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OptionsIn finance, an option is a contract between a buyer and a seller that gives the buyer the rightbut not the obligationto buy or to sell a particular asset (theunderlying asset) at a later day atan agreed price. In return for granting the option, the seller collects a payment (thepremium)from the buyer. A calloption gives the buyer the right to buy the underlying asset; a putoptiongives the buyer of the option the right to sell the underlying asset. If the buyer chooses toexercise this right, the seller is obliged to sell or buy the asset at the agreed price. The buyer maychoose not to exercise the right and let it expire. The underlying asset can be a piece of property,or shares of stock or some othersecurity, such as, among others, afutures contract. For example,buying a call option provides the right to buy a specified quantity of a security at a set agreedamount, known as the 'strike price' at some time on or before expiration, while buying aputoption provides the right to sell. Upon the option holder's choice toexercise the option, the party

who sold, or wrote the option, must fulfill the terms of the contract.

The theoretical value of an option can be evaluated according to several models. These models,which are developed by quantitative analysts, attempt to predict how the value of the option willchange in response to changing conditions. Hence, the risks associated with granting, owning, ortrading options may be quantified and managed with a greater degree of precision, perhaps, thanwith some other investments. Exchange-traded options form an important class of options whichhave standardized contract features and trade on public exchanges, facilitating trading amongindependent parties. Over-the-counteroptions are traded between private parties, often well-capitalized institutions that have negotiated separate trading and clearing arrangements with eachother. Another important class of options, particularly in the U.S., areemployee stock options,

which are awarded by a company to their employees as a form of incentive compensation. Othertypes of options exist in many financial contracts, for example real estate options are often usedto assemble large parcels of land, andprepayment options are usually included in mortgageloans. However, many of the valuation and risk management principles apply across all financialoptions.

Contents

1 Contract specifications 2 Types of options

o 2.1 Option styles 3 Valuation models

o 3.1 Black Scholes

o 3.2 Stochastic volatility models

4 Model implementationo 4.1 Analytic techniques

o 4.2 Binomial tree pricing model

o 4.3 Monte Carlo models

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contracts are standardized, accurate pricing models are often available. Exchange tradedoptions include:

1. stock options,2. commodity options,3.

bond options and otherinterest rate options4. stock market index options or, simply, index options and5. options on futures contracts

Over-the-counter options (OTC options, also called "dealer options") are tradedbetween two private parties, and are not listed on an exchange. The terms of an OTCoption are unrestricted and may be individually tailored to meet any business need. Ingeneral, at least one of the counterparties to an OTC option is a well-capitalizedinstitution. Option types commonly traded over the counter include:

1. interest rate options

2. currency cross rate options, and3. options onswaps orswaptions.

Employee stock options are issued by a company to its employees as compensation.

Option styles

Main article: Option style

Naming conventions are used to help identify properties common to many different types ofoptions. These include:

European option - an option that may only be exercised on expiration. American option - an option that may be exercised on any trading day on or before

expiration. Bermudan option - an option that may be exercised only on specified dates on or before

expiration. Barrier option - any option with the general characteristic that the underlying security's

price must pass a certain lever or "barrier" before it can be excercised Exotic option - any of a broad category of options that may include complex financial

structures. Vanilla option - by definition, any option that is not exotic.

Valuation models

Main article: Valuation of options

The value of an option can be estimated using a variety of quantitative techniques based on theconcept ofrisk neutral pricing and using stochastic calculus. The most basic model is the Black-Scholes model. More sophisticated models are used to model the volatility smile. These models

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are implemented using a variety of numerical techniques. In general, standard option valuationmodels depend on the following factors:

The current market price of the underlying security, the strike priceof the option, particularly in relation to the current market price of the

underlier (in the money vs. out of the money), the cost of holding a position in the underlying security, including interest and dividends, the time to expiration together with any restrictions on when exercise may occur, and an estimate of the futurevolatility of the underlying security's price over the life of the

option.

More advanced models can require additional factors, such as an estimate of how volatilitychanges over time and for various underlying price levels, or the dynamics of stochastic interestrates.

The following are some of the principal valuation techniques used in practice to evaluate option

contracts.

