For a non-dividend paying European option, the option premium, \(c_t\), is always greater than \(S_t-Ke^{-r\tau}\) where \(\tau=T-t\). This can be shown with some no-arbitrage arguments.

Now for an American option, since it offers everything a European option does and a bit more, the premium at any time should be at least that of the European option, that is \(C_t \geq S_t-Ke^{-r\tau}\). Early exercise of the non-dividend paying option would give a pay off of the intrinsic value \(S_t-K\) but the holder could just sell the option instead for at least \(C_t \geq S_t-Ke^{-r\tau}\) or another way of looking at it is that the option has value of \(C_t \geq S_t-Ke^{-r\tau}\). Since \(C_t \geq S_t-Ke^{-r\tau} >S_t-K\), its value is always greater than the intrinsic value.

So it would always make sense to just hold the option because it has value that is always greater than the intrinsic value.

edited May 31 by ErichMaritz

100%.

You can refer you to chapter 12, p23 (from 5.1 onwards) and p12.26 (from the top) for the textbook answer, which is similar to the above.

Another angle is that when exercising early (paying from your cash and holding the received equity in equity), you pay the K early. There is no advantage of getting S_t early, as it turns into S_T eventually. So not exercising early protects you against paying early and against S_T becoming smaller than K.

But what if I exercise early and transfer the money into cash? What you do with your investments does not change its current value or price (i.e. the fact that we keep equity in equity and cash in cash and get one portfolio as being inferior to another, is a valid method).