Forward Par Swap Rate yn,N(t) is defined

Pn+1,N(t) is called the present value of a basis point (PVBP).
A swaption gives the holder the right not the obligation to enter into a particular swap contract. A swaption with option maturity Tn and swap maturity TN is termed a Tn × TN-swaption. The total time-swap associated with the swaption is then Tn+ TN. A payer swaption gives the holder the right not the obligation to enter into a payer swap and can be seen as a call option on a swap rate. The option has the payoff at time Tn, the option maturity, of

where κ denotes the strike rate of the swaption. The second line follows directly from the definition of the forward swap rate. We let Bt = exp(ˆ«t0 rsds) be the money market account at time t. Assuming absence of arbitrage, the value of a payer swaption at time t

a. Use Pn+1,N(t) as a numeraire to find a new probability measure, Pn+1,N, that we call swap measure.
b. Under the swap measure show that

Note that under this swap measure the corresponding swap rate, yn,N(t), is a martingale. The change of numeraire shows explicitly why swaptions can be viewed as options on swap rates.

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P(t, T) P(t, TN) P(t, T)P(t, TN) +N(t) Payer j=n+1 n+1,NIn Bt BT In в,