Bermudan Swaption

Question

Is there an equation of the kind of call-put parity for Bermudean swaptions ? (maybe an inequality )

Is there an intuitive description of what would be an optimal exercise moment ? Intuitively I would say it is when the swaption worths less than the underlying swap. When such condition arises ?

Can we assume that if it is not optimal to exercise the payer swaption then the receiver swaption exercise is optimal. (Intuitively if such preposition is true than the call put parity exists)

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EDIT

An other question, let say I have a callable payer swap by both counterpaties. Hence I am long a payer swap + receiver bermuda and short a payer bermuda to my cpty. Is the fact that if one cpty exercises the call, the option of the other will definitly expire worthless, allows me to say that there is call-put parity in this case and I am in a position of a payer swap and forward receiver ? In a nutshul I have all in all a short maturity swap with maturity equals the first exercise date ? Or there is still a world state where it is not optimal to exercise for both cpties ?

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Answer 1

There is no put call parity for Bermudan swaptions. There are some necessary (but not sufficient ) conditions for exercise of a Bermudan swaption. For example , consider a Bermudan receiver option exerciseable every year into a swap with remaining maturity of 10 years. Then for optimal exercise it is necessary that the spot starting swap rate with maturity i, for all i = 1,2,3,...10 is less than the fixed rate on the swap. For a proof and further details , consult Blyth “An Introduction to Quantitative Finance” , chapter on Bermudan swaptions.