The investor holding the put option (buyer) pays the counterparty issuing the put option (writer) a premium in exchange for the right to sell a security. The date at which this transaction can take place is called maturity, and the price at which the security can be sold is called the strike price. Taking for example a put option with a strike price of USD 25 that costs USD 3.
The buyer has no obligation to exercise this right. He will choose to exercise this right based on whether the option is in the money or out of the money at maturity.
A put option is in the money if the strike price (K = USD 25) is greater than the spot price (S = USD 20) at maturity. The pay-off to the option owner is (K – S).
A put option is out of 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 0, as he simply will not exercise the put. Therefore the pay-off to the option holder is the maximum of (K – S – premium) and (0 – premium).
By being long a put option, the most an investor stands to lose is the premium paid for the option (in this case, USD 3), whereas his maximum gain is the strike price less the premium paid for the option (in this case USD 25 – USD 3 = USD 22).
The pay-off for the seller, which is short the put, is the exact opposite. The maximum pay-off of a short put 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 buy the stock from the buyer at strike price K. Thus the pay-off to being short a put is (S – K + the premium received).