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 of the American and European call with strike K and maturity T on a stock with price (Sr20. Assume that the stock does not pay divi- dends. (1) Argue that for any is T,P(x(t, T) 2 Cx(t,T)) = 1. (11) Show that if the American option is not exercised before time T, then Cx(1,7) = C(1,1). (iii) Suppose that the American option is exercised at some time s 11,T). Show that Cx(s, T) = S, - KSS,-KZ(S, T) s CK(s, t). (1) where Z(s, T) denotes the value at s of ZCB maturing at T. (iv) Show that if C (1,7) > C (1,7), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that Cx(t,T) = Ck(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? 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 of the American and European call with strike K and maturity T on a stock with price (S)rzo. Assume that the stock does not pay divi- dends. (1) Argue that for any is T,P(x(t, T) 2 Cx(t,T)) = 1. (11) Show that if the American option is not exercised before time T, then Cxt,) = C(1,1). (iii) Suppose that the American option is exercised at some time se [1,T). Show that Cx(s, T) = S, - KSS,-KZ(T) s CK(s, t). (1) where Z(s, T) denotes the value at s of ZCB maturing at T. (iv) Show that if C (1,7) > C (1,7), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that C (1,1) = Ck(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? 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 of the American and European call with strike K and maturity T on a stock with price (Sr20. Assume that the stock does not pay divi- dends. (1) Argue that for any is T,P(x(t, T) 2 Cx(t,T)) = 1. (11) Show that if the American option is not exercised before time T, then Cx(1,7) = C(1,1). (iii) Suppose that the American option is exercised at some time s 11,T). Show that Cx(s, T) = S, - KSS,-KZ(S, T) s CK(s, t). (1) where Z(s, T) denotes the value at s of ZCB maturing at T. (iv) Show that if C (1,7) > C (1,7), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that Cx(t,T) = Ck(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? 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 of the American and European call with strike K and maturity T on a stock with price (S)rzo. Assume that the stock does not pay divi- dends. (1) Argue that for any is T,P(x(t, T) 2 Cx(t,T)) = 1. (11) Show that if the American option is not exercised before time T, then Cxt,) = C(1,1). (iii) Suppose that the American option is exercised at some time se [1,T). Show that Cx(s, T) = S, - KSS,-KZ(T) s CK(s, t). (1) where Z(s, T) denotes the value at s of ZCB maturing at T. (iv) Show that if C (1,7) > C (1,7), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that C (1,1) = Ck(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