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In finance options are types of derivative contracts, including call options and put options, where the future payoffs to the buyer and seller of the contract are determined by the price of another security, such as a common stock. More specifically, a call option is an agreement in which the buyer (holder) has the right (but not the obligation) to exercise by buying an asset at a set price (strike price) on (for a European style option) or not later than (for an American style option) a future date (the exercise date or expiration); and the seller (writer) has the obligation to honor the terms of the contract. A put option is an agreement in which the buyer has the right (but not the obligation) to exercise by selling an asset at the strike price on or before a future date; and the seller has the obligation to honor the terms of the contract.

Since the option gives the buyer a right and the writer an obligation, the buyer pays the option premium to the writer. The buyer is considered to have a long position, and the seller a short position. For every open contract there is a buyer and a seller. Traders in exchange-traded options do not usually interact directly, but through a clearing house such as, in the U.S., the Options Clearing Corporation (OCC) or in Germany and Luxemburg Clearstream International. The clearing house guarantees that an assigned writer will fulfill his obligation if the option is exercised. Options/Derivatives are not rated and/or are below investment grade; however the OCC's clearing process is considered AAA rated.

The contract specifies
whether it is a put option or call option. Put options give the holder the right to sell the asset at the strike price. Call options give the holder the right to purchase the asset at the strike price.
the underlying security (e.g. XYZ Co.)
the strike price or exercise price. It can be specified, or based on a reference rate, or at a strike price 54.00 to expire Dec 2007 with a multiplier of 100. 1 JNJ Dec Call 54

Types of options
Real option (real option) is a choice that an investor has when investing in the real economy (i.e. in the production of goods or services, rather than in financial contracts). This option may be something as simple as the opportunity to expand production, or to change production inputs. Real options are an increasingly influential tool in corporate finance. They are typically difficult or impossible to trade, and lack the liquidity of exchange-traded options.
Traded options (also called "Exchange-Traded Options" or "Listed Options") is a class of Exchange traded derivatives. As for other classes of exchange traded derivatives, trade options have standardized contracts, quick systematic pricing, and are settled through a clearing house (ensuring fulfillment). Trade options include
stock options, discussed below,
commodity options,
bond options,
interest rate options
index (equity) options,
currency cross rate options, and
swaption.
Vanilla options are 'simple', well understood, and traded options; Exotic options are more complex, or less easily understood. Asian options, lookback options, barrier options are considered to be exotic, especially if the underlying instrument is more complex than simple equity or debt.
Employee stock options (employee stock option) are issued by a company to its employees as compensation.

Valuation
The premium for an option contract is ultimately determined by supply and demand, but is influenced by five principal factors:

The price of the underlying security in relation to.
The strike price. Options will be in-the-money when there is a positive intrinsic value; when the strike price is above/below (put/call) the security's current price. They will be at-the-money when the strike price equals the security's current price. They will be out-of-the-money when the strike price is below/above (put/call) the security's current price. Options at-the-money or out-of-the-money have an intrinsic value of zero.
The cumulative cost required to hold a position in the security (including interest + dividends).
The time to expiration. The time value decreases to zero at its expiration date. The option style determines when the buyer may exercise the option. Generally the contract will either be
American style — which allows exercise up to the expiration date — or
European style — where exercise is only allowed on the expiration date — or
Bermudan style — where exercise is allowed on several, specific dates up to the expiration date.
European contracts are easier to value. Due to the "American" style option having the advantage of an early exercise day (i.e. at any time on or before the options expiry date), they are always at least as valuable as the "European" style option (only exercisable at the expiration date).

The estimate of the future volatility of the security's price. This is perhaps the least-known input into any pricing model for options, therefore traders often look to the marketplace to see what the implied volatility of an option is — meaning that given the price of an option and all the other inputs except volatility you can solve for that value.
Pricing models include the Binomial options pricing model for American options and the Black-Scholes model for European options. Even though there are pricing models, the value of an option is a personal decision, requiring multiple trade offs and depending on the investment objective. See the Excel model [1] for the metrics of a call option.

