Z-Score Normalization and Dynamic Thresholding in Statistical Arbitrage

The Z-score is a simple yet effective tool in statistical arbitrage for identifying trading opportunities. It measures how many standard deviations a spread is from its historical mean. A large positive Z-score indicates that the spread is unusually high and likely to revert downwards, while a large negative Z-score indicates the opposite.

Calculating the Z-Score

The Z-score of a spread at time t is calculated as:

Z(t) = (Spread(t) - Mean(Spread)) / StdDev(Spread)

Where Mean(Spread) and StdDev(Spread) are the moving average and moving standard deviation of the spread over a specified lookback window.

A common trading rule is to go short the spread when the Z-score exceeds a certain positive threshold (e.g., 2.0) and go long the spread when it falls below a certain negative threshold (e.g., -2.0). The position is then closed when the Z-score reverts to zero.

The Problem with Static Thresholds

While simple to implement, using static thresholds for the Z-score can be suboptimal. The volatility of the spread can change over time, meaning that a Z-score of 2.0 in a low-volatility regime may be more significant than a Z-score of 2.0 in a high-volatility regime. This can lead to either missed trading opportunities or taking on too much risk.

Dynamic Thresholding

A more advanced approach is to use dynamic thresholds that adapt to the changing volatility of the spread. One way to do this is to use a GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model to forecast the conditional volatility of the spread. The GARCH model can capture the volatility clustering that is often observed in financial time series.

Another approach is to use a rolling standard deviation of the Z-score itself to adjust the thresholds. For example, the entry threshold could be set to 2 * (1 + rolling_stddev(Z-score)). This would cause the thresholds to widen during periods of high volatility and narrow during periods of low volatility.*

By using dynamic thresholds, a trader can create a more robust and adaptive statistical arbitrage strategy that is better able to navigate changing market conditions.