When deriving the LIBOR-based swap rate formula in any interest rate model, expressions of the following types appear naturally:
Literature tells us that, switching to the – forward neutral measure, it is equal to:
where the expressions and represent the time-t forward and time-T spot Libor rates respectively.
On one hand, performing the change of measure using the time- Radon-Nikodym derivative leads directly to the desired result.
However, it seems to me that we are not entitled to do so because is defined only up to time T and it wouldn’t make sense to apply to a security that is no longer defined at time , would it?
On the other hand, performing the change of measure using the time-T Radon-Nikodym derivative makes more sense to me, but leads to the presence of terms that I don't know how to simplify in the thus obtained forward-neutral expectation.
Hence the following questions:
Should the change of measure be done using or ?
If it should be done using , how is it compatible with the fact that is no longer defined fot t > T?
If it should be done using , how can we simplify the expression ?
Thanks in advance for your help.