Interpreting Units of Short Rate Parameters

Question

I've estimated the parameters for the Vasicek model $$ dr(t) = a(b - r(t))dt + \sigma dW(t) $$ and the CIR model $$ dr(t) = a(b - r(t))dt + \sigma\sqrt{r(t)} dW(t) $$ to one-year Treasury yield data from 1974 (which were around 8% then!). Let's say the estimates I got were $$ Vasicek: a = 3.2, b = 8.1, \sigma = 6.0 \qquad (1)\\ CIR: a = 3.2, b = 8.1, \sigma = 2.3. \qquad (2) $$ N.b. these values correspond to $r(t)$ in percent, not decimal. So, my dimensions are short rate level measured in percent (%), and time, say, measured in seconds (s). The units of the parameters are then $$ a = s^{-1}, b = \%, \sigma = \%/\sqrt{s} $$ My question is, how does one intuitively interpret the estimated values in (1) and (2)? I.e., I'm trying to think of the process as a physical process, and so what does a "mean reversion speed of $3.2 / s$" mean, e.g.? Actually, it seems that calling $a$ a "speed" is a misnomer, given the units.

Any insights welcome!

Answer 1

The processes revert towards their mean with the speed $E(dr(t)/dt) =a*(b-r(t))$ so $a$ is not the speed itself, only one factor of it. If $\sigma$ would vanish then $a$ would be $ln (2)$ times the inverse of the time - hence its unit $s^{-1}$- it would take for $b-r(t)$ to halve. Think of a as the decay-factor of the deviation from the mean!