In

strike price
In finance, the strike price (or exercise price) of an option (finance), option is a fixed price at which the owner of the option can buy (in the case of a call option, call), or sell (in the case of a put option, put), the underlying Security (fin ...

of the option, particularly in relation to the current market price of the underlying (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

Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the partial differential equation in the model, known as ...

and the Black model. The resulting solutions are readily computable, as are their Greeks (finance), "Greeks". Although the Roll–Geske–Whaley model applies to an American call with one dividend, for other cases of American options, closed form solutions are not available; approximations here include Barone-Adesi and Whaley, Bjerksund and Stensland and others.

Journal of Political Economy

', 81 (3), 637–654 (1973). * Feldman, Barry and Dhuv Roy. "Passive Options-Based Investment Strategies: The Case of the CBOE S&P 500 BuyWrite Index.

''The Journal of Investing''

(Summer 2005). * Hagen Kleinert, Kleinert, Hagen, ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, World Scientific (Singapore, 2004); Paperback ''(also available online

PDF-files

'' * Hill, Joanne, Venkatesh Balasubramanian, Krag (Buzz) Gregory, and Ingrid Tierens. "Finding Alpha via Covered Index Writing.

Financial Analysts Journal

(Sept.-Oct. 2006). pp. 29–46. * * Moran, Matthew. “Risk-adjusted Performance for Derivatives-based Indexes – Tools to Help Stabilize Returns.”

The Journal of Indexes

'. (Fourth Quarter, 2002) pp. 34–40. * Reilly, Frank and Keith C. Brown, Investment Analysis and Portfolio Management, 7th edition, Thompson Southwestern, 2003, pp. 994–5. * Schneeweis, Thomas, and Richard Spurgin. "The Benefits of Index Option-Based Strategies for Institutional Portfolios"

The Journal of Alternative Investments

', (Spring 2001), pp. 44–52. * Whaley, Robert. "Risk and Return of the CBOE BuyWrite Monthly Index"

The Journal of Derivatives

', (Winter 2002), pp. 35–42. * Bloss, Michael; Ernst, Dietmar; Häcker Joachim (2008): Derivatives – An authoritative guide to derivatives for financial intermediaries and investors Oldenbourg Verlag München * Espen Gaarder Haug & Nassim Nicholas Taleb (2008)

"Why We Have Never Used the Black–Scholes–Merton Option Pricing Formula"

{{Authority control Options (finance), Contract law

finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in the left corn ...

, an option is a contract which conveys to its owner, the ''holder'', the right, but not the obligation, to buy or sell an underlying
In finance, the underlying of a derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...

asset
In financial accounting
Financial accounting is the field of accounting
Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic entities such a ...

or instrument
Instrument may refer to:
Science and technology
* Flight instruments
two-seat light airplane. The flight instruments are visible on the left of the instrument panel
Flight instruments are the instruments in the cockpit of an aircraft that pro ...

at a specified strike price
In finance, the strike price (or exercise price) of an option (finance), option is a fixed price at which the owner of the option can buy (in the case of a call option, call), or sell (in the case of a put option, put), the underlying Security (fin ...

on or before a specified date, depending on the style
Style is a manner of doing or presenting things and may refer to:
* Architectural style
An architectural style is a set of characteristics and features that make a building or other structure notable or historically identifiable. It is a sub-cla ...

of the option. Options are typically acquired by purchase, as a form of compensation, or as part of a complex financial transaction. Thus, they are also a form of asset and have a valuation that may depend on a complex relationship between underlying asset value, time until expiration, market volatility, and other factors. Options may be traded between private parties in ''over-the-counter
Over-the-counter (OTC) drugs are medicine
Medicine is the Art (skill), art, science, and Praxis (process) , practice of caring for a patient and managing the diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment or Palliat ...

'' (OTC) transactions, or they may be exchange-traded in live, orderly markets in the form of standardized contracts.
Definition and application

An option is a contract that allows the holder the right to buy or sell an underlying asset or financial instrument at a specified strike price on or before a specified date, depending on the form of the option. The strike price may be set by reference to thespot price
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availa ...

(market price) of the underlying security or commodity on the day an option is issued, or it may be fixed at a discount or at a premium. The issuer has the corresponding obligation to fulfill the transaction (to sell or buy) if the holder "exercises" the option. An option that conveys to the holder the right to buy at a specified price is referred to as a call
Call or Calls may refer to:
Arts, entertainment, and media Games
* Call, a type of betting in poker
* Call, in the game of contract bridge, a bid, pass, double, or redouble in the bidding stage
Music and dance
* Call (band), from Lahore, Pakis ...