Because options are derivatives, they can be combined with different combinations of

other options
risk-free T-bills
the underlying security, and
futures contracts on that security
to create a risk neutral portfolio (zero risk, zero cost, zero return). In a liquid market, arbitrageurs ensure that the values of all these assets are 'self-leveling', i.e. they incorporate the same assumptions of risk/reward. In theory traders could buy cheap options and sell expensive options (relative to their theoretical prices), in quantities such that the overall delta is zero, and expect to make a profit. Nevertheless, implementing this in practice may be difficult because of "stale" stock prices, large bid/ask spreads, market closures and other symptoms of stock market illiquidity. If stock market prices do not follow a random walk (due, for example, to insider trading) this delta neutral strategy or other model-based strategies may encounter further difficulties. Even for veteran traders using very sophisticated models, option trading is not an easy game to play.

History of valuation
Models of option pricing were very simple and incomplete until 1973 when Fischer Black and Myron Scholes published the Black-Scholes pricing model. Scholes received the 1997 Bank of Sweden Prize in Economic Sciences (Nobel Prize of Economics) for this work, along with Robert C. Merton. In a departure from tradition, Fischer Black was specifically mentioned in the award, even though he had died and was therefore not eligible.

The Black-Scholes model gives theoretical values for European put and call options on non-dividend paying stocks. The key argument is that traders could risklessly hedge a long options position with a short position in the stock and continuously adjust the hedge ratio (the delta value — one of the option sensitivities known as "Greeks") as needed. Assuming that the stock price follows a random walk, and using the methods of stochastic calculus, a price for the option can be calculated where there is no arbitrage profit. This price depends only on 5 factors: the current stock price, the exercise price, the risk-free interest rate, the time until expiration, and the volatility of the stock price. Eventually, the model was adapted to be able to price options on dividend paying stocks as well.

The availability of a good estimate of an option's theoretical price contributed to the explosion of trading in options. Other option pricing models have since been developed for other markets and situations using similar arguments, assumptions, and tools, including the Black model for options on futures, Monte Carlo methods, Path Integrals, and Binomial options models.

Risks
Risk is concerned with the unknown. Upside risk is the possibility of gain. Downside risk is the possibility of loss. One half the reasons to use options (like other derivatives) is to reduce risk. Certainty is exchanged with other players who assume the risk in hope of big gains. It is wrong to state that "options are risky."

reduce risk: The seller of a covered call exchanges his upside risk (gains above the strike price) for the certainty of cash in hand (the premium). The buyer of a covered put limits his downside risk for a price — just like buying fire insurance for your house.
increase risk: The buyer of a call wants the upside risk of an asset, but will only pay a small percentage of its current value, so his returns are leveraged. The seller of a put accepts the downside risk of locking in his purchase price of an asset, in exchange for the premium.
To understand risk, look at the four standard graphs of options (put-call-buy-sell). The value of the options in the interim between purchase and expiration will not be exactly like these graphs, but close enough. In all cases, the premium was a certainty.

Buyers start out-of-pocket. But going forward, the option buyer has no downside risk. The graph either flat lines or goes up on either side of the spot price.
Sellers start with a gain. Going forward, they have no upside risk. These graphs either flat line or go down on either side of the spot price.
The extent of risk varies. Buyers/sellers of calls have unlimited upside/downside risk as the asset price increases. Buyers/sellers of puts have upside/downside risk limited to the spot price of the asset (less the premium).

Pin risk
A special situation called pin risk can arise when the underlier closes at or very close to the option's strike value on the last day the option is traded prior to expiration. The option writer (seller) may not know with certainty whether or not the option will actually be exercised or be allowed to expire worthless. Therefore, the option writer may end up with a large, unwanted residual position in the underlier when the markets open on the next trading day after expiration, regardless of their best efforts to avoid such a residual.

Trading
The most common way to trade stock options is trading standardized options contracts that are listed by various futures and options exchanges — there are currently six exchanges in the United States that list standardized options contracts based on underlying stocks — The Philadelphia Stock Exchange (PHLX), American Stock Exchange (AMEX) and NYSE Arca in New York City, and the Chicago Board Options Exchange (CBOE) which are all open-outcry marketplaces, and the International Securities Exchange (ISE) and Boston Options Exchange (BOX) are electronic marketplaces. However, even for the non-electronic exchanges, competition and the introduction of automated execution (AutoEx) has led, by late 2006, to hybridization where all but the largest trades are executed electronically. In Europe the main exchanges where stock options are traded are Euronext.liffe and Eurex.

There are also over-the-counter options contracts that are traded not on exchanges, but between two independent parties. At least one of those parties is usually a large financial institution with a balance sheet big enough to underwrite such a contract.