Put option
From Academic Kids

A put option (sometimes simply called a "put") is a financial contract between two parties, the buyer and the seller of the option. The put allows the buyer the right but not the obligation to sell a commodity or financial instrument (the underlying instrument) to the seller of the option at a certain time for a certain price (the strike price). The seller has the obligation to purchase at that strike price, if the buyer does choose to exercise the option.
Note that the seller of the option is agreeing to buy the underlying instrument if the buyer of the option so decides! In exchange for having this option, the buyer pays the seller a fee (the premium).
Exact specifications may differ depending on option style. A European put option allows the holder to exercise the put option on the delivery date only. An American put option allows exercise at any time during the life of the option.
The most widelyknown 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.
Example of a put option on a stock
 I purchase a put option to sell a share in XYZ Corp. on June 1, 2006, for $50. The current price is $55, and I pay a premium of $5.
 Assume that the XYZ Corp. share price is actually $40 on that day. Then I would exercise my option, by purchasing a share of the stock in the open market (for $40) and then selling it to the counterparty at the strike price of $50. (In practice, the seller of the put option could simply pay me the $10 difference.) My profit would be $10 minus the fee (of $5) that I paid for the option. So I have doubled my money (began with $5 to purchase the call option; ended with $10 in my pocket).
 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, when it is more valuable in the open market?) My option would be worthless and I would have lost my whole investment, the fee (premium) for the option, $5.
 Thus, in any future state of the world, my loss is limited to the fee I have paid (in this case $5), while my profit depends on how much the stock price falls (consider, for example, if the stock sold at $20 on the exercise date).
In general, the buyer of a put option expects the price of stock to fall significantly, but does not want to sell the stock short because that could result in large losses. (With a put option, the loss is limited to the purchase price of the option.) The seller of the put option generally feels that the stock in question is reasonably priced, and should the price fall, the seller may be willing to become the owner of the stock at a lower price, considering it to be a bargain. (On the other hand, the seller of the put may be merely gambling.)
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 "inthemoney", the payoff for a put option is
 Max[ (KS) ; 0 ] or formally, <math>(KS)^{+}<math>
 where <math>(x)^+ =\{^{x\ if\ x\geq 0}_{0\ otherwise}<math>
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 inthemoney". 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 BlackScholes formula. Whatever the formula used, the buyer and seller must agree this value initially.