Problem 5 (American call with discrete dividend). (6 pts) For an American call option on a stock with only one discrete dividend, it is possible to derive a formula to determine whether early exercise is optimal. Let t be the time when the dividend is paid. T is the maturity time, and D is the cash amount of the dividend. S(t+)is the stock price right after the dividend is paid. S(t)is the stock price right before the dividend is paid. Thus, S(t)=S(t+)+D. At time t, we have two choices: a) No early exercise at time t. Then the stock right after t becomes non-dividend paying; b) Early exercise right before the dividend is paid. The value of the American call at time tshould be the maximum from these two choices. Show that the value is max{S(t+)+DK,Europeancallprice(strikeK,timetomaturityTt+,stockpriceS(t+))} Therefore, if choice b) generates a higher value, then we should early exercise