(Swap frequency) We have assumed that the payment dates for the fixed and floating legs of a swap contract are the same. However, in practice the frequency may differ for example, in the US swap market the fixed leg is usually semi-annual ( = 0.5), versus a floating let of three-month LIBOR paid quarterly ( = 0.25).

(a) Prove that the theoretical value of a swap is independent of the floating frequency for example, three-month LIBOR paid quarterly has the same value as six-month LIBOR paid semi-annually.

(b) Prove that the value of a swap with fixed rate K is not indepen- dent of the fixed leg frequency. In particular, does Pt[T0,Tn] increase or decrease if changes from 0.25 to 1? Deduce directly whether the annual swap rate for T0 to Tn is higher or lower than the quarterly swap rate for the same period.