When you buy an option, the risk is limited to the premium that you pay. Selling an option is the equivalent of acting as the insurance company. When you sell an option, all you can earn is the premium that you initially receive. The potential for losses is unlimited. The best hedge for an option is another option on the same asset as options act similarly over time.
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.
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The less time there is until expiry, the less value an option will have. This is because the chances of a price move in the underlying stock diminish as we draw closer to expiry. This is why an option is a wasting asset. If you buy a one-month option that is out of the money, and the stock doesn’t move, the option becomes less valuable with each passing day. Since time is a component to the price of an option, a one-month option is going to be less valuable than a three-month option. This is because with more time available, the probability of a price move in your favor increases, and vice versa.
Options belong to the larger group of securities known as derivatives. A derivative's price is dependent on or derived from the price of something else. As an example, wine is a derivative of grapes ketchup is a derivative of tomatoes, and a stock option is a derivative of a stock. Options are derivatives of financial securities—their value depends on the price of some other asset. Examples of derivatives include calls, puts, futures, forwards, swaps, and mortgage-backed securities, among others.

A call option is a contract that gives the investor the right to buy a certain amount of shares (typically 100 per contract) of a certain security or commodity at a specified price over a certain amount of time. For example, a call option would allow a trader to buy a certain amount of shares of either stocks, bonds, or even other instruments like ETFs or indexes at a future time (by the expiration of the contract). 
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