An option is a financial contract that gives its holder the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a fixed strike price on or before a certain date, depending on the type of option.
Options are usually acquired through purchase, used as part of compensation, or included in more complex financial transactions. Because of this, an option can be treated as either an asset or a contingent liability. Its value depends on several factors, including the price of the underlying asset, time remaining until expiration, market volatility, the risk-free rate of interest, and the strike price.
Options may be traded privately between parties in over-the-counter markets, or they may be traded on exchanges in the form of standardized contracts.
Definition and Application
An option is a contract that gives 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. The exact meaning depends on the style of the option. The seller of the option has the corresponding obligation to complete the transaction if the holder chooses to exercise the option.
An option that gives the right to buy is called a call option. An option that gives the right to sell is called a put option. The strike price may be linked to the spot price of the underlying asset when the option is issued, or it may be set at a discount or premium.
The buyer may receive an option as part of another transaction, such as a share issue or employee compensation plan, or may pay a premium directly to the issuer. A call option is usually exercised when the market price of the underlying asset is above the strike price, while a put option is usually exercised when the market price is below the strike price.
When an option is exercised, the holder must pay the strike price, along with any premium that was originally paid. If the option expires without being exercised, the holder loses the premium. In that case, the premium becomes income for the issuer and a loss for the holder.
An option holder may also sell the option to another party before expiration. This can happen in an exchange market or in an over-the-counter market, depending on the contract. However, owning an option does not usually give the holder voting rights or dividend rights in the underlying asset.
History
Early Uses of Options
Contracts similar to options have existed since ancient times. One of the earliest reported examples is associated with Thales of Miletus, who reportedly gained rights to olive presses in advance of a large harvest and later used those rights profitably.
A famous early reference to option-like trading appears in the 1688 book Confusion of Confusions, which described option trading on the Amsterdam stock exchange. In London, puts and calls became more widely known in the 1690s.
In the nineteenth century, option-like contracts were also used in the United States under the name “privileges.” These contracts were sold over the counter and were not traded in secondary markets.
Options have also long been used in real estate, where buyers may secure the right to purchase land without being obligated to complete the purchase immediately. Similar ideas appear in film rights, where producers buy the right to adapt a book or script, and in lending, where a line of credit gives a borrower the right to borrow within a specified period.
Many bond contracts also include embedded options. For example, some bonds can be converted into shares, while others can be called back by the issuer. Mortgage borrowers may also have the right to repay loans early.
Modern Stock Options
Modern options trading became much more organized after the Chicago Board Options Exchange was established in 1973. This introduced standardized contract terms and clearing through a guaranteed clearinghouse.
Today, many options are standardized and traded on regulated exchanges, while others remain custom-made bilateral contracts in the over-the-counter market. Options are a major part of the broader derivatives market.
Types of Options
According to the Option Right
Call options give the holder the right to buy the underlying asset at a specified price for a specified period. Put options give the holder the right to sell the underlying asset at a specified price for a specified period.
According to Delivery Type
A physical delivery option requires the actual delivery of the underlying asset. A cash-settled option is settled through a cash payment rather than physical delivery.
According to the Underlying Asset
Options may be based on equities, bonds, futures, indices, commodities, currencies, swaps, and other instruments.
Other Option Types
A major category of options is employee stock options, which companies use as incentive compensation. Other options also appear in real estate transactions and mortgage contracts. Although these contracts may differ in structure, many of the same valuation and risk principles still apply.
Option Styles
Options are commonly divided into styles. An American option may be exercised on any trading day up to expiration. A European option may be exercised only on the expiration date.
Other styles include Bermudan options, which may be exercised only on certain dates before expiration, and Asian options, whose payoff is based on the average underlying price over a period of time. Barrier options depend on whether the price of the underlying asset crosses a certain level. Binary options pay a fixed amount if a condition is met at expiration and nothing if it is not. Exotic options is a broad term for more complex structures.
Valuation
The value of an option is difficult to calculate because it depends on many variables beyond the current price of the underlying asset. Valuation models generally rely on rational pricing, moneyness, time value, and put-call parity.
A pricing model combines a mathematical description of how the underlying asset behaves with a method for calculating the option premium. Common models include Black-Scholes for equities, Heston for stochastic volatility, and Heath-Jarrow-Morton for interest rate products.
The value of an option is often split into two parts. The first is intrinsic value, which is the difference between the underlying market price and the strike price. The second is time value, which reflects the possibility that the option may gain additional value before expiration.
Valuation Models
Standard option valuation models usually take into account the current market price of the underlying asset, the strike price, interest rates, dividends, time to expiration, and expected volatility. More advanced models may also include changing volatility, changing interest rates, or other market dynamics.
