# 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? – Juan Ignacio Gil 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. – PBD10017 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? – Juan Ignacio Gil 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. – PBD10017 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. – Juan Ignacio Gil Apr 1 '17 at 10:09