For example, if you bought a long call option (remember, a call option is a contract that gives you the right to buy shares later on) for 100 shares of Microsoft stock at $110 per share for December 1, you would have the right to buy 100 shares of that stock at $110 per share regardless of if the stock price changed or not by December 1. For this long call option, you would be expecting the price of Microsoft to increase, thereby letting you reap the profits when you are able to buy it at a cheaper cost than its market value. However, if you decide not to exercise that right to buy the shares, you would only be losing the premium you paid for the option since you aren't obligated to buy any shares. 
As shown above, a long options trade has unlimited profit potential, and limited risk, but a short options trade has limited profit potential and unlimited risk. However, this is not a complete risk analysis, and in reality, short options trades have no more risk than individual stock trades (and actually have less risk than buy and hold stock trades).
American options can be exercised at any time between the date of purchase and the expiration date. European options are different from American options in that they can only be exercised at the end of their lives on their expiration date. The distinction between American and European options has nothing to do with geography, only with early exercise. Many options on stock indexes are of the European type. Because the right to exercise early has some value, an American option typically carries a higher premium than an otherwise identical European option. This is because the early exercise feature is desirable and commands a premium.

In terms of valuing option contracts, it is essentially all about determining the probabilities of future price events. The more likely something is to occur, the more expensive an option would be that profits from that event. For instance, a call value goes up as the stock (underlying) goes up. This is the key to understanding the relative value of options.
The fee you are paying to buy the call option is called the premium (it's essentially the cost of buying the contract which will allow you to eventually buy the stock or security). In this sense, the premium of the call option is sort of like a down-payment like you would place on a house or car. When purchasing a call option, you agree with the seller on a strike price and are given the option to buy the security at a predetermined price (which doesn't change until the contract expires). 
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