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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?

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  • 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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    $\begingroup$ Some FOs I've spoken to say "mean reversion for hull white is 5%" - in the context of your example, this means rates take $\ln(2)/0.05 = 13.86$ years to travel from current level halfway to equilibrium. Is this the correct interpretation? $\endgroup$ – crunch Jul 2 '15 at 15:16
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    $\begingroup$ @crunch Strictly speaking, it's only correct in the absence of volatility, but this is the right conceptual framework to think about things. $\endgroup$ – Helin Jul 8 '15 at 15:15
  • $\begingroup$ of course, lots of things are nice in the absence of volatility! $\endgroup$ – crunch Jul 9 '15 at 6:39

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