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?
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.
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.