# Long put option wiki

Put—call parity is a static replication , and thus requires minimal assumptions, namely the existence of a forward contract. In the absence of traded forward contracts, the forward contract can be replaced indeed, itself replicated by the ability to buy the underlying asset and finance this by borrowing for fixed term e.

These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than those of the Black—Scholes model , which requires dynamic replication and continual transaction in the underlying. Replication assumes one can enter into derivative transactions, which requires leverage and capital costs to back this , and buying and selling entails transaction costs , notably the bid-ask spread.

The relationship thus only holds exactly in an ideal frictionless market with unlimited liquidity. However, real world markets may be sufficiently liquid that the relationship is close to exact, most significantly FX markets in major currencies or major stock indices, in the absence of market turbulence.

The left side corresponds to a portfolio of long a call and short a put, while the right side corresponds to a forward contract. The assets C and P on the left side are given in current values, while the assets F and K are given in future values forward price of asset, and strike price paid at expiry , which the discount factor D converts to present values.

In this case the left-hand side is a fiduciary call , which is long a call and enough cash or bonds to pay the strike price if the call is exercised, while the right-hand side is a protective put , which is long a put and the asset, so the asset can be sold for the strike price if the spot is below strike at expiry. Both sides have payoff max S T , K at expiry i. Note that the right-hand side of the equation is also the price of buying a forward contract on the stock with delivery price K.

Thus one way to read the equation is that a portfolio that is long a call and short a put is the same as being long a forward. In particular, if the underlying is not tradeable but there exists forwards on it, we can replace the right-hand-side expression by the price of a forward.

However, one should take care with the approximation, especially with larger rates and larger time periods. When valuing European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes:. We can rewrite the equation as:. We will suppose that the put and call options are on traded stocks, but the underlying can be any other tradeable asset. The ability to buy and sell the underlying is crucial to the "no arbitrage" argument below.

First, note that under the assumption that there are no arbitrage opportunities the prices are arbitrage-free , two portfolios that always have the same payoff at time T must have the same value at any prior time. To prove this suppose that, at some time t before T , one portfolio were cheaper than the other.

Then one could purchase go long the cheaper portfolio and sell go short the more expensive. At time T , our overall portfolio would, for any value of the share price, have zero value all the assets and liabilities have canceled out. The profit we made at time t is thus a riskless profit, but this violates our assumption of no arbitrage. We will derive the put-call parity relation by creating two portfolios with the same payoffs static replication and invoking the above principle rational pricing.

Consider a call option and a put option with the same strike K for expiry at the same date T on some stock S , which pays no dividend. We assume the existence of a bond that pays 1 dollar at maturity time T.

The bond price may be random like the stock but must equal 1 at maturity. Let the price of S be S t at time t. Now assemble a portfolio by buying a call option C and selling a put option P of the same maturity T and strike K.

In this way the buyer of the put will receive at least the strike price specified, even if the asset is currently worthless. 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 any time until the option's maturity time T.

The put yields a positive return only if the security price falls below the strike when the option is exercised. A European option can only be exercised at time T rather than any time 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.

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: Puts may also be combined with other derivatives as part of more complex investment strategies, and in particular, may be useful for hedging.

By put-call parity , a European put can be replaced by buying the appropriate call option and selling an appropriate forward contract. 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 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 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. 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.

The seller's potential loss on a naked put can be substantial. If the stock falls all the way to zero bankruptcy , his loss is equal to the strike price at which he must buy the stock to cover the option minus the premium received. The potential upside is the premium received when selling the option: