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This is more a conceptual question around calibration. My objective is to calibrate a 1-factor Hull White model, and my question relates to calibrating a and sigma (both constants) to swaptions.

Let's say that I have a set of ATM swaption prices (this is my market prices) coming from a model based on OIS discounting. When doing the calibration, i.e. minimizing the diff between model (HW) and market prices, should I do a transformation of the swaption prices to get some sort of proxy for prices under, say, LIBOR discounting? The reason I started thinking about this, is because the HW closed-form formulas that I use are based on the fact that fixing and discount curve is identical.

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  • $\begingroup$ assuming a LIBOR-OIS deterministic spread, you work with one curve. $\endgroup$
    – Canardini
    Nov 26 '20 at 5:44
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As the comment by @Canardini says, ideally you should extend your Hull-White model to include LIBOR-OIS deterministic spread, which is not that difficult and well-covered in the literature.

If that is not possible, you have (at least) two options. One is to adjust input swaption prices for discounting. Basically you just need to multiply OIS-based swaption quotes by the ratio of Libor Annuity to OIS Annuity for the tenor of the underlying swap. If your prices are given as forward premiums (as often the case), these should be forward annuities

Another option is to calibrate to implied volatiltiies rather than prices. If you have access to these volatilities directly, then you can just calibrate to them in your single-curve model. If you do not, but have a two-curve-enabled vanilla model for swaptions (such as SABR or even Black), then you should imply vols from your prices using that model and then calibrate HW to the vols

Both of these are approximations but should work ok for ATM swaptions, esp. if they are not super-long-dated

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  • $\begingroup$ Thanks a lot, that answers my question very nicely. My own thought was to do something similar to what you write in option 1, scaling the swaption prices. Regarding your first comment, do you have some good examples on specific literature for HW with LIBOR-OIS deterministic spread? $\endgroup$
    – Bohlke
    Nov 26 '20 at 14:18
  • $\begingroup$ @Bohlke a quick Google search brings up a few things, eg janroman.dhis.org/finance/Swaptions/HullWhite%20Swaptions2.pdf does not look too bad and is free. there are some books eg springer.com/gp/book/9783319253831 $\endgroup$
    – piterbarg
    Nov 26 '20 at 14:34

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