In finance, an option is a contract between a buyer and a seller that gives the buyer of the option the right, but not the obligation, to buy or to sell a specified asset (underlying) on or before the option's expiration time, at an agreed price, the strike price.
In return for granting the option, the seller collects a payment (the premium) from the buyer. Granting the option is also refered to as "selling" or "writing" the option.
If the buyer chooses to exercise this right, the seller is obliged to sell or buy the asset at the agreed price.^{[1]}^{[2]} The buyer may choose not to exercise the right and let it expire. The underlying asset can be a piece of property, a security (stock or bond), or a derivative instrument, such as a futures contract.
The theoretical value of an option is evaluated according to several models. These models, which are developed by quantitative analysts, attempt to predict how the value of an option changes in response to changing conditions. Hence, the risks associated with granting, owning, or trading options may be quantified and managed with a greater degree of precision, perhaps, than with some other investments. Exchangetraded options form an important class of options which have standardized contract features and trade on public exchanges, facilitating trading among independent parties. Overthecounter options are traded between private parties, often wellcapitalized institutions that have negotiated separate trading and clearing arrangements with each other.
Another important class of options, particularly in the U.S., are employee stock options, 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 loans. However, many of the valuation and risk management principles apply across all financial options.
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Every financial option is a contract between the 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:^{[3]}
The primary types of financial options are:
Naming conventions are used to help identify properties common to many different types of options. These include:
The value of an option can be estimated using a variety of quantitative techniques based on the concept of risk neutral pricing and using stochastic calculus. The most basic model is the BlackScholes model. More sophisticated models are used to model the volatility smile. These models are implemented using a variety of numerical techniques.^{[7]} In general, standard option valuation models depend on the following factors:
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.
In the early 1970s, Fischer Black and Myron Scholes made a major breakthrough by deriving a differential equation that must be satisfied by the price of any derivative dependent on a nondividendpaying stock. By employing the technique of constructing a risk neutral portfolio that replicates the returns of holding an option, Black and Scholes produced a closedform solution for a European option's theoretical price.^{[8]} At the same time, the model generates hedge parameters necessary for effective risk management of option holdings. While the ideas behind the BlackScholes model were groundbreaking and eventually led to Scholes and Merton receiving the Swedish Central Bank's associated Prize for Achievement in Economics (often mistakenly referred to as the Nobel Prize),^{[9]} the application of the model in actual options trading is clumsy because of the assumptions of continuous (or no) dividend payment, constant volatility, and a constant interest rate. Nevertheless, the BlackScholes 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.^{[10]}
Since the 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 is stochastic, varying both for time and for the price level of the underlying security. Stochastic volatility models have been developed including one developed by S.L. Heston.^{[11]} One principal advantage of the Heston model is that it can be solved in closedform, while other stochastic volatility models require complex numerical methods.^{[11]}
Once a valuation model has been chosen, there are a number of different techniques used to take the mathematical models to implement the models.
In some cases, one can take the mathematical model and using analytical methods develop closed form solutions such as BlackScholes and the Black model. The resulting solutions are useful because they are rapid to calculate.
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.^{[12]} ^{[13]} It models the dynamics of the option's theoretical value for discrete time intervals over the option's duration. The model starts with a binomial tree of discrete future possible underlying stock prices. By constructing a riskless portfolio of an option and stock (as in the BlackScholes 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 BlackScholes 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.
For many classes of options, traditional valuation techniques are intractable due to 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 generates 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.^{[14]}
The equations used to model the option are often expressed as partial differential equations (see for example Black–Scholes PDE). Once expressed in this form, a finite difference model can be derived, and the valuation obtained. This approach is useful for extensions of the option pricing model, such as changing assumptions as to dividends, that are not able to be represented with a closed form analytic solution. A number of implementations of finite difference methods exist for option valuation, including: explicit finite difference, implicit finite difference and the CrankNicholson method. A trinomial tree option pricing model can be shown to be a simplified application of the explicit finite difference method.
Other numerical implementations which have been used to value options include finite element methods.
As with all securities, trading options entails the risk of the option's value changing over time. However, unlike traditional securities, the return from holding an option varies nonlinearly with the value of the underlier 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 Ito's lemma as:
where the greeks Δ, Γ, κ and θ are the standard hedge parameters calculated from an option valuation model, such as BlackScholes, and dS, dσ and dt are unit changes in the underlier price, the underlier volatility and time, respectively.
