Negative Mean Reversion in the Hull–White One-Factor Model

Summary

The document explains what a negative mean-reversion estimate implies in the Hull–White one-factor short-rate model. It gives the model’s expected-rate expression and describes how a negative mean-reversion parameter causes expected rates to move away from the model’s reference level over time, rather than converge toward it. The direction of divergence depends on the starting rate relative to that level.

The discussion connects this behavior to calibration: a negative estimate may indicate that the observed data do not exhibit mean reversion, making Hull–White a poor fit. The author also reports a practical consequence: their interest-rate tree could not be generated with the negative estimate. The note is brief and does not explain calibration choices, numerical implementation fixes, or compare alternative models. Its conclusion should therefore be read as a diagnostic interpretation, not a full treatment of model selection or of every possible implementation.

Key ideas

• A negative mean-reversion parameter makes expected rates diverge from the model’s reference level over time.
• The direction of divergence depends on whether the initial rate is above or below that level.
• A negative calibration estimate may indicate that the data do not support mean-reverting dynamics.
• The author reports that their interest-rate tree could not be generated with a negative estimate.

Tags

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


# 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.

## Answer by Juan Ignacio Gil (score 4)

https://quant.stackexchange.com/a/24655

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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