Black Scholes

Main article: BlackScholes

In the early 1970s, Fischer Blackand Myron Scholes made a major breakthrough by deriving adifferential equation that must be satisfied by the price of any derivative dependent on a non-dividend-paying stock. By employing the technique of constructing a risk neutral portfolio thatreplicates the returns of holding an option, Black and Scholes produced a closed-form solutionfor a European option's theoretical price.]At the same time, the model generates hedge

parameters necessary for effective risk management of option holdings. While the ideas behindthe Black-Scholes model were ground-breaking and eventually led toScholes and Mertonreceiving the Swedish Central Bank's associated Prize for Achievement in Economics (oftenmistakenly referred to as theNobel Prize), the application of the model in actual options tradingis clumsy because of the assumptions of continuous (or no) dividend payment, constantvolatility, and a constant interest rate. Nevertheless, the Black-Scholes model is still one of themost important methods and foundations for the existing financial market in which the result iswithin the reasonable range.

Stochastic volatility models

Main article: Heston model

Since the market crash of 1987, it has been observed that market implied volatility for options oflower strike prices are typically higher than for higher strike prices, suggesting that volatility isstochastic, varying both for time and for the price level of the underlying security.Stochasticvolatility models have been developed including one developed byS.L. Heston. One principaladvantage of the Heston model is that it can be solved in closed-form, while other stochasticvolatility models require complex numerical methods.[

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Model implementation

Once a valuation model has been chosen, there are a number of different techniques used to takethe mathematical models to implement the models.

Analytic techniques

In some cases, one can take the mathematical model and using analytic methods develop closedform solutions. The resulting solutions are useful because they are rapid to calculate.

Binomial tree pricing model

Main article: Binomial options pricing model

Closely following the derivation of Black and Scholes, John Cox, Stephen Ross and MarkRubinstein developed the original version of thebinomial options pricing model. It models thedynamics of the option's theoretical value for discrete time intervals over the option's duration.The model starts with a binomial tree of discrete future possible underlying stock prices. Byconstructing a riskless portfolio of an option and stock (as in the Black-Scholes model) a simpleformula can be used to find the option price at each node in the tree. This value can approximatethe theoretical value produced by Black Scholes, to the desired degree of precision. However, thebinomial model is considered more accurate than Black-Scholes because it is more flexible, e.g.discrete future dividend payments can be modeled correctly at the proper forward time steps, andAmerican options can be modeled as well as European ones. Binomial models are widely usedby professional option traders.

Monte Carlo models

Main article: Monte Carlo methods for option pricing

For many classes of options, traditional valuation techniques are intractable due to thecomplexity of the instrument. In these cases, a Monte Carlo approach may often be useful.Rather than attempt to solve the differential equations of motion that describe the option's valuein relation to the underlying security's price, a Monte Carlo model generates random price pathsof the underlying asset, each of which results in a payoff for the option. The average of thesepayoffs can be discounted to yield an expectation value for the option.[14]

Finite difference models

The equations used to value options can often be expressed in terms ofpartial differentialequations, and once expressed in this form, a finite difference model can be derived.

Other models

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Other numerical implementations which have been used to value options include finite elementmethods.

Risks

As with all securities, trading options entails the risk of the option's value changing over time.However, unlike traditional securities, the return from holding an option varies non-linearly withthe value of the underlier and other factors. Therefore, the risks associated with holding optionsare more complicated to understand and predict.

In general, the change in the value of an option can be derived from Ito's lemma as:

where the greeks, , and are the standard hedge parameters calculated from an optionvaluation model, such as Black-Scholes, and dS, d and dtare unit changes in the underlier price,the underlier volatility and time, respectively.

Thus, at any point in time, one can estimate the risk inherent in holding an option by calculatingits hedge parameters and then estimating the expected change in the model inputs, dS, d and dt,provided the changes in these values are small. This technique can be used effectively tounderstand and manage the risks associated with standard options. For instance, by offsetting aholding in an option with the quantity of shares in the underlier, a trader can form a deltaneutralportfolio that is hedged from loss for small changes in the underlier price. Thecorresponding price sensitivity formula for this portfolio is:

Example

A call option expiring in 99 days on 100 shares of XYZ stock is struck at $50, with XYZcurrently trading at $48. With future realized volatility over the life of the option estimated at25%, the theoretical value of the option is $1.89. The hedge parameters , , , are (0.439,0.0631, 9.6, and -0.022), respectively. Assume that on the following day, XYZ stock rises to$48.5 and volatility falls to 23.5%. We can calculate the estimated value of the call option by

applying the hedge parameters to the new model inputs as:

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Under this scenario, the value of the option increases by $0.0614 to $1.9514, realizing a profit of$6.14. Note that for a delta neutral portfolio, where by the trader had also sold 44 shares of XYZstock as a hedge, the net loss under the same scenario would be ($15.81).

