# Consequence of negative mean reversion of hull white one factor model

I tried to calibrate the data for hull-white one-factor model. Sometimes, I get negative estimate of mean reversion factor after the calibration process. When I plug the negative mean reversion factor into the hull-white one factor model, the interest rate tree cannot be generated.

I just wonder the theoretical consequence of hull-white one-factor model. Can anyone provide the meaning of negative mean reversion of hull-white one-factor model. If the mean reversion factor is negative, can the model be implemented properly?

Thanks.

A negative mean reversion makes the dynamics of the asset explode. If the model is:

$$dr=[\theta-\alpha r]dt+\sigma dW$$

The expected value in this model is:

$$\mathbb{E}(r)= r(0) e^{-\alpha t} + \frac{\theta}{\alpha} (1-e^{-\alpha t} )$$

If $\alpha<0$ $\mathbb{E}(r)$ goes to $\infty$ or $-\infty$, depending on if $r(0)$ is above or below the "long term mean" $\frac{\theta}{\alpha}$ (so our long term mean is not the long term mean here).

If you get $\alpha<0$ when calibrating your model, it is a sign that there is not a mean reversion in your data, and that the Hull-White model is not the right one here.

• Why do I get this "|" after the equations? Am I doing something wrong? Mar 2 '16 at 10:55
• Hi @juan-ignacio-gil . I intuitively agree with you, but do you by any chance have a reference (whitepaper, book?). I have the Meucci/Brigo book, but they don't talk about negative mean reversion. Mar 30 '17 at 6:03
• Nothing I remember now (and a quick search does not show anything), but both intuition and experience tell me so (I work in energy where mean reversion is usually strong). Maybe Hull's or Joshi's books? Mar 30 '17 at 6:47
• Yes, thanks. Hull talks about non-negative mean reversion within the Vasicek section, but is silent during the discussion of Hull-White, which I think is an extension of Vasicek. Trouble is this paper - planchet.net/EXT/ISFA/1226.nsf/769998e0a65ea348c1257052003eb94f/… seems to think negative mean reversion is acceptable, at least in the short term. But I'm struggling with the concept of "negative" mean reversion. Mar 30 '17 at 6:56
• While it can be acceptable in the short term, the resulting dynamics are the opposite of what we look for in a mean reversion model: the long term mean does not act as an attractor, but as a repeller (and a weird one, as the farther the price is from the repeller the more strongly it's repelled). In practice, when I calibrate a mean Hull and White model and get a negative mean reversion, I automatically change to a Black-Scholes and recalibrate with the new assumptions. Apr 1 '17 at 10:09