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Usually Hull & White is calibrated to co-terminal swaptions. When asking why specifically co-terminal, I get the response that it is just a choice and it depends on the use we intend to do with the calibrated Hull&White.

But as I read more materials on the subject, the co-terminals come up almost every time when calibration of the Hull White is discussed.

I am sure this is not just a coincidence. Can anyone help me understand why we pay so much attention to coterminal swaptions for calibration?

Also how does one choose to use 20Y-coterminal, 10Y-coterminal ... or any other maturity? what is the reasoning behind?

Thank you

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Hull & White is often use to value Bermudan swaptions, given a market for European swaptions. The idea is, at given mean reversion speed, to calibrate the instantaneous volatility to the set of coterminal european swaptions that correspond to each Bermudan exercise date. Hence the Bermudan swaption price becomes a function of its coterminal European swaptions prices and a single parameter, the mean reversion speed.

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  • $\begingroup$ Nice answer. Can I pop a follow up question, since this is very interesting to me. Assuming you'd calibrate this model (so mean rev and vol structure) using the coterminals of same strike of the Bermuda (assume they are liquid enough), would model prices be in line with market prices of payer vs. receiver bermudans? Intuitively, I would say no since their exercise boundaries would involve fitting the swaption smile. But maybe this effect is negligible in practice? Or maybe since you can't fit any perfectly you reach some kind of tradeoff? $\endgroup$ – Quantuple Nov 7 '19 at 14:27
  • $\begingroup$ As you said I don't think a single mean rev would fit all, not even bermudan receiver/payer swaptions on the same swap. But given market prices for bermudans one can estimate a mean rev and then use the model to compute european coterminals vega hedges $\endgroup$ – Antoine Conze Nov 8 '19 at 11:41
  • $\begingroup$ Hi @Antoine Conze. Thanks for the answer, sorry for the late reply. So basically I would both a basket of both payer/receiver bermudans to calibrate the mean reversion? In other words with a single mean rev, the model is too simplistic to allow me to fit both, so I'd best try to minimise the error made on both simultaneously? $\endgroup$ – Quantuple Nov 28 '19 at 8:24
  • $\begingroup$ Yes or use two different mean revs. I don't think this will create an arbitrage, or at least not an easy one to detect. $\endgroup$ – Antoine Conze Nov 28 '19 at 15:21

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