Put option

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Buying a put option - This is a graphical interpretation of the payoffs and profits generated by a put option as seen by the buyer of the option. A lower stock price means a higher profit. Eventually, the price of the underlying security will be low enough to fully compensate for the price of the option.

Writing a put option - This is a graphical interpretation of the payoffs and profits generated by a put option as seen by the writer of the option. Profit is maximized when the price of the underlying security exceeds the strike price, because the option expires worthless and the writer keeps the premium.

A put option (sometimes simply called a "put") is a financial contract between two parties, the buyer and the writer (seller) of the option. The put allows the owner the right but not the obligation to sell a commodity or financial instrument (the underlying instrument) to the writer (seller) of the option at a certain time for a certain price (the strike price). The writer (seller) has the obligation to purchase the underlying asset at that strike price, if the buyer exercises the option.

Note that the writer of the option is agreeing to buy the underlying asset if the buyer exercises the option. In exchange for having this option, the buyer pays the writer(seller) a fee (the premium). (Note: Although option writers are frequently referred to as sellers, because they initially sell the option that they create, thus taking a short position in the option, they are not the only sellers. An option holder can also sell his long position in the option. However, the difference between the two sellers is that the option writer takes on the legal obligation to buy the underlying asset at the strike price, whereas the option holder is merely selling his long position, and is not contractually obligated by the sold option.)

Exact specifications may differ depending on option style. A European put option allows the holder to exercise the put option for a short period of time right before expiration. An American put option allows exercise at any time during the life of the option.

The most widely-known put option is for stock in a particular company. However, options are traded on many other assets: financial - such as interest rates (see interest rate floor) - and physical, such as gold or crude oil.

The buyer of the put either expects the price of the underlying asset to fall or to protect a long position in the asset. The advantage of buying a put over shorting the asset is that the risk is limited to the premium. The put writer does not expect the price of the underlying to fall, and so writes the put to collect the premium. Puts can also be used to limit portfolio risk, and may be part of an option spread.

[edit] Example of a put option on a stock

Buy A Put  : Buyer thinks price of stock will decrease. 
             Pay a Premium which buyer will never get back. 
             The buyer has the right to sell the stock 
             at strike price.
Write a put: Writer receives a premium, if Buyer exercises the 
             option Writer will buy the stock at strike price, 
             If Buyer does not exercise the option, 
             writer's profit is Premium

I purchase a put contract to sell 100 shares of XYZ Corp. for 50. The current price is 55, and I pay a premium of 5. If the price of XYZ stock falls to 40 per share right before expiration, then I can exercise my put by buying 100 shares for 4,000, then selling it to a put writer for 5,000. My total profit would equal 500 (5,000 from put writer - 4,000 for buying the stock - 500 for buying the put contract of 100 shares at 5 per share, excluding commissions).
If, however, the share price never drops below the strike price (in this case, 50), then I would not exercise the option. (Why sell a stock to someone at 50, the strike price, if it would cost me more than that to buy it?) My option would be worthless and I would have lost my whole investment, the fee (premium) for the option contract, 500 (5 per share, 100 shares per contract). My total loss is limited to the cost of the put premium plus the sales commission to buy it.

This example illustrates that the put option has positive monetary value when the underlying instrument has a spot price (S) below the strike price (K). Since the option will not be exercised unless it is "in-the-money", the payoff for a put option is

max[(K − S) ; 0] or formally,

(K - S) +

where : $(x)^+ =\begin{cases} x & \mathrm{if}\ x \geq 0, \\ 0 & \mathrm{otherwise.} \end{cases}$

Prior to exercise, the option value, and therefore price, varies with the underlying price and with time. The put 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 this value initially.