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:

Borrow (S1 − K) at t = 1, agreeing to repay the loan at time t = 2.

AND Enter into a forward contract to sell the stock one year from now at time t = 2.
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).

(c) What are the cash flows at t = 1 and t = 2 for Strategy B if S2 ≤ K?

(d) 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. 

(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.