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) )

Thanks!

Perhaps, think of a swap like this:

**Floating leg**
[$] PV_{float}(t) = \sum_{i=1}^{n}{NF(t,T_{i-1}) \tau_{i} DF(t,T_{i})}[$]

where, [$]F(t,T)[$] is the index forward rate(some IBOR rate) and [$]DF(t,T_{i})[$] = the discount factor from [$]T_{i}[$] to [$]t[$]. So, the swap rate is an economic equivalent of what the market thinks should be the value of a string of IBOR forwards.

Let's make the simplifying assumptions :

(1) The duration of the LIBOR deposit matches the coupon period.

(2) Index-forward rates are equal to discount forward rates.

Denoting the discount forward rate between the period [$](T_{i-1},T_{i})[$] by [$]R(t,T_{i-1},T_{i})[$],

[$]F(t,T_{i-1})=R(t,T_{i-1},T_{i})=\frac{1}{\tau_{i}}\left(\frac{DF(t,T_{i-1})}{DF(t,T_{i})}-1\right)[$]

Substituting this, we would get a formula similar to what you stated:

[$] PV_{float} = NDF(t,T_{0})-NDF(t,T_{n})[$]

Maybe that helps. Instead of using any of the above formulae, it's nice to price a few swaps on Excel, make a schedule for the fixed leg and the floating legs and try to find the par swap rate, to get an intuitive feel.

Post 2008, the simplifying assumptions no longer hold. Swap pricing is far more complex.