Please someone correct me if I get anything wrong.
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For an Euro option, if it expires past the ex-dividend date, and the dividends is higher than expected, the value of the call would decrease because we cannot exercise to obtain the underlying asset’s cash flows.
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Other things equal, an American call should have a higher value than an Euro call because of the free right to exercise at anytime.
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For an American option, if the dividends increase (i.e higher than expected), the value of the call should also increase because we can exercise at anytime. Honestly, I’m not sure if it should increase or stays the same here. Either way, it would have a higher price than an Euro call.
It’s a matter of timing: when has the dividend increase occurred?
If the company declares a higher than expected dividend sometime before the ex-date, the price of both American and European style options should decline, but the American style option would likely decline by a smaller amount.
Early exercise (in time to receive the dividend) is not always optimal. Early exercise is optimal only if the time value of the dividend one would receive from early exercise (i.e. dividend plus the reinvestment income earned on it over the time remaining until expiration) is greater than the time value of the option (i.e. premium – the option’s intrinsic value).
If the time value of the option is greater than the time value of the dividend, early exercise results in throwing away more value than is gained from early exercise.