European Option
European Options can only be exercised on the expiry date. European options are typically valued using the BlackScholes or Black model formula. This is a simple equation with a closedform solution that has become standard in the financial community.
American Option
This is an option that can be exercised at any time up to and including the expiry date. There are no general formulas for valuing American options, but a choice of models to approximate the price is available (for example Whaley, binomial options model, Monte Carlo and others), although there is no consensus on which is preferable.
American options are rarely exercised early. This is because all options have a nonnegative time value and are usually worth more unexercised. Owners who wish to realize the full value of their options will mostly prefer to sell them rather than exercise them early and sacrifice some of the time value.
Note that the names of these types of options are in no way related to Europe or the United States.
Moneyness
The concept of moneyness describes whether an option is in, out, at, or inthemoney by examining the position of strike vs. existing market price of the option's underlying security.
 In the Money  Any option that has intrinsic value is in the money. A call option is said to be in the money when the futures price exceeds the option's strike price. A put is in the money when the futures price is below the option's strike price. For example, a March CME euro 90 call option will be in the money if March CME euro futures are above 90, meaning that the holder has the right to buy these futures at 90, regardless of how much the price has risen. The further in the money an option, the less time value it will have.
 Deep In the Money  These options represent a larger spread between the strike and market price of an underlying security. Options that are deep in the money generally trade at or near their actual intrinsic values, calculated by subtracting the strike price from the underlying asset's market price for a call option (and vice versa for a put option). This is because options with a significant amount of intrinsic value built in have a very low chance of expiring worthless. Therefore, the primary value they provide is already priced into the option in the form of their intrinsic value. As an option moves deeper into the money, the delta approaches 100% (for call options), which means for every point change in the underlying asset's price, there will be an equal and simultaneous change in the price of the option, in the same direction. Thus, investing in the option is similar to investing in the underlying asset, except the option holder will have the benefits of lower capital outlay, limited risk, leverage and greater profit potential.
 Out of the Money  These options exist when the strike price of a call (put) is above (below) the underlying asset's market price. (Essentially, it is the inverse of an in the money option). Options that are out of the money have a high risk of expiring worthless, but they tend to be relatively inexpensive. As the time value approaches zero at expiration, out of the money options have a greater potential for total loss if the underlying stock moves in an adverse direction.
 At the Money  These options exist when the strike price of a call or put is equal to the underlying asset's market price. You can essentially think of at the money as the breakeven point (excluding transaction costs).
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Look Out!
Note that the above forms of moneyness do not take the cost of the option contract, or premium, into account.
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 Payoff  Calculated by deducting the option premium paid from the intrinsic value of the option. In this case, an inthemoney option could produce a negative payoff if the premium is greater than the intrinsic value of the option.
 Intrinsic Value  Intrinsic value in options is the inthemoney portion of the option's premium. It is the value that any given option would have if it were exercised today. It is defined as the difference between the option's strike price (X) and the stock's actual current price (CP). In the case of a call option, you can calculate this intrinsic value by taking CP  X. If the result is greater than zero (in other words, if the stock's current price is greater than the option's strike price), then the amount left over after subtracting CP  X is the option's intrinsic value. If the strike price is greater than the current stock price, then the intrinsic value of the option is zero  it would not be worth anything if it were to be exercised today (please note that an option's intrinsic value can never be below zero). To determine the intrinsic value of a put option, simply reverse the calculation to X  CP.

Time Value  The time value is any value of an option other than its intrinsic value. Time value is basically the risk premium that the seller requires to provide the option buyer with the right to buy or sell the stock up to the expiration date. While the actual calculation is complex, fundamentally, time value is related to a stock's beta or volatility. If the market does not expect the stock to move much (if it has a low beta), then the option's time value will be relatively low. Conversely, the option's time value will be high if the stock is expected to fluctuate significantly.
Time value decreases as an option gets closer and closer to expiration. This is why options are considered "wasting" assets. As an option approaches expiration, the underlying stock has less and less time to move in a favorable direction for the option buyer; therefore, if you have two identical options  one that expires in six months and one expires in 12 months  the option that expires in 12 months will have greater time value because it has a better chance of moving higher.

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