equilibrium relationship between two asset price series.

How:

  • Computed using coint() from statsmodels.tsa.stattools.
  • Inputs: Price series of both assets.
  • Outputs: Test statistic and p-value.

Interpretation:

  • p-value < 0.05: Reject the null hypothesis of no cointegration.
  • A p-value below 0.05 suggests the pair is suitable for trading, indicating that prices may diverge temporarily but tend to revert to equilibrium.

1.3 Augmented Dickey-Fuller (ADF) Test

Purpose: Test whether the spread between two assets is stationary.

How:

  • Computed using adfuller() from statsmodels.tsa.stattools.
  • Input: Log spread between two assets (i.e. log price A - log price B).
  • Outputs: Test statistic and p-value.

Interpretation:

  • p-value < 0.05: Reject the null hypothesis of non-stationarity.
  • Indicates that the spread is stationary and likely to mean-revert.
  • Suitable for pair trading when p < 0.05.

1.4 Half-Life Calculation

Purpose: Measure how quickly the spread reverts to its mean.

How:

  • Based on the Ornstein-Uhlenbeck mean-reversion process.
  • Derived from a regression of the spread change against its lagged value.

Formula:

  • Run an AR(1) regression on the spread series: \[ \Delta \text{Spread}_t = \alpha + \rho \cdot \text{Spread}_{t-1} + \varepsilon_t \]
  • Then calculate the half-life using: \[ \text{Half-life} = \frac{-\ln(2)}{\rho} \]

Interpretation:

  • 5-25 days: Ideal half-life range for pair trading.
  • < 5 days: Very rapid mean reversion, may lead to frequent trading.
  • 25-60 days: Slower mean reversion, requiring longer holding periods.
  • > 60 days or negative: May not be suitable for pair trading.

2 Trading Signals

2.1 Z-Score

Purpose: Measure how many standard deviations the current spread deviates from its mean.

How:

  • Based on the rolling mean and standard deviation of the spread.
  • Formula: \[ \text{Z-score} = \frac{\text{Current Spread} - \text{Rolling Mean}}{\text{Rolling Standard Deviation}} \]
  • Typically uses a 20-day rolling window.

Interpretation:

  • |Z-score| > 2: Typical entry signal threshold.
  • |Z-score| < 0.5: Typical exit signal threshold.
  • High absolute Z-score values indicate extreme divergence from the normal relationship.

2.2 VIX-Adjusted Thresholds

Purpose: Adapt entry thresholds based on market volatility.

How:

  • Uses VIX index levels to adjust the Z-score thresholds.
  • Higher VIX values lead to higher threshold requirements.

Interpretation:

  • VIX < 20: Entry threshold = 2.0
  • VIX 20-25: Entry threshold = 2.25
  • VIX 25-30: Entry threshold = 2.5
  • VIX > 30: Entry threshold = 3.0
  • A higher threshold during periods of high volatility reduces false signals.

3 Trade Execution

3.1 Entry Signals

Purpose: Determine when to enter pair trades.

How:

  • Based on the Z-score crossing the adjusted threshold values.
  • Signal Examples:
    • Z-score > threshold: Short asset A, Long asset B.
    • Z-score < -threshold: Long asset A, Short asset B.

Interpretation:

  • A positive Z-score indicates that the spread is wider than normal.
  • A negative Z-score indicates that the spread is narrower than normal.
  • A large absolute Z-score suggests a higher probability of mean reversion.

3.2 Exit Signals

Purpose: Determine when to exit pair trades.

How:

  • Primary exit: When the Z-score returns to the ±0.5 range.
  • Secondary exits: Based on time-based or risk-based conditions.

Additional Exit Conditions:

  • Maximum holding time: 1.5 × half-life days.
  • Stop-loss: 5% divergence between the pair.

4 Performance Metrics

4.1 Trade Success Rate

Purpose: Measure the percentage of profitable trades.