, while one that conveys the right to sell at a specified price is known as a .
The issuer may grant an option to a buyer as part of another transaction (such as a share issue or as part of an employee incentive scheme), or the buyer may pay a premium to the issuer for the option. A call option would normally be exercised only when the strike price is below the market value of the underlying asset, while a put option would normally be exercised only when the strike price is above the market value. When an option is exercised, the cost to the option holder is the strike price of the asset acquired plus the premium, if any, paid to the issuer. If the option’s expiration date passes without the option being exercised, the option expires, and the holder forfeits the premium paid to the issuer. In any case, the premium is income to the issuer, and normally a capital loss to the option holder.
The holder of an option may on-sell the option to a third party in a secondary market
The secondary market, also called the aftermarket and follow on public offering, is the financial market
A financial market is a market
Market may refer to:
*Market (economics)
*Market economy
*Marketplace, a physical marketplace or public ...

, in either an over-the-counter
Over-the-counter (OTC) drugs are medicine
Medicine is the Art (skill), art, science, and Praxis (process) , practice of caring for a patient and managing the diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment or Palliat ...

transaction or on an options exchange, depending on the option. The market price of an American-style option normally closely follows that of the underlying stock being the difference between the market price of the stock and the strike price of the option. The actual market price of the option may vary depending on a number of factors, such as a significant option holder needing to sell the option due to the expiration date approaching and not having the financial resources to exercise the option, or a buyer in the market trying to amass a large option holding. The ownership of an option does not generally entitle the holder to any rights associated with the underlying asset, such as voting rights or any income from the underlying asset, such as a dividend
A dividend is a distribution of profit
Profit may refer to:
Business and law
* Profit (accounting), the difference between the purchase price and the costs of bringing to market
* Profit (economics), normal profit and economic profit
* Profit ...

.
History

Historical uses of options

Contracts similar to options have been used since ancient times. The first reputed option buyer was theancient Greek
Ancient Greek includes the forms of the Greek language
Greek ( el, label=Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...

mathematician and philosopher Thales of Miletus
Thales of Miletus ( ; el, Θαλῆς
Thales of Miletus ( ; el, Θαλῆς (ὁ Μιλήσιος), ''Thalēs''; ) was a Greek mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (fr ...

. On a certain occasion, it was predicted that the season's olive
The olive, botanical name ''Olea europaea'', meaning "European olive", is a species
In biology, a species is the basic unit of biological classification, classification and a taxonomic rank of an organism, as well as a unit of biodivers ...

harvest would be larger than usual, and during the off-season, he acquired the right to use a number of olive presses the following spring. When spring came and the olive harvest was larger than expected, he exercised his options and then rented the presses out at a much higher price than he paid for his 'option'.
The 1688 book Confusion of Confusions describes the trading of "opsies" on the Amsterdam stock exchange, explaining that "there will be only limited risks to you, while the gain may surpass all your imaginings and hopes."
In London, puts and "refusals" (calls) first became well-known trading instruments in the 1690s during the reign of William
William is a male
Male (♂) is the sex of an organism that produces the gamete known as sperm. A male gamete can fuse with a larger female gamete, or ovum, in the process of fertilization. A male cannot sexual reproduction, reproduce sexually ...

and Mary
Mary may refer to:
People
* Mary (name)
Mary is a feminine
Femininity (also called womanliness or girlishness) is a set of attributes, behaviors, and roles generally associated with women and girls. Although femininity is socially constru ...

. Privileges were options sold over the counter in nineteenth century America, with both puts and calls on shares offered by specialized dealers. Their exercise price was fixed at a rounded-off market price on the day or week that the option was bought, and the expiry date was generally three months after purchase. They were not traded in secondary markets.
In the real estate
Real estate is property consisting of land and the buildings on it, along with its natural resources such as crops, minerals or water; immovable property of this nature; an interest vested in this (also) an item of real property, (more genera ...

market, 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 they can buy all the plots in the entire parcel.
In the motion picture industry, film or theatrical producers often buy the right — but not the obligation — to dramatize a specific book or script.
Lines of credit
A line of credit is a Credit (finance), credit facility extended by a bank or other financial institution to a government, business or Personal finance, individual customer that enables the customer to draw on the facility when the customer needs ...

give the potential borrower the right — but not the obligation — to borrow within a specified time period.
Many choices, or embedded options, have traditionally been included in bond
Bond or bonds may refer to:
Common meanings
* Bond (finance)
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of ...

contracts. For example, many bonds are convertible
A convertible or cabriolet () is a passenger car
Passenger car may refer to:
* A car (automobile) as opposed to a truck
* Passenger car (Ferris wheel), a compartment for carrying passengers
* Passenger car (rail), railway rolling stock for carr ...

into common stock at the buyer's option, or may be called (bought back) at specified prices at the issuer's option. Mortgage
A mortgage loan or simply mortgage () is a loan
In finance, a loan is the lending of money by one or more individuals, organizations, or other entities to other individuals, organizations etc. The recipient (i.e., the borrower) incurs a ...

borrowers have long had the option to repay the loan early, which corresponds to a callable bond option.
Modern stock options

Options contracts have been known for decades. TheChicago Board Options Exchange
The Chicago Board Options Exchange (CBOE), located at 433 West Van Buren Street
250px, Street in downtown Bucharest (Romania)
A street is a public thoroughfare in a built environment
In urban planning, architecture and civil engineering, ...

