This page explains differences between American and European options and their prices. We will also discuss the origin of these terms, which most sources don't mention.
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Main Difference: When They Can Be Exercised
European options can be exercised only at expiration.
American options can be exercised at any time from the moment you buy the option until its expiration.
Consider two call options on a stock. Both have the same strike (let's say $50) and same expiration date (let's say four weeks from now). One option is American and the other is European.
If the stock is currently trading at $48 and goes up to $55 in two weeks (with two more weeks left until expiration), you can exercise the American option (you get the stock for the strike = $50, can immediately sell it for $55 and make $5 per share, minus initial cost of the option). Or you can wait and hope the stock will increase further, making the exercise even more profitable. You can exercise at any time.
On the contrary, you can exercise the European option only at expiration (if the stock stays above the strike price).
American vs. European Option Prices
American options are generally more valuable than European options.
As Samuelson  first observed in 1965:
Obviously, the additional American option of early conversion can do the owner no harm, and it may help him.
An American option is never worth less than an otherwise identical (same underlying, same expiration date, same strike) European option, because it gives you the same rights plus more.
The difference between American and European price equals the value of the right to early exercise.
The Value of Early Exercise
Very often the right to early exercise is not worth much, and American and European option price is almost the same.
When you choose to exercise an option and there is still time left to expiration (like in our above example when the stock was at $55 with two weeks left to expiration), you are giving up the option's remaining time value.
For instance, the option price in our example could have been $7 ($5 intrinsic value + $2 time value).
You can do two things:
- Exercise the option, get the stock for $50 (strike price) and sell the stock in the market for $55. Profit $5 (minus initial cost of the option).
- Sell the option for $7 in the options market. Profit $7 (minus initial cost).
The second is more profitable, because in the first scenario you are giving up the option's remaining time value of $2.
The only case when early exercise is more profitable than selling the option is when the option's remaining time value is negative.
That is far less common than positive time value, but more common than most people think.
Typical cases when time value can be negative are some deep in the money put options (especially when time to expiration is relatively short), or call options on underlyings with high yield (e.g. high dividend stock or high interest currency).
Black-Scholes Model: Only for Some American Options
The best known option pricing model, Black-Scholes(-Merton), does not consider early exercise. It prices each option as if it were European. It is therefore inaccurate for some American options.
It is always accurate for American call options on underlying assets with no yield (no dividend stocks). These can't have negative time value.
For American put options or for American calls with dividends, Black-Scholes may or may not be accurate (depending on several inputs and their relationships, including time to expiration, yield, interest rate, and the option's moneyness).
Binomial Option Pricing Models
To accurately price all American options, it is better to use a binomial model (such as Cox-Ross-Rubinstein, Jarrow-Rudd, or Leisen-Reimer).
In binomial models, time to expiration is divided into a certain number of discrete steps.
- When pricing an American option, at each step you can check whether exercising the option is profitable at that moment (intrinsic value is greater than calculated option price at that step, or in other words, time value is negative).
- When pricing a European option, you simply don't do these checks.
This makes binomial models accurate for both European and American options. The logic and exact formulas are explained in the Binomial Option Pricing tutorials.
Which Options Are American / European
The difference between American and European options has nothing to do with geography. Either style can be traded on any continent.
Most exchange-traded equity (stock and ETF) options are American style. This includes most equity options traded in the US, Europe, Australia, India, China, and most other major markets. A notable exception is Japan, where stock and ETF options are European style.
Most exchange-traded index options are European style. There are again exceptions. OEX (S&P100) index options are American style, while SPX (S&P500) options are European style. VIX options are European style.
Options traded over-the-counter (outside an exchange) are mostly European.
Origin of the Terms American and European Options
The terms American and European option were first used by Paul A. Samuelson in his paper Rational Theory of Warrant Pricing (1965) .
Another paper by Robert Jarrow and Philip Protter (2004)  mentions how Samuelson arrived at these names:
This is the paper that first coined the terms "European" and "American" options. According to a private communication with R. C. Merton, prior to writing the paper, P. Samuelson went to Wall Street to discuss options with industry professionals. His Wall Street contact explained that there were two types of options available, one more complex – that could be exercised any time prior to maturity, and one more simple – that could be exercised only at the maturity date, and that only the more sophisticated European mind (as opposed to the American mind) could understand the former. In response, when Samuelson wrote the paper, he used these as prefixes and reversed the ordering.
Although this explanation sounds like a joke and there might have been other reasons, the story has also been told by the late Paul Samuelson himself in an interview he gave to the American Finance Association in 2004 (video time 11:00-11:30):
... Talking to still another one of those guys... He said: I don't understand, why are you here, what are you up to? I said I'm trying to study the science of option pricing. He said: That's hopeless. You'll never succeed. And I said: Why not? He said it takes a European kind of mind. ... So, in revenge, I gave the name European option to the simpler option.
Though primary sources are scarce, it is likely that privilege trading in the US was present from the late 18th century beginnings of trade in securities, perhaps earlier in the produce markets. Over time, this trade developed differently from Europe due to differing settlement practices.
In the US, "each day is a settling day and a clearing day for transactions of the day before ... This is a marked difference from European practice" where "trading for the account" (prolongationsgeschäfte) involves monthly or fortnightly settlement periods with allowance for continuation of the position until the next settlement date (Emery 1896, p.82).
The continuation process for a buyer seeking to delay delivery involves the immediate sale of the stock being delivered and the simultaneous repurchase for the next settlement date. As this transaction would involve the lending of money, an additional "contango" payment would typically be required.
As a consequence of these settlement differences, in the US (American) options developed with fixed exercise prices, possible exercise prior to delivery and premiums paid in advance. In Europe, premiums for (European) options would be due on the scheduled future delivery date which coincided with a regular settlement date, exercise could only take place on the delivery date and the exercise price would be adjusted to determine a market clearing "price" for the option at the time of purchase.
It is unclear whether Samuelson was aware of this and whether it played a role in his use of the terms.
 Emery, Henry C.; Speculation on the Stock and Produce Exchanges of the United States, Columbia University Press, New York (1896); reprinted by AMS Press, New York (1968).
 Jarrow, Robert and Protter, Philip; A Short History of Stochastic Integration and Mathematical Finance: The Early Years, 1880-1970, Institute of Mathematical Statistics Lecture Notes, vol. 45 (2004), pp. 75-91.
 Poitras, Geoffrey; The Early History of Option Contracts, Chapter 18 in W. Hafner and H. Zimmermann (eds.); Vincenz Bronzin's Option Pricing Models: Exposition and Appraisal, Springer-Verlag, New York (2009).
 Samuelson, Paul A.; Rational Theory of Warrant Pricing, Industrial Management Review, Vol. 6, No. 2 (Spring 1965), pp. 13-31.