This problem will demonstrate (in a different way) that early exercise of an American call on a non-dividend-paying stock is never optimal. I would calculate interest using the simple interest formula, i.e $1 -> $1(1+rt) over a time period t as it makes the formulas easier. For simplicity, let us assume that the option was purchased at t = 0 and expires at time T = 2. At time t = 1, the option is in the money with S1 > K. Consider two strategies: Strategy A: Exercise the call option at t = 1. Strategy B: o Borrow (S1 K) at t = 1, agreeing to repay the loan at time t = 2. o AND Enter into a forward contract to sell the stock one year from now at time t = 2. o AND At time t = 2, exercise the option (if it is in the money), close the forward, and pay back the loan. (a) What are the cash flows at t = 1 and t = 2 for Strategy A? Note that once the option has been exercised there are no future cash flows. (b) What are the cash flows at t = 1 and t = 2 for Strategy B if S2 > K? Let r > 0 denote the one- year interest rate quoted at time t = 1, i.e. r(1,2). (Note: You should assume that the forward price has been correctly calculated as we learned in the first part of the course - and that there are no dividends here.) (c) What are the cash flows at t = 1 and t = 2 for Strategy B if S2 K? (d) Parts (a-c) show that the cash flows of Strategy B are always at least as big as those of Strategy A, and sometimes strictly bigger. Thus Strategy A cannot be optimal. What is the effect on this argument if the stock pays a large dividend at time 2. (Note: you will have to recalculate the fair forward price for when there is a dividend) (e) Use a similar approach to determine if it would ever be optimal to exercise an American put option early. You do not have to consider dividends here.