was established in 1973, which set up a regime using standardized forms and terms and trade through a guaranteed clearing house. Trading activity and academic interest has increased since then.
Today, many options are created in a standardized form and traded through clearing houses on regulated options exchanges, while other over-the-counter
Over-the-counter (OTC) drugs are medicine
Medicine is the Art (skill), art, science, and Praxis (process) , practice of caring for a patient and managing the diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment or Palliat ...

options are written as bilateral, customized contracts between a single buyer and seller, one or both of which may be a dealer or market-maker. Options are part of a larger class of financial instruments known as derivative products
In finance, a derivative is a contract
A contract is a legally binding agreement that defines and governs the rights and duties between or among its parties
Image:'Hip, Hip, Hurrah! Artist Festival at Skagen', by Peder Severin Krøyer (18 ...

, or simply, derivatives.
Contract specifications

A financial option is a contract between two counterparties with the terms of the option specified in aterm sheet
A term sheet is a bullet-point
In typography
File:metal movable type.jpg, 225px, Movable type being assembled on a composing stick using pieces that are stored in the type case shown below it
Typography is the art and technique of types ...

. Option contracts may be quite complicated; however, at minimum, they usually contain the following specifications:
* whether the option holder has the right to buy (a call option
A call option, often simply labeled a "call", is a contract, between the buyer and the seller of the call option, to exchange a Security_(finance), security at a set price. The buyer of the call Option (finance), option has the right, but not ...

) or the right to sell (a put option
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availabl ...

)
* the quantity and class of the underlying
In finance, the underlying of a derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...

asset(s) (e.g., 100 shares of XYZ Co. B stock)
* the strike price
In finance, the strike price (or exercise price) of an option (finance), option is a fixed price at which the owner of the option can buy (in the case of a call option, call), or sell (in the case of a put option, put), the underlying Security (fin ...

, also known as the exercise price, which is the price at which the underlying transaction will occur upon exercise
Exercise is any bodily activity that enhances or maintains physical fitness
Physical fitness is a state of health
Health is a state of physical, mental and social well-being
Well-being, also known as ''wellness'', ''prudential value ...

* the expiration
Expiration or expiration date may refer to:
Expiration
Expiration may refer to:
*Death
*Exhalation of breath, breathing out
*Expiration (options), the legal termination of an option to take an action
*Shelf life, or the time after which a product e ...

date, or expiry, which is the last date the option can be exercised
* the settlement terms, for instance whether the writer must deliver the actual asset on exercise, or may simply tender the equivalent cash amount
* the terms by which the option is quoted in the market to convert the quoted price into the actual premium – the total amount paid by the holder to the writer
Option trading

Forms of trading

Exchange-traded options

Exchange-traded options (also called "listed options") are a class of exchange-traded derivatives. Exchange-traded options have standardized contracts, and are settled through aclearing house
Clearing house or Clearinghouse may refer to:
Banking and finance
* Clearing house (finance)
A clearing house is a financial institution formed to facilitate the exchange (i.e., '' clearance'') of payments, securities, or derivatives transactions ...

with fulfillment guaranteed by the Options Clearing Corporation
Options Clearing Corporation (OCC) is a United States Clearing house (finance), clearing house based in Chicago. It specializes in equity derivatives Clearing (financial), clearing, providing Central Counterparty Clearing, central counterparty ...

(OCC). Since the contracts are standardized, accurate pricing models are often available. Exchange-traded options include:
* Stock options
* Bond option
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available ...

s and other interest rate options
* Stock market index optionStock market index option is a type of option (finance), option, a financial Derivative (finance), derivative, that is based on Stock index, stock indices like the S&P 500 Index, S&P 500 or the Dow Jones Industrial Average. They give an investor the ...

s or, simply, index options and
* Options on futures contracts
* Callable bull/bear contractA callable bull/bear contract, or CBBC in short form, is a Derivative (finance), derivative financial instrument that provides investors with a Leverage (finance), leveraged investment in underlying assets, which can be a single stock, or an Index (e ...

Over-the-counter options

Over-the-counter
Over-the-counter (OTC) drugs are medicine
Medicine is the Art (skill), art, science, and Praxis (process) , practice of caring for a patient and managing the diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment or Palliat ...

options (OTC options, also called "dealer options") are traded between two private parties, and are not listed on an exchange. The terms of an OTC option are unrestricted and may be individually tailored to meet any business need. In general, the option writer is a well-capitalized institution (in order to prevent the credit risk). Option types commonly traded over the counter include:
* Interest rate options
* Currency cross rate options, and
* Options on swaps or swaption
A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swap
In finance
...

s.
By avoiding an exchange, users of OTC options can narrowly tailor the terms of the option contract to suit individual business requirements. In addition, OTC option transactions generally do 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 each other's clearing and settlement procedures.
With few exceptions, there are no secondary market
The secondary market, also called the aftermarket and follow on public offering, is the financial market
A financial market is a market
Market may refer to:
*Market (economics)
*Market economy
*Marketplace, a physical marketplace or public ...

s for employee stock options
Employee stock options (ESO) is a label that refers to compensation contracts between an employer and an employee that carries some characteristics of financial options.
Employee stock options are commonly viewed as an internal agreement provid ...