Pin risk

Main article: Pin risk

A special situation calledpin riskcan arise when the underlier closes at or very close to theoption's strike value on the last day the option is traded prior to expiration. The option writer(seller) may not know with certainty whether or not the option will actually be exercised or beallowed to expire worthless. Therefore, the option writer may end up with a large, unwantedresidual position in the underlie when the markets open on the next trading day after expiration,regardless of their best efforts to avoid such a residual.

Counterparty risk

A further, often ignored, risk in derivatives such as options is counterparty risk. In an optioncontract this risk is that the seller won't sell or buy the underlying asset as agreed. The risk can beminimized by using a financially strong intermediary able to make good on the trade, but in amajor panic or crash the number of defaults can overwhelm even the strongest intermediaries.

Trading

The most common way to trade options is via standardized options contracts that are listed byvariousfutures and options exchangesListings and prices are tracked and can be looked up byticker symbol. By publishing continuous, live markets for option prices, an exchange enablesindependent parties to engage in price discovery and execute transactions. As an intermediary toboth sides of the transaction, the benefits the exchange provides to the transaction include:

fulfillment of the contract is backed by the credit of the exchange, which typically has thehighest rating (AAA),

counterparties remain anonymous, enforcement of market regulation to ensure fairness and transparency, and maintenance of orderly markets, especially during fast trading conditions.

Over-the-counteroptions contracts are not traded on exchanges, but instead between twoindependent parties. Ordinarily, at least one of the counterparties is a well-capitalized institution.By avoiding an exchange, users of OTC options can narrowly tailor the terms of the optioncontract to suit individual business requirements. In addition, OTC option transactions generallydo not need to be advertised to the market and face little or no regulatory requirements.However, OTC counterparties must establish credit lines with each other, and conform to eachothers clearing and settlement procedures.

With few exceptions,[16] there are nosecondary marketsforemployee stock options. These musteither be exercised by the original grantee or allowed to expire worthless.

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The basic trades of traded stock options (American style)

These trades are described from the point of view of a speculator. If they are combined withother positions, they can also be used in hedging. An option contract in US markets usuallyrepresents 100 shares of the underlying security

Long call

Payoff from buying a call.

A trader who believes that a stock's price will increase might buy the right to purchase the stock(a call option) rather than just buy the stock. He would have no obligation to buy the stock, onlythe right to do so until the expiration date. If the stock price at expiration is above the exerciseprice by more than the premium (price) paid, he will profit. If the stock price at expiration islower than the exercise price, he will let the call contract expire worthless, and only lose theamount of the premium. A trader might buy the option instead of shares, because for the sameamount of money, he can obtain a much larger number of options than shares. If the stock rises,he will thus realize a larger gain than if he had purchased shares.

Long put

Payoff from buying a put.

A trader who believes that a stock's price will decrease can buy the right to sell the stock at afixed price (aput option). He will be under no obligation to sell the stock, but has the right to doso until the expiration date. If the stock price at expiration is below the exercise price by morethan the premium paid, he will profit. If the stock price at expiration is above the exercise price,he will let the put contract expire worthless and only lose the premium paid.

Short call

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Payoff from writing a call.

A trader who believes that a stock price will decrease, can sell the stock short or instead sell, or"write," a call. The trader selling a call has an obligation to sell the stock to the call buyer at thebuyer's option. If the stock price decreases, the short call position will make a profit in theamount of the premium. If the stock price increases over the exercise price by more than theamount of the premium, the short will lose money, with the potential loss unlimited.

Short put

Payoff from writing a put.

A trader who believes that a stock price will increase can buy the stock or instead sell a put. Thetrader selling a put has an obligation to buy the stock from the put buyer at the put buyer'soption. If the stock price at expiration is above the exercise price, the short put position willmake a profit in the amount of the premium. If the stock price at expiration is below the exerciseprice by more than the amount of the premium, the trader will lose money, with the potential lossbeing up to the full value of the stock.

