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 xt,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):20. Assume that the stock does not pay divi- dends. I (1) Argue that for any ts T, Pax(t,T) 2 Cx (6,1)) = 1. (ii) Show that if the American option is not exercised before time T, then x(1, 1) = Cx(1,T). (lil) Suppose that the American option is exercised at some time se 11,7). Show that x(s, T) = S, - KSS, - KZ(s, t) Cx(t, T), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that x(t, T) = Cx(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 t of the American and European call with strike K and maturity T on a stock with price (S1):20. Assume that the stock does not pay divi- dends. I (1) Argue that for any ts T,POx(t, T) 2 Cxt,)) = 1. (ii) Show that if the American option is not exercised before time 7, then x(1,7) = Cx(t, T). (iii) Suppose that the American option is exercised at some time se (1,7). Show that x(s, T) = S, -Ks S.-KZ(s, 1) s Cx(s, ). (1) where Z(s, T) denotes the value at s of ZCB maturing at T. (iv) Show that if x(1,7) > Cx(1, 1), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that x(t, T) = Cx(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 xt,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):20. Assume that the stock does not pay divi- dends. I (1) Argue that for any ts T, Pax(t,T) 2 Cx (6,1)) = 1. (ii) Show that if the American option is not exercised before time T, then x(1, 1) = Cx(1,T). (lil) Suppose that the American option is exercised at some time se 11,7). Show that x(s, T) = S, - KSS, - KZ(s, t) Cx(t, T), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that x(t, T) = Cx(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 t of the American and European call with strike K and maturity T on a stock with price (S1):20. Assume that the stock does not pay divi- dends. I (1) Argue that for any ts T,POx(t, T) 2 Cxt,)) = 1. (ii) Show that if the American option is not exercised before time 7, then x(1,7) = Cx(t, T). (iii) Suppose that the American option is exercised at some time se (1,7). Show that x(s, T) = S, -Ks S.-KZ(s, 1) s Cx(s, ). (1) where Z(s, T) denotes the value at s of ZCB maturing at T. (iv) Show that if x(1,7) > Cx(1, 1), then one can create an arbitrage opportunity. (Consider two cases when the American option is exercised before T or not.) Conclude that x(t, T) = Cx(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
