TheInfoList

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 which could ...

, 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 the
spot 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 the
ancient 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. The
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, ...
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 a
term 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 Security is freedom from, or resilience against, potential Potential generally refers to a currently un ...

) or the right to sell (a
put option 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 ...

) * 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 ...
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

Exchange-traded options (also called "listed options") are a class of exchange-traded derivatives. Exchange-traded options have standardized contracts, and are settled through a
clearing 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., ''Clearing (finance), clearance'') of payments, securities, or deri ...
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.

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 by
ticker 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.

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 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 Security is freedom from, or resilience against, potential Potential generally refers to a currently un ...

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 ...

### Long put

A trader who expects a stock's price to decrease can buy a
put option 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 ...

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 the CBOE S&P 500 PutWrite Index (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 a
butterfly 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 options owns the corresponding amount of the underlying instrument, such as shares of a stock or other securities. If a trader buys the underlying instrument at the same ti ...

, 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 option

## Other option types

Another important class of options, particularly in the U.S., are
employee 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 . Employee stock options are commonly viewed as an internal agreement providing the possibil ...
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 be
exercised 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 of
rational 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 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 ...
,
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 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 ...
, 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 concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
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 of
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 ...
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
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
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 ...
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 by
Louis 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 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, Prize for Achievement in Economics (a.k.a., the Nobel Prize 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 Black Monday (1987), market crash of 1987, it has been observed that market implied volatility 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. 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 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 Heston model#Risk-neutral measure, #Risk-neutral_measure for a discussion of the logic. Other models include the Constant elasticity of variance model, CEV and SABR volatility models. 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. An alternate, though related, approach is to apply a local volatility model, where Volatility (finance), volatility is treated as a ''deterministic'' 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 Dupire and Emanuel Derman and Iraj Kani noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options. See Local volatility#Development, #Development for discussion.

### Short-rate models

For the valuation of bond options,
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 floors (effectively options on the interest rate) various short-rate models have been developed (applicable, in fact, to interest rate derivatives generally). The best known of these are Black–Derman–Toy model, Black-Derman-Toy and Hull–White model, Hull–White. These models describe the future evolution of interest rates 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, 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 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 ...
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.

### Binomial 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: ::$dC=\Delta dS + \Gamma \frac + \kappa d\sigma + \theta dt \,$ where the Greeks (finance), Greeks $\Delta$, $\Gamma$, $\kappa$ and $\theta$ are the standard hedge parameters calculated from an option valuation model, such as Black–Scholes model, Black–Scholes, and $dS$, $d\sigma$ 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, $dS$, $d\sigma$ 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 $-\Delta$ 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 $\Pi$ is: ::$d\Pi=\Delta dS + \Gamma \frac + \kappa d\sigma + \theta dt - \Delta dS = \Gamma \frac + \kappa d\sigma + \theta dt\,$

## 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.

* 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

* 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