. These must either be exercised by the original grantee or allowed to expire.
Exchange trading

The most common way to trade options is via standardized options contracts that are listed by various futures and options exchanges. Listings and prices are tracked and can be looked up byticker symbol
A ticker symbol or stock symbol is an abbreviation
An abbreviation (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area ...

. By publishing continuous, live markets for option prices, an exchange enables independent parties to engage in price discovery
The price discovery process (also called price discovery mechanism) is the process of determining the price
A price is the (usually not negative) quantity of payment or compensation given by one party to another in return for one unit o ...

and execute transactions. As an intermediary to both 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 the highest rating
A rating is an evaluation
Evaluation is a
system
A system is a group of interacting
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the conc ...

(AAA),
* Counterparties remain anonymous,
* Enforcement of market regulation to ensure fairness and transparency, and
* Maintenance of orderly markets, especially during fast trading conditions.
Basic trades (American style)

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

A trader who expects a stock's price to increase can buy acall option
A call option, often simply labeled a "call", is a contract, between the buyer and the seller of the call option, to exchange a Security_(finance), security at a set price. The buyer of the call Option (finance), option has the right, but not ...

to purchase the stock at a fixed price (strike price
In finance, the strike price (or exercise price) of an option (finance), option is a fixed price at which the owner of the option can buy (in the case of a call option, call), or sell (in the case of a put option, put), the underlying Security (fin ...

) at a later date, rather than purchase the stock outright. The cash outlay on the option is the premium. The trader would have no obligation to buy the stock, but only has the right to do so on or before the expiration date. The risk of loss would be limited to the premium paid, unlike the possible loss had the stock been bought outright.
The holder of an American-style call option can sell the option holding at any time until the expiration date, and would consider doing so when the stock's spot price is above the exercise price, especially if the holder expects the price of the option to drop. By selling the option early in that situation, the trader can realise an immediate profit. Alternatively, the trader can exercise the option — for example, if there is no secondary market for the options — and then sell the stock, realising a profit. A trader would make a profit if the spot price of the shares rises by more than the premium. For example, if the exercise price is 100 and premium paid is 10, then if the spot price of 100 rises to only 110 the transaction is break-even; an increase in stock price above 110 produces a profit.
If the stock price at expiration is lower than the exercise price, the holder of the option at that time will let the call contract expire and lose only the premium (or the price paid on transfer).
Long put

A trader who expects a stock's price to decrease can buy aput option
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availabl ...

to sell the stock at a fixed price (strike price) at a later date. The trader is under no obligation to sell the stock, but has the right to do so on or before the expiration date. If the stock price at expiration is below the exercise price by more than the premium paid, the trader makes a profit. If the stock price at expiration is above the exercise price, the trader lets the put contract expire, and loses only the premium paid. In the transaction, the premium also plays a role as it enhances the break-even point. For example, if the exercise price is 100 and the premium paid is 10, then a spot price between 90 and 100 is not profitable. The trader makes a profit only if the spot price is below 90.
The trader exercising a put option on a stock does not need to own the underlying asset, because most stocks can be .
Short call

A trader who expects a stock's price to decrease can sell the stock or instead sell, or "write", a call. The trader selling a call has an obligation to sell the stock to the call buyer at a fixed price ("strike price"). If the seller does not own the stock when the option is exercised, they are obligated to purchase the stock in the market at the prevailing market price. If the stock price decreases, the seller of the call (call writer) makes a profit in the amount of the premium. If the stock price increases over the strike price by more than the amount of the premium, the seller loses money, with the potential loss being unlimited.Short put

A trader who expects a stock's price to increase can buy the stock or instead sell, or "write", a put. The trader selling a put has an obligation to buy the stock from the put buyer at a fixed price ("strike price"). If the stock price at expiration is above the strike price, the seller of the put (put writer) makes a profit in the amount of the premium. If the stock price at expiration is below the strike price by more than the amount of the premium, the trader loses money, with the potential loss being up to the strike price minus the premium. A benchmark index for the performance of a cash-secured short put option position is theCBOE S&P 500 PutWrite Index The CBOE S&P 500 PutWrite Index (ticker symbol PUT) is a benchmark index that measures the performance of a hypothetical portfolio that sells S&P 500
The Standard and Poor's 500, or simply the S&P 500, is a stock market index tracking the pe ...

