TheInfoList

OR:

In
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fin ...
, an option is a contract which conveys to its owner, the ''holder'', the right, but not the obligation, to buy or sell a specific quantity of an
underlying In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
asset In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value of ownership that can ...
or instrument at a specified
strike price In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security Security is protection from, or resilie ...
on or before a specified
date Date or dates may refer to: *Date (fruit), the fruit of the date palm (''Phoenix dactylifera'') Social activity *Dating, a form of courtship involving social activity, with the aim of assessing a potential partner **Group dating * Play date, a ...
, depending on the
style Style is a manner of doing or presenting things and may refer to: * Architectural style, the features that make a building or structure historically identifiable * Design, the process of creating something * Fashion, a prevailing mode of clothi ...
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 price, time until expiration, market volatility, the risk-free rate of interest, and the strike price of the option. Options may be traded between private parties in '' over-the-counter'' (OTC) transactions, or they may be exchange-traded in live, public 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. Selling or exercising an option before expiry typically requires a buyer to pick the contract up at the agreed upon price. The strike price may be set by reference to the
spot price In finance, a spot contract, spot transaction, or simply spot, is a contract of buying or selling a commodity, security or currency for immediate settlement (payment and delivery) on the spot date, which is normally two business days after the ...
(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, Pa ...
, while one that conveys the right to sell at a specified price is known as a put. 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 in which previously issued financial instruments such as stock, bonds, options, and futures are bought and sold. The initial sale of th ...
, in either an over-the-counter 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 profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed is taken to be re-inv ...
.

# 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 used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
mathematician and philosopher
Thales of Miletus Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded ...
. On a certain occasion, it was predicted that the season's olive 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 (now Euronext), 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 given name of Germanic origin.Hanks, Hardcastle and Hodges, ''Oxford Dictionary of First Names'', Oxford University Press, 2nd edition, , p. 276. It became very popular in the English language after the Norman conquest of Engl ...
and
Mary Mary may refer to: People * Mary (name), a feminine given name (includes a list of people with the name) Religious contexts * New Testament people named Mary, overview article linking to many of those below * Mary, mother of Jesus, also cal ...
. 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 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 an option giving the right – but not the obligation – to dramatize a specific book or script. Lines of credit give the potential borrower the right – but not the obligation – to borrow within a specified time period. Many choices, or embedded options, have traditionally been included in bond contracts. For example, many bonds are convertible into common stock at the buyer's option, or may be called (bought back) at specified prices at the issuer's option. Mortgage borrowers have long had 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 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 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, 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. 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 In finance, 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 at a set price. The buyer of the call option has the right, but not the obligation, to buy ...
) or the right to sell (a put option) * the quantity and class of the
underlying In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
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 is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security Security is protection from, or resilie ...
, also known as the exercise price, which is the price at which the underlying transaction will occur upon
exercise Exercise is a body activity that enhances or maintains physical fitness and overall health and wellness. It is performed for various reasons, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic ...
* the expiration 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 with fulfillment guaranteed by the Options Clearing Corporation (OCC). Since the contracts are standardized, accurate pricing models are often available. Exchange-traded options include: * Stock options * Bond options and other interest rate options *
Stock market index option Stock market index option is a type of option, a financial derivative, that is based on stock indices like the S&P 500 or the Dow Jones Industrial Average. They give an investor the right to buy or sell the underlying stock index for a defined time ...
s or, simply, index options and * Options on futures contracts * Callable bull/bear contract

### Over-the-counter options

Over-the-counter 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 swaps. Types of ...
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 in which previously issued financial instruments such as stock, bonds, options, and futures are bought and sold. The initial sale of th ...
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 prov ...
. 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 used to uniquely identify publicly traded shares of a particular stock on a particular stock market. In short, ticker symbols are arrangements of symbols or characters (generally Latin letters or ...
. By publishing continuous, live markets for option prices, an exchange enables independent parties to engage in price discovery 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 or assessment of something, in terms of quality, quantity, or some combination of both. Rating or ratings may also refer to: Business and economics * Credit rating, estimating the credit worthiness of an individual, c ...
(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 In finance, 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 at a set price. The buyer of the call option has the right, but not the obligation, to buy ...
to purchase the stock at a fixed price (
strike price In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security Security is protection from, or resilie ...

### Short call

A trader who expects a stock's price to decrease can sell the stock short or instead sell, or "write", a call. The trader selling a call has an obligation to sell the stock to the call buyer at 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 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. A
condor Condor is the common name for two species of New World vultures, each in a monotypic genus. The name derives from the Quechua ''kuntur''. They are the largest flying land birds in the Western Hemisphere. They are: * The Andean condor (''V ...
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, a straddle strategy involves two transactions in options on the same underlying, with opposite positions. One holds long risk, the other short. As a result, it involves the purchase or sale of particular option derivatives that all ...
(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 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, in which a trader buys a stock (or holds a previously-purchased long stock position), and sells a call. (This can be contrasted with a
naked call A naked option or uncovered option is an options contract where the option writer (i.e., the seller) does not hold the underlying security position to cover the contract in case of assignment (like in a covered option). Nor does the seller hold ...

# 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 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 (), in civil law jurisdicions known also as a hypothec loan, is a loan used either by purchasers of real property to raise funds to buy real estate, or by existing property owners to raise funds for any pu ...
s. However, many of the valuation and risk management principles apply across all financial options.

