Ornstein–Uhlenbeck Processes for Mean-Reverting Models and Simulation
Summary
The document introduces the Ornstein–Uhlenbeck (OU) process as a continuous-time model for a variable that is pulled toward a long-run mean while exposed to random shocks. It explains the roles of the mean, reversion speed, and volatility, contrasts OU dynamics with Brownian motion and geometric Brownian motion, and connects the process to the Vasicek interest-rate model. Suggested applications include modeling rates, yields, volatility, and mean-reverting spreads in pairs trading.
It outlines an integrating-factor solution to the stochastic differential equation, though the equations are embedded as images and are not readable in the text provided. For simulation, it presents an Euler–Maruyama discretization and Python code that generates and plots a sample path. That path illustrates reversion from an initial value toward the specified mean, but it is only a simulated example, not empirical evidence. The article does not address parameter estimation, calibration, trading costs, or whether a real spread remains stationary; OU assumptions and a sample path alone do not establish a profitable strategy.
Key ideas
- The OU process models random movement with a drift that pulls values toward a long-run mean.
- Its reversion speed and volatility govern how quickly and widely the process fluctuates around that mean.
- Euler–Maruyama discretization provides a straightforward way to simulate sample paths computationally.
- OU models can describe interest rates or candidate mean-reverting spreads, but real data must support the assumptions.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.