The Ornstein-Uhlenbeck Process for Mean-Reverting Models and Simulation

Summary

This tutorial introduces the Ornstein-Uhlenbeck (OU) process as a continuous-time model for a variable that fluctuates around a long-term mean. It explains the roles of the mean, reversion speed, and volatility, and contrasts mean-reverting behavior with Brownian motion and geometric Brownian motion. The article connects the process to the Vasicek interest-rate model and describes possible uses in interest-rate modeling, bond yields, and pairs trading, where a spread may be modeled as mean-reverting.

For simulation, it presents an Euler-Maruyama discretization of the stochastic differential equation and illustrates generating and plotting a sample path with NumPy and Matplotlib. The example shows a path moving toward its specified mean, but it is an illustration rather than empirical evidence that market spreads or rates follow this process. The tutorial does not address parameter estimation, model diagnostics, transaction costs, or the risks of assuming a stable mean. Practical trading use therefore requires testing whether the mean-reversion assumption fits the chosen data and remains reliable.

Key ideas

• The OU process models random fluctuations around a long-term mean, with a parameter controlling reversion speed.
• Euler-Maruyama discretization provides a basic way to simulate sample paths from the continuous-time model.
• The process underlies the Vasicek interest-rate model and can be applied to rates and yields.
• A pairs trader may model a spread as mean-reverting to inform entry and exit signals.
• A simulated path illustrates the model but does not establish that a real market series is mean-reverting.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.