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?
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2 Answers
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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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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.
