[jason1990]

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These two properties are all one needs in order to conclude that it is never optimal to exercise the option before expiry.

This is wrong. Somtimes it is correct to exercise an option early. If it wouldn't be correct, then it would be easy to price american options.

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Let me just repeat myself:

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Just a point that might be of interest to some. The value of the call option when it is exercised is h(S(t)), where

h(x)=max{x-K,0}.

This function is convex and has h(0)=0. These two properties are all one needs in order to conclude that it is never optimal to exercise the option before expiry. Notice that the corresponding function for the put is still convex, but it is not 0 at 0.

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Nowhere did I contradict the fact that "somtimes it is correct to exercise an option early." So what are you saying?