# Put Research Materials

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THE LEGEND OF ZELDA RAP [MUSIC VIDEO]
David Guetta - Titanium ft. Sia
From the album Nothing But The Beat Ultimate - Download on iTunes here: http://smarturl.it/NBTBiTunes?IQid=vevo Featuring Sia, Ne-Yo, Akon, Nicki Minaj, Flo ...
MACKLEMORE & RYAN LEWIS - THRIFT SHOP FEAT. WANZ (OFFICIAL VIDEO)
Thrift Shop on iTunes: http://itunes.apple.com/us/album/thrift-shop-feat.-wanz-single/id556955707 The Heist physical deluxe edition: http://www.macklemoremer...
David Guetta - She Wolf (Falling To Pieces) ft. Sia
From the album Nothing But The Beat Ultimate - Download on iTunes here: http://smarturl.it/NBTBiTunes?IQid=vevo Featuring Sia, Ne-Yo, Akon, Nicki Minaj, Flo ...
Rihanna - Rehab ft. Justin Timberlake
Music video by Rihanna performing Rehab. YouTube view counts pre-VEVO: 19591123. (C) 2007 The Island Def Jam Music Group.
Man of Steel - Official Trailer 3 [HD]
http://manofsteel.com http://www.facebook.com/manofsteel In theaters June 14th. From Warner Bros. and Legendary Pictures comes "Man of Steel", starring Henry...
P!nk - Just Give Me A Reason ft. Nate Ruess
From the Grammy Nominated album The Truth About Love available now - http://smarturl.it/tal Music video by P!nk featuring Nate Ruess performing Just Give Me ...
FIRETRUCK! (Official Music Video)
BLOOPERS: http://bit.ly/FiretruckBloopers GET THE SONG: http://smo.sh/WMZv7l MILKSHAKE MUSIC VIDEO: http://bit.ly/MilkyMilkshake CHECK OUT THIS FIRETRUCK TEE...
PEWDIEPIE Song - Dj Fortify
Due to so many request I decided to upload this epic track as well. Enjoy it bros!
One Direction - Little Things
TAKE ME HOME The brand new album out now! Featuring Live While We're Young and Little Things. iTunes: http://smarturl.it/takemehome1D Amazon: http://amzn.to/...
Rihanna - Stay ft. Mikky Ekko
Download "Stay" from Unapologetic now: http://smarturl.it/UnapologeticDlx Music video by Rihanna performing Stay ft. Mikky Ekko. © 2013 The Island Def Jam Mu...
David Guetta - Just One Last Time ft. Taped Rai
"Just One Last Time" feat. Taped Rai. Available to download on iTunes including remixes of : Tiësto, HARD ROCK SOFA & Deniz Koyu http://smarturl.it/DGJustOne...
MACKLEMORE & RYAN LEWIS - CAN'T HOLD US FEAT. RAY DALTON (OFFICIAL MUSIC VIDEO)
Macklemore & Ryan Lewis present the official music video for Can't Hold Us feat. Ray Dalton. Can't Hold Us on iTunes: https://itunes.apple.com/us/album/cant-...
FUNNY GAMING MONTAGE
I'm just a guy from Sweden who likes to laugh and make other people laugh. Sharing gaming moments on YouTube with my bros! Why not join us? :D Become a bro t...
Drive Thru Invisible Driver Prank
Draw My Life- Jenna Marbles
This video accidentally turned out kind of sad, ME SO SOWWY IT NOT POSED TO BE SAD WHO WANTS HUGS AND COOKIES? Also, FYI for anyone attempting this, it takes...
Rihanna - Diamonds
Pre-order new album Unapologetic, out worldwide Monday, November 19: http://smarturl.it/UnapologeticDlx Music video by Rihanna performing Diamonds. ©: The Is...
Rihanna - Pon de Replay (Internet Version)
Music video by Rihanna performing Pon de Replay. YouTube view counts pre-VEVO: 4166822. (C) 2005 The Island Def Jam Music Group.
Rihanna - Only Girl (In The World)
Music video by Rihanna performing Only Girl (In The World). (C) 2010 The Island Def Jam Music Group #VEVOCertified on February 16, 2011. http://www.vevo.com/...
Pokemon Theme Song REVENGE!
BLOOPERS & DELETED SCENES: http://bit.ly/POKEMONbloopers DOWNLOAD THE SONG: http://smo.sh/WMZqR5 CHECK OUT THIS POKESMOSH SHIRT: http://smo.sh/YMRJ5e COLLECT...

A put or put option is a contract between two parties to exchange an asset (the underlying), at a specified price (the strike), by a predetermined date (the expiry or maturity). One party, the buyer of the put, has the right, but not an obligation, to re-sell the asset at the strike price by the future date, while the other party, the seller of the put, has the obligation to repurchase the asset at the strike price if the buyer exercises the option.

If the strike is K, and at time t the value of the underlying is S(t), then in an American option the buyer can exercise the put for a payout of K-S(t) up until the option's maturity time T. The put yields a positive return only if the spot price falls below the strike when the option is exercised. A European option can only be exercised at time T rather than any time up until T, and a Bermudan option can be exercised only on specific dates listed in the terms of the contract. If the option is not exercised by maturity, it expires worthless. (Note that the buyer will not exercise the option at an allowable date if the price of the underlying is greater than K.)