Thus, at any point in time, one can estimate the risk inherent in holding an option by calculating its hedge parameters and then estimating the expected change in the model inputs, dS, dσ 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 − Δ of shares in the underlier, a trader can form a delta neutral portfolio that is hedged from loss for small changes in the underlier price. The corresponding price sensitivity formula for this portfolio Π is:
A call option expiring in 99 days on 100 shares of XYZ stock is struck at $50, with XYZ currently trading at $48. With future realized volatility over the life of the option estimated at 25%, the theoretical value of the option is $1.89. The hedge parameters Δ, Γ, κ, θ are (0.439, 0.0631, 9.6, and 0.022), respectively. Assume that on the following day, XYZ stock rises to $48.5 and volatility falls to 23.5%. We can calculate the estimated value of the call option by applying the hedge parameters to the new model inputs as:
Under this scenario, the value of the option increases by $0.0614 to $1.9514, realizing a profit of $6.14. Note that for a delta neutral portfolio, where by the trader had also sold 44 shares of XYZ stock as a hedge, the net loss under the same scenario would be ($15.86).
A special situation called pin risk can arise when the underlier 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 worthless. Therefore, the option writer may end up with a large, unwanted residual position in the underlier when the markets open on the next trading day after expiration, regardless of their best efforts to avoid such a residual.
A further, often ignored, risk in derivatives such as options is 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.
The most common way to trade options is via standardized options contracts that are listed by various futures and options exchanges. ^{[15]} Listings and prices are tracked and can be looked up by ticker symbol. 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:
Overthecounter options contracts are not traded on exchanges, but instead between two independent parties. Ordinarily, at least one of the counterparties is a wellcapitalized institution. 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 others clearing and settlement procedures.
With few exceptions,^{[16]} there are no secondary markets for employee stock options. These must either be exercised by the original grantee or allowed to expire worthless.
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.^{[17]}
A trader who believes that a stock's price will increase might buy the right to purchase the stock (a call option) rather than just buy the stock. He would have no obligation to buy the stock, only the right to do so until the expiration date. If the stock price at expiration is above the exercise price by more than the premium (price) paid, he will profit. If the stock price at expiration is lower than the exercise price, he will let the call contract expire worthless, and only lose the amount of the premium. A trader might buy the option instead of shares, because for the same amount of money, he can control (leverage (finance)) a much larger number of shares.
A trader who believes that a stock's price will decrease can buy the right to sell the stock at a fixed price (a put option). He will be under no obligation to sell the stock, but has the right to do so until the expiration date. If the stock price at expiration is below the exercise price by more than the premium paid, he will profit. If the stock price at expiration is above the exercise price, he will let the put contract expire worthless and only lose the premium paid.
A trader who believes that a stock price will decrease, can sell the stock short or instead sell, or "write," a call. The trader selling a call has an obligation to sell the stock to the call buyer at the buyer's option. If the stock price decreases, the short call position will make a profit in the amount of the premium. If the stock price increases over the exercise price by more than the amount of the premium, the short will lose money, with the potential loss unlimited.
A trader who believes that a stock price will increase can buy the stock or instead sell a put. The trader selling a put has an obligation to buy the stock from the put buyer at the put buyer's option. If the stock price at expiration is above the exercise price, the short put position will make a profit in the amount of the premium. If the stock price at expiration is below the exercise price by more than the amount of the premium, the trader will lose money, with the potential loss being up to the full value of the stock. A benchmark index for the performance of a cashsecured short put option position is the CBOE S&P 500 PutWrite Index (ticker PUT).
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.
An Iron condor 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 (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 likelihood of profit in the trade.
One wellknown strategy is the covered call, in which a trader buys a stock (or holds a previouslypurchased 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 trader will lose money on his stock position, but this 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 putcall parity and offers insights for financial theory. A benchmark index for the performance of a buywrite strategy is the CBOE S&P 500 BuyWrite Index (ticker symbol BXM).
Contracts similar to options are believed to have been used since ancient times. 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 he can buy all the plots in the entire parcel. Film or theatrical producers often buy the right — but not the obligation — to dramatize a specific book or script. Lines of credit give the potential borrower the right — but not the obligation — to borrow 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.
In London, puts and "refusals" (calls) first became wellknown trading instruments in the 1690s during the reign of William and Mary.^{[18]}
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 roundedoff 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.
Supposedly the first option buyer in the world was the ancient Greek mathematician and philosopher Thales. On a certain occasion, it was predicted that the season's olive harvest would be larger than usual, and during the offseason 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 much higher price than he paid for his 'option'.^{[19]}^{[citation needed]}

