Problem 4. Consider an arbitrage free bond market with term structure P(t,T), and let 0 t s < T. A forward contract negotiated at time t delivers a T-bond at time s. The equivalent continuously compounded rate of return over the interval [s,T] is denoted by f(t,s,T).

(i) What is the price of the forward contract at time t to deliver a T-bond at time s.

(ii) Show that the equivalent continuously compounded rate of return of the forward contract is given by f(t,s,T) = logP(t,T) logP(t,s) / Ts

A forward rate agreement (FRA) with principal K has a cash flow of K at time s and K expR(T s) at time T, where the rate R is agreed upon at time t.

(iii) Determine the present value (t) of the FRA-agreement in terms of zero coupon prices for 0 t s.

(iv) Prove that (t) = 0, if and only if R = f(t,s,T).