Black-Scholes Model
The Black-Scholes model was developed through the work of Fischer Black, Myron Scholes, and earlier contributions by Louis Bachelier and Robert C. Merton. It provides a closed-form solution for the theoretical price of a European option on a non-dividend-paying stock.
The model was a major breakthrough in financial economics and remains one of the most important foundations of modern options pricing. However, it relies on assumptions such as constant volatility, constant interest rates, and continuous trading, which do not always match real market conditions.
Stochastic Volatility Models
After the market crash of 1987, traders observed that implied volatility often varies by strike price and expiration date. This pattern is known as the volatility smile and volatility surface.
To address this, analysts developed stochastic volatility models in which volatility itself changes over time. The Heston model is a well-known example. Other approaches include the CEV and SABR models.
A related method is local volatility modeling, where volatility is treated as a deterministic function of time and the current asset level. This approach is a generalization of Black-Scholes.
Short-Rate Models
Bond options, swaptions, and interest rate caps and floors are often valued using short-rate models. Well-known models include Black-Derman-Toy and Hull-White. These describe how short-term interest rates evolve over time.
Another major framework is the Heath-Jarrow-Morton model, which describes the evolution of the entire yield curve rather than just the short rate. This can be useful in more complex interest rate valuation problems.
Model Implementation
Analytical Techniques
In some cases, option prices can be calculated using closed-form mathematical formulas. The Black-Scholes model is the best-known example. These methods are efficient and also provide hedge measures, often called the Greeks.
Some American-style options do not have simple closed-form solutions, so approximations are used instead.
Binomial Tree Pricing Model
The binomial option pricing model builds a tree of possible future stock prices over discrete time steps. At each node, the option value is calculated using a risk-neutral portfolio approach.
This method is often considered more flexible than Black-Scholes because it can handle American options and discrete dividend payments more easily. A trinomial tree is a related method that allows up, down, or stable price movements.
Monte Carlo Models
Monte Carlo simulation is useful for complex options that are difficult to price using standard formulas. It generates many possible future price paths for the underlying asset and calculates the payoff in each case. The average discounted payoff gives the estimated option value.
This method is flexible, but it is usually more complicated for American-style options than lattice-based methods.
Finite Difference Models
When an option pricing model is expressed as a partial differential equation, finite difference methods can be used to solve it numerically. These methods include explicit, implicit, and Crank-Nicolson approaches.
Finite difference models are especially useful when model inputs such as volatility, dividend yield, or interest rates change over time.
Risks
Trading options involves several risks because the value of an option changes non-linearly with the underlying asset and other market variables.
Hedge Parameters
The change in an option’s value can be approximated using hedge parameters such as delta, gamma, vega, and theta. These measure how the option responds to changes in the underlying price, volatility, and time.
A trader can use these measures to create hedged positions, such as delta-neutral portfolios, to reduce exposure to small price changes in the underlying asset.
Pin Risk
Pin risk occurs when the underlying asset closes very close to the strike price on the last day before expiration. In that case, the option writer may not know whether the option will be exercised, which can leave the writer with an unwanted position in the underlying asset after expiration.
Counterparty Risk
Counterparty risk is the risk that the other party in the option contract fails to meet its obligation. This risk can be reduced by using strong intermediaries, but it still exists, especially during periods of market stress.
Options Approval Levels
To control risk, brokers often restrict which option strategies a trader can use. These restrictions are usually organized into approval levels, with lower levels allowing simpler and lower-risk strategies and higher levels allowing more advanced strategies.
Broker approval depends on factors such as income, net worth, trading experience, and investment goals. A trader with limited experience and a conservative profile is generally not allowed to use high-risk strategies such as naked calls or naked puts.
Options Exchanges
Chicago Board Options Exchange
The Chicago Board Options Exchange, or CBOE, was founded in 1973 and is the first options exchange in the United States. It offers trading in options on stocks, ETFs, indexes, and volatility products. Its most famous product is options on the S&P 500 Index.
NASDAQ OMX PHLX
NASDAQ OMX PHLX, formerly the Philadelphia Stock Exchange, is one of the oldest exchanges in the United States. It offers options on equities, indexes, ETFs, and foreign currencies. It is also one of the few U.S. exchanges that lists currency options.
International Securities Exchange
The International Securities Exchange, or ISE, is an electronic options exchange based in New York City. It was the first all-electronic U.S. options exchange and offers trading in options on equities, indexes, and ETFs.
Eurex Exchange
Eurex Exchange is a derivatives exchange based in Frankfurt, Germany. It offers futures and options on interest rates, equities, indexes, and fixed-income products. It operates as an electronic exchange and is one of the major derivatives markets in Europe.
Tokyo Stock Exchange
The Tokyo Stock Exchange also offers trading in stock index futures and options. Trading is conducted electronically and through auction-based systems, and the exchange remains an important part of the Japanese financial market.