Assume that the 1-year, 2-year, 3-year, and 4-year zero rates are respectively r1 = 3%, r2 = 3.4%, r3 = 3.7%, and r4 = 3.9% with continuous compounding. Thus, the price of ZCB Z(0, t) are given by Z(0, t) = e ^(rt) for 1 t 4.

(1) Consider a European receiver swaption that gives the holder the right to enter into a 3-year receiving swap in a year, i.e., a 1-year into 3-year (or a 1-year by 4-year) receiver swaption, where a fixed rate (strike) of 4.5% is received and floating is paid. The principal is $1000 and payments are exchanged annually.

(a) Determine the forward swap rate y0[1, 4].

(b) Assume the swap rate volatility is 30%, apply the Blacks formula to calculate the price of the receiver swaption.

(2) Assume that the floor rate (strike) is also 4.5%. The principal underlying the floor is $1,000 and the reset frequency is 1 year. Assume that the volatility is 30% for all floorlets. Determine the price of floor from 1 to 4 years. Observe that the price of floor is larger than that of the receiver swaption obtained in the last problem