Calibrating Hull–White to OIS-Based Swaption Prices
Summary
The document considers how to calibrate constant mean-reversion and volatility parameters in a one-factor Hull–White model when market swaption prices use OIS discounting, while the available closed-form model assumes a single curve. It describes two approximate ways to align the inputs: scale prices by the ratio of the LIBOR and OIS swap annuities, using forward annuities for forward-premium quotes, or calibrate to implied volatilities instead. For the volatility route, use quoted volatilities directly when available; otherwise derive them from prices with a two-curve vanilla model such as Black or SABR.
The preferred modeling approach is to extend Hull–White to include a deterministic LIBOR–OIS spread. The proposed adjustments are approximations, with the answer expecting them to work reasonably for at-the-money swaptions, especially at shorter maturities. The document gives conceptual guidance rather than numerical tests or a comparison of calibration errors, so the quality of either shortcut is not established across products or market conditions.
Key ideas
- A one-curve Hull–White formula may not match swaption prices built with OIS discounting and separate forwarding curves.
- A more consistent approach is to include the LIBOR–OIS spread in the model.
- An approximate price adjustment multiplies OIS-based prices by the ratio of LIBOR to OIS annuities.
- Alternatively, calibrate to implied volatilities obtained directly or from a two-curve vanilla model.
- These shortcuts are described as most suitable for at-the-money swaptions that are not very long dated.
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# Calibration Hull-White # Calibration Hull-White 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. ## Answer by piterbarg (score 3, accepted) https://quant.stackexchange.com/a/59601 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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