I have had a few experiences or chats with teammates about the Hull-White model.

The famous model has 2 parameters :

  • The volatility
  • The mean reversion

Very often I hear that the mean reversion has been fixed and that the calibration is only done on the volatility.

Why do that ? Why not fix the volatility and optimize on the mean reversion since both parameters have influence on the vanilla products ?

Moreover, why no optimize on both parameters simultaneously ?

Thanks a lot in advance for the context or opinions that help me to understand what are the justification of these practices.


1 Answer 1


Fixing the mean reversion, and parameterizing the volatility as a step function or as a piecewise linear function, the volatility can be bootstrapped exactly to a set of vanilla options sorted by expiries. This is a very stable and fast procedure, akin to the bootstrapping of a discount curve onto rate instruments.

For instance when pricing a bermuda swaption with the HW model, a mean reversion is first choosen and the volatility is then bootstrapped on the coterminal european swaptions market prices. Hence the bermuda swaption is priced in a manner consistent with the coterminal european swaptions prices (the coterminal swaptions are also the natural hedge to the bermuda swaption). The remaining degree of freedom, the mean reversion, becomes a parameter to mark the bermuda swaption (not sure if it is still the case, but I think at some point it was even contributed to Markit's Totem).

  • 1
    $\begingroup$ I understand from you answer that fixing the volatility and bootstrapping the mean reversion would be a much more complex methodology since the "inversion" of the MR is more complex than doing it for the vol... However, if I understand it right you propose to calibrate the MR at the end of the process whereas all the swpation term structure has been fitted with a fixed MR... How can it be consistent then ? $\endgroup$
    – user25844
    Feb 14, 2019 at 15:09
  • 1
    $\begingroup$ For a given MR $\lambda$ you calibrate $\sigma(t)$ to coterminal europeans and then price the bermuda. So now your bermuda price is $\text{bermuda} = f(\lambda, \text{coterminals})$. you can then calibrate $\lambda$ to quoted bermudas if there are some. If you are a market maker you can choose the $\lambda$ which you believe in... $\endgroup$ Feb 14, 2019 at 15:15

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