The no arbitrage bond price process is assumed to follow the m-dimensional Gaussian HJM model. Let t < T₀ < T₁ < ··· < T_n and α_i be the accrual factor over the interval (T_i−1,T_i],i = 1, 2, ··· ,n.

(a) A floor with preset floor rate L_f is a series of floorlets, where the payoff at T_i is α_i max(L_f − L_i−1(T_i−1), 0), i = 1, 2, ··· ,n. Find the time-t value of the floor.

(b) A caplet in arrears pays α_i max(L_i−1(T_i−1) − L_c, 0) at T_i−1, that is, the payment date coincides with the LIBOR reset date. Note that the usual caplet pays the same amount at T_i. Find the time-t value of this caplet.

(c) A cap with deferred caplets has all caplet payments made at the terminal date T_n. Find the time-t value of this deferred cap.