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While studying for the CFA, I ran into the following statement in my study materials: "American style calls are only more valuable than European style calls IF the underlying asset pays cash flows, such as dividends or interest".

Lets imagine a stock call option. Even if the underlying is a no-dividend-paying stock, its price is still going to fluctuate, so that there is a higher chance that the American call could be exercised above the strike price than the european, since there is simply a higher chance that S is going to be higher than X on any given day during the period until expiration than ONLY on the day of expiration.

Could someone help me understand this? I realize there is a simmilar question in the forum, but it is not identical. Thank you.

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Really all you need to know is that American style can be exercised at any point, European options cannot be exercised early. Read on if you want more detail.

The American style Call is worth more because it can be exercised at any point. And when the company pays a dividend, and your option is in the money, if the extrinsic value is worth less than the dividend you can be exercised early. This is not the case for a European call. You cannot be exercised until expiration. I trade a lot of options, you wont be exercised early unless the dividend scenario I mentioned happens. Or unless the extrinsic value is nothing, but even then, unless the investor really wants that position, he is more likely to just sell the call for an equivalent gain on 100 shares of stock.