(ticker PUT).
Options strategies

Combining any of the four basic kinds of option trades (possibly with different exercise prices and maturities) and the two basic kinds of stock trades (long and short) allows a variety of options strategies. Simple strategies usually combine only a few trades, while more complicated strategies can combine several. Strategies are often used to engineer a particular risk profile to movements in the underlying security. For example, buying abutterfly
Butterflies are insect
Insects (from Latin ') are pancrustacean Hexapoda, hexapod invertebrates of the class (biology), class Insecta. They are the largest group within the arthropod phylum. Insects have a chitinous exoskeleton, a three ...

spread (long one X1 call, short two X2 calls, and long one X3 call) allows a trader to profit if the stock price on the expiration date is near the middle exercise price, X2, and does not expose the trader to a large loss.
An iron condorThe iron condor is an Option (finance), option trading strategy utilizing two Vertical Spread, vertical spreads – a put spread and a call spread with the same expiration and four different strikes. A long iron condor is essentially selling both si ...

is a strategy that is similar to a butterfly spread, but with different strikes for the short options – offering a larger likelihood of profit but with a lower net credit compared to the butterfly spread.
Selling a straddle
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in ...

(selling both a put and a call at the same exercise price) would give a trader a greater profit than a butterfly if the final stock price is near the exercise price, but might result in a large loss.
Similar to the straddle
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in ...

is the strangle which is also constructed by a call and a put, but whose strikes are different, reducing the net debit of the trade, but also reducing the risk of loss in the trade.
One well-known strategy is the covered call
A covered call is a financial market transaction in which the seller of call option
A call option, often simply labeled a "call", is a contract, between the buyer and the seller of the call option, to exchange a security
Security is freedom ...

, in which a trader buys a stock (or holds a previously-purchased long stock position), and sells a call. If the stock price rises above the exercise price, the call will be exercised and the trader will get a fixed profit. If the stock price falls, the call will not be exercised, and any loss incurred to the trader will be partially offset by the premium received from selling the call. Overall, the payoffs match the payoffs from selling a put. This relationship is known as put–call parityIn financial mathematicsMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and ...

and offers insights for financial theory. A benchmark index for the performance of a buy-write strategy is the CBOE S&P 500 BuyWrite Index (ticker symbol BXM).
Another very common strategy is the , in which a trader buys a stock (or holds a previously-purchased long stock position), and buys a put. This strategy acts as an insurance when investing on the underlying stock, hedging the investor's potential losses, but also shrinking an otherwise larger profit, if just purchasing the stock without the put. The maximum profit of a protective put is theoretically unlimited as the strategy involves being long on the underlying stock. The maximum loss is limited to the purchase price of the underlying stock less the strike price of the put option and the premium paid. A protective put is also known as a married put.
Types

Options can be classified in a few ways.According to the option rights

* Call options give the holder the right—but not the obligation—to buy something at a specific price for a specific time period. * Put options give the holder the right—but not the obligation—to sell something at a specific price for a specific time period.According to the underlying assets

* Equity option * Bond option * Future option * Index option * Commodity option * Currency option * Swap optionOther option types

Another important class of options, particularly in the U.S., areemployee stock option
Employee stock options (ESO) is a label that refers to compensation contracts between an employer and an employee that carries some characteristics of financial options.
Employee stock options are commonly viewed as an internal agreement prov ...

s, which are awarded by a company to their employees as a form of incentive compensation. Other types of options exist in many financial contracts, for example real estate options are often used to assemble large parcels of land, and prepayment options are usually included in mortgage loan
A mortgage loan or simply mortgage () is a loan
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money ...

s. However, many of the valuation and risk management principles apply across all financial options. There are two more types of options; covered and naked.
Option styles

Options are classified into a number of styles, the most common of which are: * American option – an option that may beexercised
Exercise is any Human body, bodily activity that enhances or maintains physical fitness and overall health and wellness.
It is performed for various reasons, to aid growth and improve strength, preventing senescence, aging, developing muscles ...

on any trading day on or before expiration
Expiration or expiration date may refer to:
Expiration
Expiration may refer to:
*Death
*Exhalation of breath, breathing out
*Expiration (options), the legal termination of an option to take an action
*Shelf life, or the time after which a product ...

.
* European option – an option that may only be exercised on expiry.
These are often described as vanilla options. Other styles include:
* Bermudan option – an option that may be exercised only on specified dates on or before expiration.
* Asian
Asian may refer to:
* Items from or related to the continent of Asia:
** Asian people, people in or descending from Asia
** Asian culture, the culture of the people from Asia
** Asian cuisine, food based on the style of food of the people from Asi ...

option – an option whose payoff is determined by the average underlying price over some preset time period.
* Barrier
A barrier or barricade is a physical structure which blocks or impedes something.
Barrier may also refer to:
Places
* Barrier, Kentucky, a community in the United States
* Barrier, Voerendaal, a place in the municipality of Voerendaal, in southeas ...

option – any option with the general characteristic that the underlying security's price must pass a certain level or "barrier" before it can be exercised.
* Binary
Binary may refer to:
Science and technology
Mathematics
* Binary number
In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: ty ...

option – An all-or-nothing option that pays the full amount if the underlying security meets the defined condition on expiration otherwise it expires.
* Exotic
Exotic may refer to:
Mathematics and physics
*Exotic R4, Exotic R4, a differentiable 4-manifold, homeomorphic but not diffeomorphic to the Euclidean space R4
*Exotic sphere, a differentiable ''n''-manifold, homeomorphic but not diffeomorphic to th ...

option – any of a broad category of options that may include complex financial structures.
Valuation

Because the values of option contracts depend on a number of different variables in addition to the value of the underlying asset, they are complex to value. There are many pricing models in use, although all essentially incorporate the concepts ofrational pricing
Rational pricing is the assumption in financial economics
Financial economics is the branch of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), productio ...

