Calibrating Hull–White to Co-Terminal Swaptions for Bermudan Pricing

Summary

The document explains why co-terminal European swaptions are often used to calibrate the Hull–White interest-rate model when the intended application is pricing Bermudan swaptions. Each Bermudan exercise date is associated with a European swaption whose swap maturity ends on the same date. For a chosen mean-reversion speed, the model’s instantaneous volatility can then be calibrated to those market prices.

This setup makes the Bermudan price depend on the prices of the corresponding co-terminal European swaptions and on the selected mean-reversion parameter. The answer therefore gives a practical reason for the instrument choice: it connects calibration instruments to the exercise dates relevant to the Bermudan product. The document does not provide numerical examples, compare alternative calibration sets, or explain how to select the co-terminal maturity range. That choice remains tied to the product and intended use of the calibrated model.

Key ideas

• Co-terminal European swaptions align with the exercise dates of a Bermudan swaption.
• For a fixed mean-reversion speed, instantaneous volatility can be calibrated to the co-terminal swaption prices.
• This calibration links the Bermudan price to market prices of related European swaptions and the mean-reversion parameter.
• The appropriate maturity range depends on the Bermudan product being valued.

Tags

# Why co-terminal swaptions are that important?


# Why co-terminal swaptions are that important?












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

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/49557

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.