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Why are American-style options worth more than European-style options?
I'm sorry, but your math is wrong.  You are not equally likely to make as much money by waiting for expiration. Share prices are moving constantly in both directions.  Very rarely does any stock go either straight up or straight down.  Consider a stock with a share price of $12 today.  Perhaps that stock is a bad buy, and in 1 month's time it will be down to $10.  But the market hasn't quite wised up to this yet, and over the next week it rallies up to $15. If you bought a European option (let's say an at-the-money call, expiring in 1 month, at $12 on our start date), then you lost.  Your option expired worthless. If you bought an American option, you could have exercised it when the share price was at $15 and made a nice profit. Keep in mind we are talking about exactly the same stock, with exactly the same history, over exactly the same time period.  The only difference is the option contract.  The American option could have made you money, if you exercised it at any time during the rally, but not the European option - you would have been forced to hold onto it for a month and finally let it expire worthless. (Of course that's not strictly true, since the European option itself can be sold while it is in the money - but eventually, somebody is going to end up holding the bag, nobody can exercise it until expiration.) The difference between an American and European option is the difference between getting N chances to get it right (N being the number of days 'til expiration) and getting just one chance.  It should be easy to see why you're more likely to profit with the former, even if you can't accurately predict price movement.