Minimum Value of American Call Option?

Can anyone help explain this concept, I’m not quite understanding it. Why is it that the American Put minimum value is Max[0, X-S] whereas the minimum value for an American Call is Max[0,S- x/(1+Rf)^t] (as opposed to Max[0,S-X]? Thanks for the help.

American option must be worth at least as much as the european option and will usually be more valuable due to the early exercise option. Since the strike price of european call is discounted to present value terms, the american option must similarly be brought to this minimum value.

Since S - X < S - X/(1+rf)^t the inequality you give C > Max(0, S-X) is also true, it’s just that with a little thought you can do better.

yea since American call options are FLEXIBLE they oughta value MORE than EUROPEAN ones But using the discount factor [@ RFR], lower bound for EUROPEAN ones is stuck at: S - X/(1+rfr)^t So the min value for the American ones too has to rise ‘up’ to the same [since S - X < S - X/(1+rfr)^t … as mentioned above]

Ok that makes sense, thank you.