Option strategies

Main article: Option strategies

Payoffs from buying a butterfly spread.

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Payoffs from selling a straddle.

Payoffs from a covered call.

Combining any of the four basic kinds of option trades (possibly with different exercise pricesand maturities) and the two basic kinds of stock trades (long and short) allows a variety ofoptions strategies. Simple strategies usually combine only a few trades, while more complicatedstrategies can combine several.

Strategies are often used to engineer a particular risk profile to movements in the underlyingsecurity. For example, buying abutterflyspread (long one X1 call, short two X2 calls, and longone X3 call) allows a trader to profit if the stock price on the expiration date is near the middleexercise price, X2, and does not expose the trader to a large loss.

An Iron condoris a strategy that is similar to a butterfly spread, but with different strikes for theshort options - offering a larger likelihood of profit but with a lower net credit compared to thebutterfly spread.

Selling a straddle (selling both a put and a call at the same exercise price) would give a trader agreater profit than a butterfly if the final stock price is near the exercise price, but might result ina large loss.

Similar to the straddle is the strangle which is also constructed by a call and a put, but whosestrikes are different, reducing the net debit of the trade, but also reducing the likelihood of profit

in the trade.

One well-known strategy is the covered call, in which a trader buys a stock (or holds apreviously-purchased long stock position), and sells a call. If the stock price rises above theexercise price, the call will be exercised and the trader will get a fixed profit. If the stock pricefalls, the trader will lose money on his stock position, but this will be partially offset by thepremium received from selling the call. Overall, the payoffs match the payoffs from selling a put.This relationship is known asput-call parityand offers insights for financial theory.

http://en.wikipedia.org/wiki/Butterfly_(options)http://en.wikipedia.org/wiki/Butterfly_(options)http://en.wikipedia.org/wiki/Iron_condorhttp://en.wikipedia.org/wiki/Straddlehttp://en.wikipedia.org/wiki/Strangle_(options)http://en.wikipedia.org/wiki/Covered_callhttp://en.wikipedia.org/wiki/Covered_callhttp://en.wikipedia.org/wiki/Put-call_parityhttp://en.wikipedia.org/wiki/Put-call_parityhttp://en.wikipedia.org/wiki/File:Covered_Call.jpghttp://en.wikipedia.org/wiki/File:Covered_Call.jpghttp://en.wikipedia.org/wiki/File:Short_straddle_option.svghttp://en.wikipedia.org/wiki/File:Short_straddle_option.svghttp://en.wikipedia.org/wiki/Butterfly_(options)http://en.wikipedia.org/wiki/Iron_condorhttp://en.wikipedia.org/wiki/Straddlehttp://en.wikipedia.org/wiki/Strangle_(options)http://en.wikipedia.org/wiki/Covered_callhttp://en.wikipedia.org/wiki/Put-call_parity

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Historical uses of options

Contracts similar to options are believed to have been used since ancient times. In the real estatemarket, call options have long been used to assemble large parcels of land from separate owners,e.g. a developer pays for the right to buy several adjacent plots, but is not obligated to buy these

plots and might not unless he can buy all the plots in the entire parcel. Film or theatricalproducers often buy the right but not the obligation to dramatize a specific book or script.Lines of credit give the potential borrower the right but not the obligation to borrow withina specified time period.

Many choices, or embedded options, have traditionally been included inbondcontracts. Forexample many bonds areconvertible into common stock at the buyer's option, or may be called(bought back) at specified prices at the issuer's option.Mortgage borrowers have long had theoption to repay the loan early, which corresponds to a callable bond option.

In London, puts and "refusals" (calls) first became well-known trading instruments in the 1690s

during the reign ofWilliam and Mary.

Privileges were options sold over the counter in nineteenth century America, with both puts andcalls on shares offered by specialized dealers. Their exercise price was fixed at a rounded-offmarket price on the day or week that the option was bought, and the expiry date was generallythree months after purchase. They were not traded in secondary markets.

See also

American Stock Exchange

Chicago Board Options Exchange Eurex Euronext.liffe International Securities Exchange NYSE Arca Philadelphia Stock Exchange LEAPS (finance) Real options analysis

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