Speed of mean reversion of an interest rate model

Question

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 1

• 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.