Interpreting Mean-Reversion Speed in Interest Rate Models

Summary

The document gives an intuitive interpretation of the mean-reversion parameter in Vasicek- or CIR-style interest rate models. It relates the parameter to the time needed for a rate to close half the distance between its current level and its long-run equilibrium. A larger positive speed implies faster convergence. The answer suggests positive mean-reversion speed as a sensible assumption for interest rates because persistent explosive behavior is not typical, though it does not formally rule out other dynamics.

It also explains a common estimation route: fit an AR(1) process to observed rates and translate its persistence into a mean-reversion speed under the physical, or real-world, measure. That estimate is not automatically the parameter needed for derivatives valuation. Risk-neutral dynamics must instead be calibrated to market prices of benchmark instruments. The response provides intuition and a distinction between measures, but does not derive the AR(1) mapping, compare model variants, or discuss estimation uncertainty and sampling frequency.

Key ideas

• Mean-reversion speed describes how quickly a rate approaches its long-run level.
• The half-life expresses this speed as the time needed to halve the distance to equilibrium.
• An AR(1) fit can be used to estimate mean-reversion behavior under the physical measure.
• Derivative pricing requires risk-neutral parameters calibrated to market instrument prices.

Tags

# Speed of mean reversion of an interest rate model


# Speed of mean reversion of an interest rate model












I would like to have a bit more of intuition about the concept of "speed of mean reversion" for an interest rate model, e.g. Vasicek or CIR. In particular, is a negative speed of mean reversion possible? What's the connection between a mean reverting process and an AR(1) process? Does explosive AR(1) imply negative speed of mean reversion?

## Answer by Helin (score 9, accepted)

https://quant.stackexchange.com/a/18622

- Mean reversion speed $\kappa$ is better interpreted with the concept of half-life, which can be calculated from $\text{HL} = \ln(2) / \kappa$. For example, if the mean reversion coefficient is $\kappa = 1.5$, then the half-life of the process is $\ln(2) / 1.5 = 0.46209812$ years, or about 6 months. Let's assume that the current interest rate is 1% and the equilibrium level is 5%. Then you'd expect interest rate to travel half the distance toward the equilibrium level (i.e., 2%) in about 6 months. Generally speaking, $\kappa$ should be positive, since interest rates do not tend to explode.

- It is not uncommon to estimate mean reversion speed using an AR(1) process. In the context of interest rate modeling, this procedure gives you the mean reversion speed $\kappa$ in the physical measure ("real world"). For derivatives pricing, however, you need $\kappa$ in the risk-neutral measure, which can be obtained by fitting the model to market prices of some benchmark instruments.

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