(i.e. risk neutral
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods an ...

ity), moneyness
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in ...

, option time valueIn finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available w ...

, and put–call parityIn financial mathematicsMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and ...

.
The valuation itself combines a model of the behavior () of the underlying price with a mathematical method which returns the premium as a function of the assumed behavior.
The models range from the (prototypical) Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the partial differential equation in the model, known as ...

for equities, to the Heath–Jarrow–Morton framework for interest rates, to the Heston model
In finance, the Heston model, named after Steven Heston, is a mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules ...

where volatility itself is considered stochastic
Stochastic () refers to the property of being well described by a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...

. See Asset pricing
:'' This article is theory focused: for the corporate finance
Corporate finance is the area of finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with ...

for a listing of the various models here.
Basic decomposition

In its most basic terms, the value of an option is commonly decomposed into two parts: * The first part is the intrinsic value, which is defined as the difference between the market value of theunderlying
In finance, the underlying of a derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...

, and the strike price of the given option
* The second part is the time value, which depends on a set of other factors which, through a multi-variable, non-linear interrelationship, reflect the discounted
Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor
A creditor or lender is a party (e.g., person, organization, company, or government) that has a claim on the services of a second party. ...

expected value
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...

of that difference at expiration.
Valuation models

As above, the value of the option is estimated using a variety of quantitative techniques, all based on the principle ofrisk-neutral
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods an ...

pricing, and using stochastic calculus
Stochastic calculus is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (math ...

in their solution. The most basic model is the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the partial differential equation in the model, known as ...

. More sophisticated models are used to model the . These models are implemented using a variety of numerical techniques. In general, standard option valuation models depend on the following factors:
* The current market price of the underlying security
* The expiration
Expiration or expiration date may refer to:
Expiration
Expiration may refer to:
*Death
*Exhalation of breath, breathing out
*Expiration (options), the legal termination of an option to take an action
*Shelf life, or the time after which a product e ...

together with any restrictions on when exercise may occur
* an estimate of the future volatility 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 volatility changes over time and for various underlying price levels, or the dynamics of stochastic interest rates.
The following are some of the principal valuation techniques used in practice to evaluate option contracts.
Black–Scholes

Following early work byLouis Bachelier
Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study ...

and later work by , Fischer Black
Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American economist
An economist is a practitioner in the social sciences, social science discipline of economics.
The individual may also study, develop, and apply theories an ...

and Myron Scholes
Myron Samuel Scholes ( ; born July 1, 1941) is a Canadian
Canadians (french: Canadiens) are people identified with the country of Canada. This connection may be residential, legal, historical or cultural. For most Canadians, many (or all) of ...

made a major breakthrough by deriving a differential 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 that replicates the returns of holding an option, Black and Scholes produced a closed-form solution for 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 behind the Black–Scholes model were ground-breaking and eventually led to Scholes and Merton
Merton may refer to:
People
* Merton (surname)
* Merton (given name)
* Merton (YouTube), American YouTube personality
Fictional characters
* Merton Matowski, an alternate name for "Moose" Mason, an Archie Comics character
* List of Downton Ab ...

receiving the Swedish Central Bank's associated Prize for Achievement in Economics (a.k.a., the Nobel Prize
The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel
Alfred Bernhard Nobel ( , ; 21 October 1833 – 10 December 1896) was a Swedish chemist, engineer, inventor, busines ...

in Economics), the application of the model in actual options trading is clumsy because of the assumptions of continuous trading, constant volatility, and a constant interest rate. Nevertheless, the Black–Scholes model is still one of the most important methods and foundations for the existing financial market in which the result is within the reasonable range.
Stochastic volatility models

Since the market crash of 1987, it has been observed that marketimplied volatilityIn financial mathematicsMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and ...

for options of lower strike prices are typically higher than for higher strike prices, suggesting that volatility varies both for time and for the price level of the underlying security a so-called ; and with a time dimension, a volatility surface
Volatility or volatile may refer to:
Chemistry
* Volatility (chemistry)
In chemistry, volatility is a material quality which describes how readily a substance vaporizes. In at a given temperature and pressure, a substance with high volatility is ...

.
The main approach here is to treat volatility as stochastic
Stochastic () refers to the property of being well described by a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...

, with the resultant Stochastic volatility
In statistics, stochastic volatility models are those in which the variance
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, ...

models, and the Heston model
In finance, the Heston model, named after Steven Heston, is a mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules ...

as prototype; see #Risk-neutral_measure for a discussion of the logic. Other models include the CEV and SABR volatility modelIn mathematical financeMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and e ...

s. One principal advantage of the Heston model, however, is that it can be solved in closed-form, while other stochastic volatility models require complex numerical methods
Numerical analysis is the study of algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as ...

