Skip to content
Calculus I / Ordinary Differential Equations

Mean-Reversion and the Ornstein-Uhlenbeck ODE

When interest rates spike, they tend to fall back toward a long-run level. When a spread widens beyond its historical range, it tends to compress. This mean-reversion behaviour — pull toward equilibrium, decaying to zero as the state approaches the target — is captured in its simplest deterministic form by the Ornstein-Uhlenbeck ODE:
dxdt=a(b−x).\frac{dx}{dt} = a(b - x).
The solution to this ODE is the drift term inside the Vasicek interest rate model, the Hull-White model, and the Ornstein-Uhlenbeck stochastic process. Understanding the deterministic ODE is the prerequisite for making sense of why those models produce mean-reverting sample paths. This lesson derives the solution using the integrating factor method, identifies what the parameters aa and bb control, and examines the OU ODE in three financial contexts: interest rates (Vasicek), commodity prices, and volatility.
Learning objectives. By the end of this note you should be able to:
  • rewrite the OU ODE x′=a(b−x)x' = a(b-x) in linear standard form and solve it with the integrating factor;
  • identify bb as the long-run equilibrium and aa as the mean-reversion speed (inverse half-life);
  • compute the time for a shock to halve;
  • compare the OU/Vasicek model to the Black-Scholes model (which has no mean reversion) and the logistic model (which has a capacity constraint but not a fixed target).

In this lesson

  1. The OU ODE and its solution
  2. The two parameters
  3. Qualitative analysis
  4. The OU ODE in three financial contexts
  5. Remark 2 (OU vs Black-Scholes vs logistic)

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.