Exercise 1. In this exercise, we will show that the value of an American call and European call on a stock without dividend are the same. Let x(t, T) and Cx(t, T) denote the value at time t of the American and European call with strike K and maturity T on a stock with price (S1)120. Assume that the stock does not pay divi- dends. (1) Argue that for any tst, Plk(t,T) 2 Ck(t, T)) = 1. (ii) Show that if the American option is not exercised before time T, then k(t, T) = Cx(t, T). (iii) Suppose that the American option is exercised at some time se [t,T). Show that x(s, 1) = Ss-KsSs-KZ(s, T) s Cx(s, t), (1) where Z(s, T) denotes the value at s of ZCB maturing at T. (iv) Show that if k(t, T) >Ck(t, T), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that k(t, 1) = Cr(t, T). Why does this result makes sense? (v) Suppose the stock pays dividends. Is it still true that the American and European call options have the same price