The investor holding the call option (buyer) pays the counterparty issuing the call option (writer) a premium in exchange for the right to buy a security. The date at which this transaction can take place is called maturity, and the price at which the stock can be
bought is called the strike price.
The buyer has no obligation to exercise this right. He will choose to do so based on whether the option is in the money or out of the money at maturity. Taking for example a call option with a strike price of USD 25 that costs USD 3.
A call option is in the money if the spot price (S = USD 30) at maturity is greater than the strike price (K = USD 25). The pay-off to the option owner is USD 30 – 25 – 3 = 2.
A call option is out of the money if the spot price (S = USD 20) at maturity is less than the strike price (K = USD 25). The pay-off to the option owner is 0, as he simply will not exercise the call.
Therefore the pay-off to the option holder is the maximum of (S – K – premium received) and (0 – premium received).
By being long a call option, the most an investor stands to lose is the premium paid for it (in this case USD 3), whilst profits are unlimited as long as the share price is above strike at maturity.
The pay-off for the seller, which is short the call, is the exact opposite. The maximum pay-off of a short call option is the value of the premium received from the buyer. If the call is in the money at maturity, the seller of the option has the obligation to sell the stock at strike price K. Thus the pay-off to being short a call is the minimum of the premium received and (K – S + premium received).
By being short a call option, the most an investor stands to make is the premium paid for the option, whereas the maximum loss is unlimited, as long as the share price is above strike at maturity.