.
An alternate, though related, approach is to apply a local volatilityA local volatility model, in mathematical financeMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical ...

model, where volatility is treated as a ''deterministic
Determinism is the philosophical
Philosophy (from , ) is the study of general and fundamental questions, such as those about existence
Existence is the ability of an entity to interact with physical or mental reality
Reality is the ...

'' function of both the current asset level $S\_t$ and of time $t$. As such, a local volatility model is a generalisation of the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the partial differential equation in the model, known as ...

, where the volatility is a constant. The concept was developed when Bruno DupireBruno Dupire is a researcher and lecturer in quantitative finance. He is currently Head of Quantitative Research at Bloomberg LP. He is best known for his contributions to local volatility modeling and Functional Ito Calculus. He is also an Instructo ...

and Emanuel Derman
Emanuel Derman (born 1945) is a South African-born academic, businessman and writer. He is best known as a quantitative analyst, and author of the book ''My Life as a Quant: Reflections on Physics and Finance''.
He is a co-author of Black–Derman ...

and Iraj Kani
Iraj ( fa, ایرج - ʾīraj; Middle Persian, Pahlavi: ērič; from Avestan language, Avestan: 𐬀𐬌𐬭𐬌𐬌𐬀 airiia, literally "Aryan") is seventh Shah of the Pishdadian dynasty of ''Shahnameh''. Based on Persian mythology, Iranian myth ...

noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options. See for discussion.
Short-rate models

For the valuation ofbond option
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available ...

s, swaption
A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swap
In finance
...

s (i.e. options on swaps), and interest rate cap and floor An interest rate cap is a type of interest rate derivative
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money ...

s (effectively options on the interest rate) various short-rate modelA short-rate model, in the context of interest rate derivatives, is a mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of ...

s have been developed (applicable, in fact, to interest rate derivatives
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available ...

generally). The best known of these are Black-Derman-Toy and Hull–White.
These models describe the future evolution of interest rates
An interest rate is the amount of interest
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and inves ...

by describing the future evolution of the short rate. The other major framework for interest rate modelling is the Heath–Jarrow–Morton framework (HJM). The distinction is that HJM gives an analytical description of the ''entire'' yield curve
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money avai ...

, rather than just the short rate. (The HJM framework incorporates the Brace–Gatarek–Musiela model and market models. And some of the short rate models can be straightforwardly expressed in the HJM framework.) For some purposes, e.g., valuation of mortgage-backed securities, this can be a big simplification; regardless, the framework is often preferred for models of higher dimension. Note that for the simpler options here, i.e. those mentioned initially, the Black model can instead be employed, with certain assumptions.
Model implementation

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

In some cases, one can take the mathematical model and using analytical methods, develop Closed-form expression, closed form solutions such as theBinomial tree pricing model

Closely following the derivation of Black and Scholes, John Carrington Cox, John Cox, Stephen Ross (economist), Stephen Ross and Mark Rubinstein developed the original version of the binomial options pricing model. It models the dynamics of the option's theoretical value for Lattice model (finance), discrete time intervals over the option's life. The model starts with a binomial tree of discrete future possible underlying stock prices. By constructing a riskless portfolio of an option and stock (as in the Black–Scholes model) a simple formula can be used to find the option price at each node in the tree. This value can approximate the theoretical value produced by Black–Scholes, to the desired degree of precision. However, the binomial 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, and American options can be modeled as well as European ones. Binomial models are widely used by professional option traders. The Trinomial tree is a similar model, allowing for an up, down or stable path; although considered more accurate, particularly when fewer time-steps are modelled, it is less commonly used as its implementation is more complex. For a more general discussion, as well as for application to commodities, interest rates and hybrid instruments, see Lattice model (finance).Monte Carlo models

For many classes of options, traditional valuation techniques are Tractable problem, intractable because of the complexity 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 value in relation to the underlying security's price, a Monte Carlo model uses Monte Carlo simulation, simulation to generate random price paths of the underlying asset, each of which results in a payoff for the option. The average of these payoffs can be discounted to yield an expectation value for the option. Note though, that despite its flexibility, using simulation for american option, American styled options is somewhat more complex than for lattice based models.Finite difference models

The equations used to model the option are often expressed as partial differential equations (see for example Black–Scholes equation). Once expressed in this form, a Finite difference method, finite difference model can be derived, and the valuation obtained. A number of implementations of finite difference methods exist for option valuation, including: explicit method, explicit finite difference, implicit method, implicit finite difference and the Crank–Nicolson method. A trinomial tree option pricing model can be shown to be a simplified application of the explicit finite difference method. Although the finite difference approach is mathematically sophisticated, it is particularly useful where changes are assumed over time in model inputs – for example dividend yield, risk-free rate, or volatility, or some combination of these – that are not tractable problem, tractable in closed form.Other models

