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.