Calibrating Ornstein–Uhlenbeck Parameters from Historical Data

Summary

This document presents a method for estimating the parameters of an Ornstein–Uhlenbeck process from an equally spaced historical time series. It defines the process through a mean-reverting rate, a long-run level, and a volatility term, then gives formulas for estimating each parameter. The level is estimated as the sample mean; the rate uses squared successive changes relative to the series’ squared deviations from that mean; and volatility is calculated using the estimated rate and deviations from the mean.

The approach is presented for modeling interest rates and foreign exchange rates, but the document offers no worked example, empirical comparison, or discussion of estimation uncertainty. Its formulas assume equally spaced observations and a constant mean-reverting level and volatility. The notes therefore describe a compact historical calibration procedure, rather than establishing how well the fitted process predicts future behavior or whether its assumptions suit a particular dataset.

Key ideas

• The Ornstein–Uhlenbeck model represents a variable that tends to revert toward a long-run level while receiving random shocks.
• The long-run level is estimated using the historical sample mean.
• The mean-reversion rate is estimated from successive changes and deviations from the sample mean.
• Volatility is estimated using the fitted rate and the series’ dispersion around its mean.
• The stated calibration assumes equally spaced observations.

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# How to calibrate an Ornstein-Uhlenbeck process based on historical data?


# How to calibrate an Ornstein-Uhlenbeck process based on historical data?












Background: I have been working on my master thesis project for the last few months and gave the final presentation on the 2023-06-01. As a part of the master thesis project, I did a complete derivation of the historical calibration of an Ornstein-Uhlenbeck process when it is used to model interest rates and foreign exchange rates. The derivation process was excluded from the final report as the report became too long. Still, I'd like to share the knowledge with anyone might need it. So, enjoy! =)

Specifying an Ornstein-Uhlenbeck process as follows:

$$ d{y(t)} = -\kappa(y(t) - \theta)d{t} + \sigma d{W(t)},$$

where $\kappa$ is the mean-reverting rate, $\theta (t)$ is the mean-reverting level, $\sigma$ is the volatility parameter, and $W(t)$ is a Wiener process.

Suppose there is a historical time series of $y(t_i)$ with $t_i \in \{t_1,t_2,\ldots, t_N\}$. The historical time points are equally spaced with $t_1 < t_2 < \dots < t_N$ and $\delta := t_{i+1} - t_i$. The parameters of the O-U process can be calibrated based on historical data by using the formulas below.

Mean-reverting level $$\hat{\theta} = \frac{1}{N}\sum_{i=1}^{N}y(t_i)$$

Mean-reverting rate

$$ \hat{\kappa} = \frac{1}{2\delta}\frac{\sum\limits_{i=2}^{N}(y(t_i) - y(t_{i-1}))^2}{\sum\limits_{i=1}^{N}(y(t_i) - \hat{\theta})^2} $$

Volatility

$$ \hat{\sigma} = \sqrt{\frac{2\hat{\kappa} - \hat{\kappa}^2 \delta}{N-1} \sum\limits_{i=1}^{N} \left( y(t_i) - \hat{\theta} \right)^2} $$

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