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Interpreting Mean-Reversion Speed in Short-Rate Models

Article Quant Q&A · Author: bcf
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Summary

This note explains how to interpret the mean-reversion parameter in the Vasicek and Cox–Ingersoll–Ross short-rate models. The parameter is not itself the instantaneous rate of change: the expected drift also depends on the gap between the current short rate and its long-run level. Its units are inverse time, while the long-run level and volatility scale depend on the rate and time units chosen.

In the deterministic limit, with volatility set to zero, the distance from the long-run level decays exponentially. The parameter therefore sets the decay rate; the deviation’s half-life is the inverse parameter multiplied by the natural logarithm of two. The document gives fitted parameter examples for historical one-year Treasury yields, but does not explain the estimation procedure or assess the models’ fit. Its interpretation applies to these mean-reverting equations and depends on consistent units for time and interest rates.

Key ideas

  • The mean-reversion parameter has units of inverse time, so it is not a rate change by itself.
  • The expected drift depends on both the parameter and the gap between the current rate and its long-run level.
  • With volatility removed, the deviation from the long-run level decays exponentially.
  • The parameter determines the deviation’s half-life through the natural logarithm of two.

Tags

Fixed incomeMean reversionStatistics
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# Interpreting Units of Short Rate Parameters


# Interpreting Units of Short Rate Parameters












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 by Mats Lind (score 2)

https://quant.stackexchange.com/a/29526

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!

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