The most obvious use of a put is as a type of insurance. In the protective put strategy, the investor buys enough puts to cover his holdings of the underlying so that if a drastic downward movement of the underlying's price occurs, he has the option to sell the holdings at the strike price. Another use is for speculation: an investor can take a short position in the underlying stock without trading in it directly.

Puts may also be combined with other derivatives as part of more complex investment strategies, and in particular, may be useful for hedging. Note that by put-call parity, a European put can be replaced by buying the appropriate call option and selling an appropriate forward contract.

## Instrument models

The terms for exercising the option's right to sell it 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, while an American put option allows exercise at any time before expiration.

The most widely-traded put options are on equities, but they are traded on many other instruments such as interest rates (see interest rate floor) or commodities.

The put buyer either believes that the underlying asset's price will fall by the exercise date or hopes to protect a long position in it. The advantage of buying a put over short selling the asset is that the option owner's risk of loss is limited to the premium paid for it, whereas the asset short seller's risk of loss is unlimited (its price can rise greatly, in fact, in theory it can rise infinitely, and such a rise is the short seller's loss). The put buyer's prospect (risk) of gain is limited to the option's strike price less the underlying's spot price and the premium/fee paid for it.

The put writer believes that the underlying security's price will rise, not fall. The writer sells the put to collect the premium. The put writer's total potential loss is limited to the put's strike price less the spot and premium already received. Puts can be used also to limit the writer's portfolio risk and may be part of an option spread.

The put buyer is short on the underlying asset of the put, but long on the put option itself. That is, the buyer wants the value of the put option to increase by a decline in the price of the underlying asset below the strike price. The writer (seller) of a put is long on the underlying asset and short on the put option itself. That is, the seller wants the option to become worthless by an increase in the price of the underlying asset above the strike price. Generally, a put option that is purchased is referred to as a long put and a put option that is sold is referred to as a short put.

A naked put, also called an uncovered put, is a put option whose writer (the seller) does not have a position in the underlying stock or other instrument. This strategy is best used by investors who want to accumulate a position in the underlying stock, but only if the price is low enough. If the buyer fails to exercise the options, then the writer keeps the option premium as a "gift" for playing the game.

If the underlying stock's market price is below the option's strike price when expiration arrives, the option owner (buyer) can exercise the put option, forcing the writer to buy the underlying stock at the strike price. That allows the exerciser (buyer) to profit from the difference between the stock's market price and the option's strike price. But if the stock's market price is above the option's strike price at the end of expiration day, the option expires worthless, and the owner's loss is limited to the premium (fee) paid for it (the writer's profit).

## Example of a put option on a stock

Payoff from writing a put.

A buyer thinks the price of a stock will decrease. He pays a premium which he will never get back, unless it is sold before it expires. The buyer has the right to sell the stock at the strike price.

Writing a put

• "Trader A" (Put Buyer) purchases a put contract to sell 100 shares of XYZ Corp. to "Trader B" (Put Writer) for $50 per share. The current price is$55 per share, and Trader A pays a premium of $5 per share. If the price of XYZ stock falls to$40 a 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. The sale of the 100 shares of stock at a strike price of$50 to Trader B = $5,000 (P). The purchase of 100 shares of stock at$40 = $4,000 (Q). The put option premium paid to trader B for buying the contract of 100 shares at$5 per share, excluding commissions = $500 (R). Thus 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 (because selling a stock to Trader B at $50 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 per share, 100 shares per contract). Trader A's total loss is limited to the cost of the put premium plus the sales commission to buy it.

A put option is said to have intrinsic value when the underlying instrument has a spot price (S) below the option's strike price (K). Upon exercise, a put option is valued at K-S if it is "in-the-money", otherwise its value is zero. Prior to exercise, an option has time value apart from its intrinsic value. The following factors reduce the time value of a put option: shortening of the time to expire, decrease in the volatility of the underlying, and increase of interest rates. Option pricing is a central problem of financial mathematics.

## Payoff of a put

These examples lead to the following formal reasoning. Fix $\mathcal{O}$ an underlying financial instrument. Let $\Pi$ be a put option for this instrument, purchased at time $0$, expiring at time $T\in\mathbb{R}^{+}$, with exercise (strike) price of $K\in\mathbb{R}$; and let $S:[0,T]\to\mathbb{R}$ be the price of the underlying instrument.

Assume the owner of the option $\Pi$, wants to not take a loss, and does not want to actually possess the underlying instrument, $\mathcal{O}$. Then either (i) the person will purchase $\mathcal{O}$ at expiry, and then immediately exercise the selling option; or (ii) the person will not exercise the option (which subsequently becomes worthless).

In (i), the pay-off would be $-S_{T}+K$; in (ii) the pay-off would be $0$. So if $K-S_{T}\geq 0$ (i) or (ii) occurs; if $K-S_{T}<0$ then (ii) occurs.

Hence the pay-off, i.e. the value of the put option at expiry, is

$\operatorname{max}\{K-S_{T},0\}$

which is alternatively written $(K-S_{T})\vee 0$ or $(K-S_{T})^{+}$.

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