Thanks for this clarification. I still have a couple of questions:
1). I am not sure if I understand why you say the "relevant forward period is identical". For the swaption case, the vol is the forward swap rate in one years time (relative to today) for a future two year period [1, 3]. For the cap case, we have two forward rates, one expiring at one year for the period [1,2], the other expiring in two years for the period [2,3]. Is this right?
2). I can intuitively understand why swap rate is an average of forward libors. However, how can I show this mathematically. I am asking because the swap rate formula is a ratio of zero coupon bond prices:
swap rate = ( P(t, Ta) - P(t, Tb) ) / sum ( tau * P(t, Ti) )