The Ornstein-Uhlenbeck Model for Idiosyncratic Residuals

After stripping out systematic risk via the factor-residual decomposition, the accumulated idiosyncratic residual $X^i_t$ should be stationary and mean-reverting. But stationarity alone does not give a tradeable signal — the trading rules (entry level, sizing, expected time to convergence, stop-loss horizon) all require a parametric model of how fast the residual reverts and how wide its fluctuations are.

The Ornstein-Uhlenbeck (OU) process is the canonical mean-reverting diffusion: it is the continuous-time analogue of the discrete AR(1) model, with a well-defined equilibrium variance and a mean-reversion speed that can be estimated by OLS from the time series of accumulated residuals. Its parameters — the speed of reversion $\kappa$, the long-run mean $m$, and the instantaneous volatility $\sigma$ — connect directly to trading design: entry band width, expected half-life of a position, and stop-loss horizon.

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