## Option styles

Options are classified into a number of styles, the most common of which are: * American option – an option that may be exercised on any trading day on or before expiration. * 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 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, Netherla ...
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, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
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, a differentiable 4-manifold, homeomorphic but not diffeomorphic to the Euclidean space R4 *Exotic sphere, a differentiable ''n''-manifold, homeomorphic but not diffeomorphic to the ordinar ...
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 that asset prices - and hence asset pricing models - will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is us ...
(i.e. risk neutrality), moneyness,
option time value In finance, the time value (TV) (''extrinsic'' or ''instrumental'' value) of an option is the premium a rational investor would pay over its ''current'' exercise value ( intrinsic value), based on the probability it will increase in value before e ...
, and put–call parity. The valuation itself combines a model of the behavior ( "process") 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 investment instruments. From the parabolic partial differential equation in the model, known as the Bl ...
for equities, to the Heath–Jarrow–Morton framework for interest rates, to the Heston model where volatility itself is considered
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themsel ...
. See
Asset pricing In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspo ...
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, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
, 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, for a defined period of time, in exchange for a charge or fee.See "Time Value", "Discount", "Discount Yield", "Compound Interest", "Efficie ...
expected value 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 and finance, risk neutral preferences are preferences that are neither risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is ind ...
pricing, and using
stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created ...
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 investment instruments. From the parabolic partial differential equation in the model, known as the Bl ...
. More sophisticated models are used to model the
volatility smile Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...
. 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 is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security Security is protection from, or resilie ...
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 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 at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as pa ...
and later work by Robert C. Merton,
Fischer Black Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American economist, best known as one of the authors of the Black–Scholes equation. Background Fischer Sheffey Black was born on January 11, 1938. He graduated from Harvard ...
and
Myron Scholes Myron Samuel Scholes ( ; born July 1, 1941) is a Canadian- American financial economist. Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel Laureate in Economic Sciences, and co-origi ...
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 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's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfre ...
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 market
implied volatility In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equ ...
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
volatility smile Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given exp ...
; 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 probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themsel ...
, with the resultant stochastic volatility models, and the Heston model as prototype; see #Risk-neutral_measure for a discussion of the logic. Other models include the 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 Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
. An alternate, though related, approach is to apply a local volatility model, where volatility is treated as a ''
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
'' 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 investment instruments. From the parabolic partial differential equation in the model, known as the Bl ...
, 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 #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 swaps. Types of ...
s (i.e. options on swaps), and
interest rate cap and floor An interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for ...
s (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 and 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 In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or ye ...
, 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 A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
and using analytical methods, develop 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 investment instruments. From the parabolic partial differential equation in the model, known as the Bl ...
and the Black model. The resulting solutions are readily computable, as are their "Greeks". Although the Roll–Geske–Whaley model applies to an American call with one dividend, for other cases of
American option In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options ...
s, 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 Cox, 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 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 option In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options ...
s 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) In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is ...
.

### Monte Carlo models

For many classes of options, traditional valuation techniques are 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
simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the s ...
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 In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of ...
for the option. Note though, that despite its flexibility, using simulation for 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 In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE ...
). Once expressed in this form, a finite difference model can be derived, and the valuation obtained. A number of implementations of finite difference methods exist for option valuation, including: explicit finite difference, 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 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
return Return may refer to: In business, economics, and finance * Return on investment (ROI), the financial gain after an expense. * Rate of return, the financial term for the profit or loss derived from an investment * Tax return, a blank document or ...
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 $\Delta$, $\Gamma$, $\kappa$ and $\theta$ are the standard hedge parameters calculated from an option valuation model, such as 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 In finance, delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged when small changes occur in the value of the underlying security. Such a portfolio typically contains options and their ...
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 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 risk. In an option contract this risk is that the seller will not 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.

## Options approval levels

To limit risk, brokers use access control systems to restrict traders from executing certain options strategies that would not be suitable for them. Brokers generally offer about four or five approval levels, with the lowest level offering the lowest risk and the highest level offering the highest risk. The actual numbers of levels, and the specific options strategies permitted at each level, vary between brokers. Brokers may also have their own specific vetting criteria, but they are usually based on factors such as the trader's annual salary and net worth, trading experience, and investment goals (capital preservation, income, growth, or speculation). For example, a trader with a low salary and net worth, little trading experience, and only concerned about preserving capital generally would not be permitted to execute high-risk strategies like
naked call A naked option or uncovered option is an options contract where the option writer (i.e., the seller) does not hold the underlying security position to cover the contract in case of assignment (like in a covered option). Nor does the seller hold ...
s and naked puts. Traders can update their information when requesting permission to upgrade to a higher approval level.

*
American Stock Exchange NYSE American, formerly known as the American Stock Exchange (AMEX), and more recently as NYSE MKT, is an American stock exchange situated in New York City. AMEX was previously a mutual organization, owned by its members. Until 1953, it was know ...
* Area yield options contract * Ascot (finance) * Chicago Board Options Exchange * Dilutive security *
Eurex Eurex Exchange is an international exchange which primarily offers trading in European based derivatives. It is the largest European futures and options market. The products traded on this exchange vary from German and Swiss debt instruments to E ...
* Euronext.liffe * International Securities Exchange *
NYSE Arca NYSE Arca, previously known as ArcaEx, an abbreviation of Archipelago Exchange, is an exchange on which both stocks and options are traded. It was owned by Intercontinental Exchange. It merged with the New York Stock Exchange in 2006 and now op ...
* Philadelphia Stock Exchange * LEAPS (finance) * Options backdating * Options Clearing Corporation * Options spread * Options strategy * Option symbol * Real options analysis *
PnL Explained In investment banking, PnL Explained (also called P&L Explain, P&L Attribution or Profit and Loss Explained) is an income statement with commentary that attributes or ''explains'' the daily fluctuation in the value of a portfolio of trades to th ...
* Pin risk (options) *
XVA An X-Value Adjustment (XVA, xVA) is an umbrella term referring to a number of different “valuation adjustments” that banks must make when assessing the value of derivative contracts that they have entered into. The purpose of these is twofold: ...

# 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). * 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"
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