How:

  • Calculated as: \[ \text{Trade Success Rate} = \left(\frac{\text{Number of Profitable Trades}}{\text{Total Number of Trades}}\right) \times 100\% \]

Interpretation:

  • Higher percentages indicate a more reliable strategy.
  • Should be evaluated alongside the average profit or loss per trade.

4.2 Total Return

Purpose: Measure the overall profitability of the strategy.

How:

  • Calculated as: \[ \text{Total Return} = \left(\frac{\text{Final Portfolio Value} - \text{Initial Portfolio Value}}{\text{Initial Portfolio Value}}\right) \times 100\% \]

Interpretation:

  • Positive values indicate a profitable strategy.
  • Should be compared to benchmark returns over the same period.

4.3 Alpha & Beta

Purpose: Measure performance relative to a benchmark and sensitivity to market movements.

How:

  • Perform a linear regression of strategy returns ( r_s ) against benchmark returns ( r_b ).
  • Formula: \[ r_s = \alpha + \beta r_b + \varepsilon \] where:
    • ( ) = Alpha (strategy’s excess return independent of the market)
    • ( ) = Beta (sensitivity to market movements)
    • ( ) = Residuals (errors)

Interpretation:

  • Alpha > 0: Outperformance vs. benchmark.
  • Beta ≈ 0: Market-neutral; low market risk exposure.
  • Beta > 1: Strategy more volatile than market.

4.4 Volatility

Purpose: Measure the degree of variation in strategy returns.

How:

  • Computed as the standard deviation of periodic returns ( r_t ).

  • Formula: \[ \text{Volatility} = \sqrt{\frac{1}{n-1} \sum_{t=1}^{n} (r_t - \bar{r})^2} \] where:

    • ( r_t ) = return at time ( t )
    • ( {r} ) = average return
    • ( n ) = number of periods

Interpretation:

  • Higher volatility indicates larger swings in returns.
  • Prefer lower volatility for more stable strategies.

4.5 Geometric Mean Rate of Return (GMRR)

Purpose: Capture the compounded average rate of return per period.

How:

  • Computed as: \[ \text{GMRR} = \left(\prod_{i=1}^n (1 + r_i)\right)^{\frac{1}{n}} - 1 \] where (r_i) are the periodic returns.

Interpretation:

  • Provides a more accurate measure of return over time, accounting for compounding.

4.6 Sharpe Ratio

Purpose: Measure the risk-adjusted return.

How:

  • Computed as: \[ \text{Sharpe Ratio} = \frac{\text{Average Return} - \text{Risk-Free Rate}}{\text{Standard Deviation of Returns}} \]

Interpretation:

  • Higher Sharpe ratios indicate better risk-adjusted returns.

4.7 Average Return per Trade

Purpose: Measure the average profitability of each trade.

How:

  • Computed as: \[ \text{Average Return per Trade} = \frac{\text{Total Return}}{\text{Number of Trades}} \]

Interpretation:

  • Higher values indicate more profitable trades.

4.8 Average Number of Trades per Year

Purpose: Annualize trading activity based on backtest data.

How:

  • Computed by scaling the number of trades by the ratio of 12 months to backtest months.
  • Example: 35 trades over 3 months → (35 = 140) trades per year.

Interpretation:

  • Helps assess strategy activity and transaction cost implications.

5 Summary of Screening Criteria

Indicator Test Threshold Purpose
Correlation Pearson ( > 0.7 ) Co-movement detection
Cointegration Coint p < 0.05 Long-term equilibrium
ADF Test ADF p < 0.05 Mean reversion confirmation
Half-Life AR(1) 5–60 days Timeframe viability
Z-Score Rolling ±2.0 to ±3.0 Entry/exit signal strength

These indicators form the statistical and performance backbone of our market-neutral pairs trading framework. Each filter and metric adds a layer of validation to improve trade quality, reduce noise, and enhance strategy evaluation.