The S-Score: Normalizing the Residual into a Trading Signal

The OU model gives each stock an idiosyncratic residual $X^i_t$ with its own equilibrium mean $m^i$ and equilibrium standard deviation $\sigma^i_\text{eq}$. For a universe of 500 stocks, these are 500 different spread series with different scales: a deviation of $+0.05$ log-points is large for one stock (if $\sigma_\text{eq} = 0.02$) and small for another (if $\sigma_\text{eq} = 0.08$). A unified entry rule — "enter when the deviation exceeds threshold $T$" — cannot be applied uniformly across stocks without normalisation.

The S-score solves this by expressing each residual in units of its own equilibrium standard deviation. A stock with $s_t = 2$ is exactly 2 equilibrium standard deviations above its long-run mean, regardless of whether its absolute residual is 2 cents or 8 log-points. The S-score is the final step in the Avellaneda-Lee pipeline: it converts the output of the OU calibration into a single cross-sectionally comparable signal that drives position entry, exit, and stop-loss.

Learning objectives. By the end of this note you should be able to: