# Minimum Value of an American Call Option

I understand that the minimum value of an American Put Option is Max (0,X-St) But was hoping someone could shed some light on that reason that the minimum value of an American Call is Max(0, St - X(1+RFR) ^T-t rather than simply Max (0,St-X)

I understand why you are saying Max (0,St-X) as against (0, St - X(1+RFR) ^T-t for mimimum value of AMERICAN Call Option, because you are thinking an AMERICAN Option can be exercised any time during the strike period. But, take it this way. Given that you can buy a stock at a FIXED price of \$30, either today or 3 months from now. What would you choose? You would rather choose to buy it 3 months from now. Because, you would rather keep that money with you and earn some interest. That is why, even though you have a choice of buying it today, you will postpone it till the maximum allowed time. The general explanation is, when you have to SPEND money (on a non-consumable item), you would rather spend it LATER than sooner and when you have to GET money, you would rather get it SOONER than later. Hope this clarifies.

A European Call Option’s min value is Max(0, St - X(1+RFR) ^T-t. No question about that. Comparing an American Call to European Call, the former will be worth at least as much as the latter, ceteris paribus, due to the ability of exercising any time. Hence, AC’s min value must not be less than EC’s min value. Max(0, St - X(1+RFR) ^T-t > Max (0,St-X)

Yes revenant. Perfectly correct.

But if buying an \$30 gives the good pay off i.e. lets say when the stock is trading at \$60, you would like to by at \$30 and then sell it immediately