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Calibrating Bermudan Swaptions to Coterminal and Off-Diagonal Volatility

Article Quant Q&A · Author: Arshdeep
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Summary

The discussion explains why calibrating a Bermudan swaption model only to coterminal European swaptions may miss important market drivers. A Bermudan’s exercise value can depend on swap rates and swaption prices across many expiries and tenors, including off-diagonal instruments that are not direct coterminals.

For a Bermudan with multiple exercise dates, the answer illustrates how the range of remaining swap maturities at each date creates this broader dependency. One suggested compromise is to calibrate to coterminals plus short-tenor swaptions, then check whether the model interpolates the rest of the volatility surface reasonably. The responses note that a one-factor model may achieve this by adjusting mean reversion, while a LIBOR market model can also be used. The discussion is conceptual rather than a comparative empirical study; it gives no quantitative calibration results, and the adequacy of interpolation must be checked for the specific model and trade.

Key ideas

  • A Bermudan swaption’s value can depend on off-diagonal swaption volatilities as well as coterminal instruments.
  • The exercise decision at each date can involve a range of remaining swap maturities.
  • Calibrating only to coterminals may leave sensitivity to other market instruments unexplained.
  • Coterminals plus short-tenor swaptions are suggested as a practical calibration set.
  • Model interpolation across the remaining volatility surface should be checked.

Tags

Fixed incomeOptionsVolatilityDerivatives pricing
Full text
# Local v/s global calibration for a Bermudan Option (calibrate co-terminals vs entire matrix)


# Local v/s global calibration for a Bermudan Option (calibrate co-terminals vs entire matrix)












I am quite new to rates modeling and I have a question on the pros and cons of calibrating to larger set of vanilla instruments v/s calibrating to an exotic's 'natural' hedges. For example, I could value a Bermudan with a 1F model calibrating to co-terminal Europeans; or I could use a LIBOR market model to calibrate to the entire swaption matrix. What are the things to keep in mind while making this choice?

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/73529

Calibrating to coterminal swaptions only is not good enough, since we know that the value of a Bermudan swaption depends on a wider set. Consider a 5yr Bermudan receiver swaption exerciseable into a 25 year swap (thus a 30yr final maturity) with annual exercise dates from year 5 to year 29. Then a necessary condition for exercise at year 5 is that all of the 1yr, 2yr,…… 24yr swap rates are less than the strike K (otherwise it is better to receive a swap rate in the market and exercise the swaption later). Thus, there is some dependency on the set of swaptions 5yr into n yr (n=1 to 24). The same analysis applies to all the exercise dates, so we have dependency on the set (m yr into n year) where m= 5 to 24 and n = 1 to 29-m.

One common approach is to calibrate to the coterminals plus the swaptions where n=1 (for annually exerciseable swaptions). A good model will self-interpolate the rest of the swaption set in a reasonable way , although it needs to be checked. I believe that this calibration can be achieved even in a one factor model by varying the mean reversion parameter, and it obviously can be achieved in a Libor market model.

## Answer by JUW (score -1)

https://quant.stackexchange.com/a/72239

Firstly, we should acknowledge the fact that off-diagonal vols should be part of the drivers of the Berm value. Pls refer to the Bible: Andersen and Piterbarg's book.

By calibrating the model solely to co-terminal European swaptions, the model price for the Bermudan is a fun of co-terminal euro prices. Consider the scenario that the co-terminal swaption vols remain unchanged and while the off-diagonal vols vary a lot. Then your model price is invariant and the change of the Berm value caused by off-diagonal vols is attributed to unexplained P&L.

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