Assume that the T-maturity discounted bond price process B(t,T) follows the one-factor Gaussian HJM under the risk neutral measure Q:

A caption is a call option on a cap, whose terminal payoff at time T is given by

Here, Ci(T ; T_i−1,T_i) is the time-T value of a caplet with payment on the LIBOR L_i−1 at time T_i and X is the strike price, i = 1, 2, ··· ,n. Since a cap can be visualized as a series of put options on the zero-coupon bonds, a caption is seen as a compound call on a put. By applying Jamshidian’s decomposition technique (Jamshidian, 1989) for an option on a coupon-bearing bond, find the time-t value of the caption, t < T.

Transcribed Image Text:

dB (t, T) B(t, T) =r, dt - oB(t, T) dZ₁.