# Down and out binary call options b less than k

In finance, a put or put option is a stock market device which gives the owner of a put the right, but not the obligation, to sell an asset the underlyingat a specified price the strikeby a predetermined date the expiry or maturity to a given party the seller of the put.

The purchase of a put option is interpreted as a negative sentiment about the future value of the underlying. Put options are most commonly used in the stock market to protect against the decline of the price of a stock below a specified price. 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 Kand at time t the value of the underlying is S tthen 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 Tand 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 paritya 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 putalso called an uncovered putis 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 bankruptcyhis 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: During the option's lifetime, if the stock moves lower, the option's premium may increase depending on how far the stock falls and how much time passes.

If it does, it becomes more costly to close the position repurchase the put, sold earlierresulting in a loss. If the stock price completely collapses before the put position is closed, the put writer potentially can face catastrophic loss.

In order to protect the put buyer from default, the put writer is required to post margin. The put buyer does not need to post margin because the buyer would not exercise the option if it had a negative payoff.

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.

The writer receives a premium from the buyer. If the buyer exercises his option, the writer will buy the stock at the strike price. If the buyer does not exercise his option, the writer's profit is the premium. 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: Option pricing is a central problem of financial mathematics.

Trading options involves a constant monitoring of the option value, which is affected by changes in the base asset price, volatility and time decay. Moreover, the dependence of the put option value to those factors is not linear — which makes the analysis even more complex.

The graphs clearly shows the non-linear dependence of the option value to the base asset price. From Wikipedia, the free encyclopedia. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources.

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For one thing, there are equally many parameters (binary choice probabilities) as there are empirical cells (binary choice proportions), yet we can tell from the figures that the models can be extremely restrictive, especially in highdimensional spaces, and hence, must be testable. As explained in Davis-Stober and Brown (2011) one cannot simply count parameters to evaluate the complexity of these types of models.

The second reason, returning to data like those of DM1 and DM 13 in Figure 3 is that the best fitting parameters, i.

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