Call option

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Buying a call option - This is a graphical interpretation of the payoffs and profits generated by a call option as seen by the buyer. A higher stock price means a higher profit. Eventually, the price of the underlying security will be high enough to fully compensate for the price of the option.

Writing a call option - This is a graphical interpretation of the payoffs and profits generated by a call option as seen by the writer of the option. Profit is maximized when the strike price exceeds the price of the underlying security, because the option expires worthless and the writer keeps the premium.

A call option is a financial contract between two parties, the buyer and the seller of this type of option. Often it is simply labeled a "call". The buyer of the option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument (the underlying instrument) from the seller of the option at a certain time (the expiration date) for a certain price (the strike price). The seller (or "writer") is obligated to sell the commodity or financial instrument should the buyer so decide. The buyer pays a fee (called a premium) for this right.

The buyer of a call option wants the price of the underlying instrument to rise in the future; the seller either expects that it will not, or is willing to give up some of the upside (profit) from a price rise in return for (a) the premium (paid immediately) plus (b) retaining the opportunity to make a gain up to the strike price (see below for examples).

Call options are most profitable for the buyer when the underlying instrument is moving up, making the price of the underlying instrument closer to the strike price. When the price of the underlying instrument surpasses the strike price, the option is said to be "in the money".

The initial transaction in this context (buying/selling a call option) is not the supplying of a physical or financial asset (the underlying instrument). Rather it is the granting of the right to buy the underlying asset, in exchange for a fee - the option price or premium.

Exact specifications may differ depending on option style. A European call option allows the holder to exercise the option (i.e., to buy) only on the option expiration date. An American call option allows exercise at any time during the life of the option.

Call options can be purchased on many financial instruments other than stock in a corporation - options can be purchased on futures on interest rates, for example (see interest rate cap) - as well as on commodities such as gold or crude oil. A tradeable call option should not be confused with either Incentive stock options or with a warrant. An incentive stock option, the option to buy stock in a particular company, is a right granted by a corporation to a particular person (typically executives) to purchase treasury stock. When an incentive stock option is exercised, new shares are issued. Incentive stock options are not traded on the open market. In contrast, when a call option is exercised, the underlying asset is transferred from one owner to another.

[edit] Example of a call option on a stock

Buy a call: buyer expects that the price may go up. 
            Pays a premium that buyer will never get back.
            Buyer has the right to exercise the option at the strike price.
Write a call: writer receives the premium. 
              if buyer decides to exercise the option,
              writer has to sell the stock at the strike price.

An investor buys a call on a stock with a strike price of 50 and an option expiration date of June 16, 2006 and pays a premium of 5 for this call option. The current price is 40.
Assume that the share price (the spot price) rises, and is 60 on the strike date. The investor would exercise the option (i.e., buy the share from the counter-party), and could then hold the share, or sell it in the open market for 60. The profit would be 10 minus the fee paid for the option, 5, for a net profit of 5. The investor has thus doubled his money, having paid 5 and ending up with 10.
If however the share price never rises to 50 (that is, it stays below the strike price) up through the exercise date, then the option would expire as worthless. The investor loses the premium of 5.
Thus, in any case, the loss is limited to the fee (premium) initially paid to purchase the stock, while the potential gain is theoretically unlimited (consider if the share price rose to 100).
From the viewpoint of the seller, if the seller thinks the stock is a good one, (s)he is better (in this case) by selling the call option, should the stock in fact rise. However, the strike price (in this case, 50) limits the seller's profit. In this case, the seller does realize the profit up to the strike price (that is, the 10 rise in price, from 40 to 50, belongs entirely to the seller of the call option), but the increase in the stock price thereafter goes entirely to the buyer of the call option.

From the above, it is clear that a call option has positive monetary value when the underlying instrument has a spot price (S) above the strike price (K). Since the option will not be exercised unless it is "in-the-money", the payoff for a call option is

$\,Max[ (S-K) ; 0 ]$ or formally, $\,(S-K)^{+}$

where $(x)^+ =\{^{x\ if\ x\geq 0}_{0\ otherwise}$

Prior to exercise, the option value, and therefore price, varies with the underlying price and with time. The call price must reflect the "likelihood" or chance of the option "finishing in-the-money". The price should thus be higher with more time to expiry (except in cases when a significant dividend is present) and with a more volatile underlying instrument. The science of determining this value is the central tenet of financial mathematics. The most common method is to use the Black-Scholes formula. Whatever the formula used, the buyer and seller must agree on the initial value (the premium), otherwise the exchange (buy/sell) of the option will not take place.