Exercise . Show that the no-arbitrage price of a European call on a zero-coupon bond will satisfy max 

, BS t − KB T t



≤ CK,T,S t ≤ BS t ( − K)

provided that all interest rates are non-negative. Here, T is the maturity date of the option, K is the exercise price, and S is the maturity date of the underlying zerocoupon bond. Compare with the corresponding bounds for a European call on a stock, see, for example (Hull, , Ch. ). Derive similar bounds for a European call on a coupon bond.

Exercise . Show the put–call parity for options on coupon bonds by a replication argument, that is form two portfolios that have the same payoffs and conclude from their prices that (.) must hold.