4
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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).

$\endgroup$
  • $\begingroup$ 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 ? $\endgroup$ – StudentInFinance Feb 14 at 15:09
  • $\begingroup$ 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... $\endgroup$ – Antoine Conze Feb 14 at 15:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.