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A contract that gives the owner the right, but not the obligation, to buy or sell a security at a fixed price in the future.

In finance, an option is a contract which gives the buyer (the owner or holder) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on or before a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – that is to sell or buy – if the buyer (owner) "exercises" the option. An option that conveys to the owner the right to buy something at a specific price is referred to as a call; an option that conveys the right of the owner to sell something at a specific price is referred to as a put. Both are commonly traded, but for clarity, the call option is more frequently discussed.

The simplest option gives the holder the right to trade in the future at a previously agreed price but takes away the obligation. So if the stock falls, we don’t have to buy it after all. The European option is one of the simplest options.Indeeda European-type call option on a security $S_t$ is the right to buy the security at a present strike price $K$. This right maybe exercised at the expiration date T of the option. The call option can be purchased for a price of Ct dollars,called the premium, at time $t < T$. A European put option is similar, but gives the owner the right to sell an asset at a specified price at expiration.In contrast to European options, American options can be exercised any time between the writing and the expiration of the contract.

There are several reasons that traders and investors may want to calculate the arbitrage-free price, $C_t$, of a call option Before the option is first written at time $t$,$C_t$ is not known. A trader may want to obtain some estimate of what this price will be if the option is written. If the option is an exchange-traded security, it will start trading and a market price will emerge. If the option trades over-the-counter, it may also trade heavily and a price can be observed.