Other numerical implementations which have been used to value options include finite element methods.Risks

As with all securities, trading options entails the risk of the option's value changing over time. However, unlike traditional securities, the Stock option return, return from holding an option varies non-linearly with the value of the underlying and other factors. Therefore, the risks associated with holding options are more complicated to understand and predict. In general, the change in the value of an option can be derived from Itô's lemma as: ::$dC=\backslash Delta\; dS\; +\; \backslash Gamma\; \backslash frac\; +\; \backslash kappa\; d\backslash sigma\; +\; \backslash theta\; dt\; \backslash ,$ where the Greeks (finance), Greeks $\backslash Delta$, $\backslash Gamma$, $\backslash kappa$ and $\backslash theta$ are the standard hedge parameters calculated from an option valuation model, such as Black–Scholes model, Black–Scholes, and $dS$, $d\backslash sigma$ and $dt$ are unit changes in the underlying's price, the underlying's volatility and time, respectively. Thus, at any point in time, one can estimate the risk inherent in holding an option by calculating its hedge parameters and then estimating the expected change in the model inputs, $dS$, $d\backslash sigma$ and $dt$, provided the changes in these values are small. This technique can be used effectively to understand and manage the risks associated with standard options. For instance, by offsetting a holding in an option with the quantity $-\backslash Delta$ of shares in the underlying, a trader can form a delta neutral portfolio that is hedged from loss for small changes in the underlying's price. The corresponding price sensitivity formula for this portfolio $\backslash Pi$ is: ::$d\backslash Pi=\backslash Delta\; dS\; +\; \backslash Gamma\; \backslash frac\; +\; \backslash kappa\; d\backslash sigma\; +\; \backslash theta\; dt\; -\; \backslash Delta\; dS\; =\; \backslash Gamma\; \backslash frac\; +\; \backslash kappa\; d\backslash sigma\; +\; \backslash theta\; dt\backslash ,$Pin risk

A special situation called Pin risk (options), pin risk can arise when the underlying closes at or very close to the option'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 be allowed to expire. Therefore, the option writer may end up with a large, unwanted residual position in the underlying when the markets open on the next trading day after expiration, regardless of his or her best efforts to avoid such a residual.Counterparty risk

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

* American Stock Exchange * Area yield options contract * Ascot (finance) *Chicago Board Options Exchange
The Chicago Board Options Exchange (CBOE), located at 433 West Van Buren Street
250px, Street in downtown Bucharest (Romania)
A street is a public thoroughfare in a built environment
In urban planning, architecture and civil engineering, ...

* Dilutive security
* Eurex
* Euronext.liffe
* International Securities Exchange
* NYSE Arca
* Philadelphia Stock Exchange
* LEAPS (finance)
* Options backdating
* Options Clearing Corporation
Options Clearing Corporation (OCC) is a United States Clearing house (finance), clearing house based in Chicago. It specializes in equity derivatives Clearing (financial), clearing, providing Central Counterparty Clearing, central counterparty ...

* Options spread
* Options strategy
* Option symbol
* Real options analysis
* PnL Explained
* Pin risk (options)
* XVA
References

Further reading

* Fischer Black and Myron S. Scholes. "The Pricing of Options and Corporate Liabilities,"Journal of Political Economy

', 81 (3), 637–654 (1973). * Feldman, Barry and Dhuv Roy. "Passive Options-Based Investment Strategies: The Case of the CBOE S&P 500 BuyWrite Index.

''The Journal of Investing''

(Summer 2005). * Hagen Kleinert, Kleinert, Hagen, ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, World Scientific (Singapore, 2004); Paperback ''(also available online

PDF-files

'' * Hill, Joanne, Venkatesh Balasubramanian, Krag (Buzz) Gregory, and Ingrid Tierens. "Finding Alpha via Covered Index Writing.

Financial Analysts Journal

(Sept.-Oct. 2006). pp. 29–46. * * Moran, Matthew. “Risk-adjusted Performance for Derivatives-based Indexes – Tools to Help Stabilize Returns.”

The Journal of Indexes

'. (Fourth Quarter, 2002) pp. 34–40. * Reilly, Frank and Keith C. Brown, Investment Analysis and Portfolio Management, 7th edition, Thompson Southwestern, 2003, pp. 994–5. * Schneeweis, Thomas, and Richard Spurgin. "The Benefits of Index Option-Based Strategies for Institutional Portfolios"

The Journal of Alternative Investments

', (Spring 2001), pp. 44–52. * Whaley, Robert. "Risk and Return of the CBOE BuyWrite Monthly Index"

The Journal of Derivatives

', (Winter 2002), pp. 35–42. * Bloss, Michael; Ernst, Dietmar; Häcker Joachim (2008): Derivatives – An authoritative guide to derivatives for financial intermediaries and investors Oldenbourg Verlag München * Espen Gaarder Haug & Nassim Nicholas Taleb (2008)

"Why We Have Never Used the Black–Scholes–Merton Option Pricing Formula"

{{Authority control Options (finance), Contract law