Ornstein Uhlenbeck Processes and Mean Reversion Modelling

The Ornstein-Uhlenbeck process reverts to a long-run mean at a set speed, and it underlies interest rate models and pairs trading spreads.

The Ornstein-Uhlenbeck process models a quantity that reverts toward a long-run mean at a speed proportional to its current distance from that mean, and it is the standard building block for interest rate models and for pairs trading strategies built on a mean-reverting spread. Its defining equation writes the differential of X as kappa times the long-run mean minus X, all times dt, plus sigma times dW, where kappa controls the speed of reversion.

The drift term, kappa times the long-run mean minus X, pulls the process back toward the mean whenever X is above it and pushes it up whenever X is below it, with the strength of the pull proportional to the current gap. A larger kappa produces faster reversion and a tighter distribution around the mean over any fixed horizon, while a smaller kappa produces a process that wanders further before correcting. The volatility term, sigma times dW, is constant and additive, in contrast to geometric Brownian motion where volatility scales with the level of the process, a distinction covered in Geometric Brownian Motion and Its Limitations.

Because the drift depends linearly on X itself, the process is one of the few stochastic differential equations with a fully explicit closed-form solution, obtained by multiplying through by an integrating factor of e to the kappa times t and integrating directly, a technique that parallels the method used to solve first-order linear ordinary differential equations.

As time grows without bound, the Ornstein-Uhlenbeck process converges to a stationary normal distribution with mean equal to the long-run mean parameter and variance equal to sigma squared divided by two kappa. This stationary variance is the number that determines how far the process typically strays from its mean once it has settled, and it shrinks as the reversion speed kappa increases, which makes intuitive sense, since a process pulled back harder settles into a tighter range.

The half-life of the process, the time it takes for the expected deviation from the mean to fall by half, equals the natural logarithm of two divided by kappa. This is the number traders actually quote when describing how quickly a spread or a short rate is expected to revert, since it converts the abstract speed parameter into a horizon measured in days or months.