VANILLA OPTIONS : EARLY EXERCISE OF AMERICAN OPTIONS

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European options can only be exercised on their expiration date, but American options allow for early exercise. 

In practice most options are either exercised on their expiration date or expire worthless.

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It can be shown that American call option exercise is not optimal before expiration (in the absence of dividends).

The intuition is that there is always a positive probability that the underlying will fall below the exercise price before expiration, and exercising the option early foregoes that value.

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For American calls written on dividend paying stocks, the expected drop in price of the stock when a dividend is paid must outweigh the time value of the call to preclude optimal exercise. This condition is most likely for deep in the money options, options with high dividend yields, or options that are close to expiring.

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Optimal exercise of American puts is possible. The intuition is that the time value of money for the in the money value of a put  (the early exercise payout) can outweigh the time value (the remaining optional value) of the option.

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Picture an underlying stock trading at zero prior to expiration, in which case the American put has reached its maximum potential payout. In this case receiving the payout today is always worth more than receiving the payout in the future, because the money received can be invested and receive a positive return.