anilla swap agreement has positive cashflows at times
calculated as
and negative cashflows at times
calculated as
where
.
The number
is some predetermined fixed rate. The swap rate
is the particular value of the parameter
that makes such contract of zero value at the time
.
Hence, the
is defined by the
relationship
Since
is a martingale with respect to the
forward
measure we
continue
Hence,
Observe
that
Therefore,


(Swap rate)

It is useful to express price of swap with any parameter
through the swap rate. Similarly to the above computations we
have
