Definitions

# Put option

A put option (sometimes simply called a "put") is a financial contract between two parties, the seller (writer) and the buyer of the option. The put allows its buyer 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 long position in the option, they are not the only sellers. An option holder can also sell his short 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 put buyer either believes it's likely the price of the underlying asset will fall by the exercise date, or hopes 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 believe the price of the underlying security is likely to fall. The writer sells the put to collect the premium. Puts can also be used to limit portfolio risk, and may be part of an option spread.

### Example of a put option on a stock

`Buy a Put: A Buyer thinks price of a stock will decrease.`
`           Pay a premium which buyer will never get back, unless`
`             it is sold before expiration.`
`           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.`

• 'Trader A' (Put Buyer) purchases a put contract to sell 100 shares of XYZ Corp. to 'Trader B' (Put Writer) for \$50/share. The current price is \$55/share, and 'Trader A' pays a premium of \$5/share. If the price of XYZ stock falls to \$40/share right before expiration, then 'Trader A' can exercise the put by buying 100 shares for \$4,000 from the stock market, then selling them to 'Trader B' for \$5,000.

`Trader A's total earnings (S) can be calculated at \$500.`
` Sale of the 100 stock at strike price of \$50 to 'Trader B' = \$5,000 (P)`
` Purchase of 100 stock at \$40 = \$4,000 (Q)`
` Put Option premium paid to Trader B for buying the contract of 100 shares @ \$5/share, excluding commissions = \$500 (R)`

` S=P-(Q+R)=\$5,000-(\$4,000+\$500)=\$500`

• If, however, the share price never drops below the strike price (in this case, \$50), then 'Trader A' would not exercise the option. (Why sell a stock to 'Trader B' at \$50, if it would cost 'Trader A' more than that to buy it?). Trader A's option would be worthless and he would have lost the whole investment, the fee (premium) for the option contract, \$500 (5/share, 100 shares per contract). Trader A's total loss are 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, $\left(K-S\right)^\left\{+\right\}$
where :$\left(x\right)^+ =begin\left\{cases\right\}$
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. Determining this value is the central problem 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.