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# Call option

A call option is a financial contract between two parties, the buyer and the seller of the 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 for a certain price (the strike price). The seller, however, is obligated to sell the commodity or financial instrument should the buyer so decide. The buyer of a call option wants the price of the commodity/instrument to rise in the future; the seller eithers expects that it will not, or is willing to give up some of the upside (profit) from a price rise in return for the premium (paid immediately) plus the price rise up to the strike price (see below).

The initial transaction in this 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 delivery date. An American call option allows exercise at any time during the life of the option.

A call option should not be confused with a stock option. A stock option, the option to buy stock in a particular company, is a right issued by a corporation to a particular person (typically, employees) to purchase treasury stock. When a stock option is exercised, new shares are issued. When a call option is exercised, if it involves shares, the shares are simply being transferred from one owner to another. Nor are stock options traded on the open market.

Call options can be purchased on many financial instruments other than stock in a corporation - options can be purchased on interest rates, for example (see interest rate cap) - as well as on physical assets such as gold or crude oil.

Example of a call option on a stock
• I buy a call on Microsoft Corporation stock with a strike price of \$50 (the future exchange price) and an exercise date of June 1 2006. I pay 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. Then I would exercise my option (i.e., buy the share from the counter-party). I could then sell it in the open market for \$60. My profit would be \$10 minus the fee I paid for the option, \$5, for a net profit of \$5. I have thus doubled my money (beginning with \$5, ending with \$10 in my pocket).
• If however the share price never rises to \$50 (that is, it stays below the strike price) up through the exercise date, then I would not exercise the option. (If I really wanted to own such a share, I could buy it in the open market for less than \$50, so why exercise the option?) The option would expire as worthless. I would have lost my entire \$5.
• Thus, in any future state of the world, my loss is limited to the fee (premium) I initially paid to purchase the stock, while my potential gain is quite large (consider if the share price rose to \$100).
• From the viewpoint of the seller, if the seller thinks the stock is a good one, he/she is \$5 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, 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.