An American option is a financial contract that gives its holder a choice to purchase or sell a financial asset at a specified exercise price at any time before the specified expiry date.
American option entitles its holder to discretion not only in exercising his option, but also in the timing of such exercise. European option on the other hand, does not allow flexibility in timing of exercise.
Reference to America and Europe in option names is just by convention and there is no connection between the options and any geographical location.
Since an American option is more flexible than a European option, it has higher value. In practice, value of an American option is taken as at least equal to the value of equivalent European option.
Value of American Option ≥ Value of European Option
Example 1:American Call Option
On 23 July 20Y3, Dona Amati, a trader with a large brokerage house bought 100 American call options (or simply American calls) on BP Plc (NYSE:BP) stock. The option has an exercise price of $42 and expiry date of 27 July 20Y3. She believes that BP price on 24th, 25th and 26th of July is expected to be $43.5, $44.5 and $43. Assuming she is very confident in her projections, what is the maximum she can gain on the options and when should she exercise them? Hint: ignore time value of money, call option value = max [0, price of the underlying stock – the option's exercise price].
Since Dona bought American options, she can exercise them at any time before 27th. Based on the projections:
Value on 24th = max [0, $43.5 – $42] = $1.5
Value on 25th = max [0, $44.5 – $42] = $2.5
Value on 26th = max [0, $43 – $42] = $1
She should exercise the options on 25th and gain $2.5 per option.
Had she bought European options, she would have been able to exercise them only on 26th July 20Y3 for a gain of $1 per option.
Example 2: American Put Option
Dona also bought 50 American put options with exercise price of $60 on Discover Financial Services (NYSE: DFS) stock. They are due to expire in 2 days. If BP share price is expected to be $58 and $61 tomorrow and day after tomorrow respectively, when it would be profitable to exercise the options?
Hint: gain on a put option = max [0, exercise price – price of the underlying stock].
Gain when exercised tomorrow = max [0, $60 – $58] = $2
Loss when exercised day after tomorrow = max [0, $60 – $61] = 0
If Dona is correct in her projections, the best strategy is to cash out $2 per option gain tomorrow. The options will be worthless day after tomorrow.