Swaption annuity factor

Question

In H. Corb's book about interest rate swaps and oder derivatives, the present value of an T into n payer swaption is given via

$A\sigma\sqrt{T}\left[\frac{1}{\sqrt{2\pi}}e^{-\frac{d^2}{2}}+d\,\mathcal{N}(d)\right]$

where

$A=\sum\limits_{t=T+1}^{T+n}\mathrm{DF}(0,t)$

is the annuity factor.

I understand the sum in such a way, that it reflects the discounting of all cash flows following the first year after option expiration ($T+1$) up to the ending of the swap ($T+n$).

However, it strikes me that this might assume an interest rate swap with only annual cash flows. What about IR swaps with distinct and different leg frequencies, like semi-annual for the fixed leg and quartlerly for the floating leg.

Summarized, is the annuity factor the sum of all discounted cash flows?

Answer 1

To build intuition, let us consider the underlying swap itself rather than a swaption.

Conceptually, you can think of the swap annuity factor as the present value of gaining 1 unit every period of the underlying swap. Scaled appropriately, the swap annuity factor is the PV01, i.e. the Present Value of a Basis Point. Adjusting for convexity gives you the DV01, i.e. the Dollar Value of a Basis Point. It is highly relevant for the pricing of off-market swaps.

Consider a situation where you have previously entered into a payer swap. You can enter into a receiver swap at market (i.e. at the par swap rate) at 0 cost. All the floating cash flows will cancel out, and you will simply be left with a series of fixed cash flows of the differences in the fixed legs of your two swaps. Clearly, you can determine the current value of these fixed cash flows (and hence the value of the initial swap) by simply multiplying this difference with the swap annuity factor!

This formula should simply sum over all relevant discount factors. Hence, if you have semi-annual payments, you simply use semi-annual discount factors. The appropriate discount factors will always be the ones corresponding to the fixed payments, the floating ones cancel out in the above argument!