# Options

An option in finance is a financial derivative product that represents a contract sold by the option writer to another party (option holder). An option contract gives the buyer the right –not the obligation – to buy or sell the underlying asset that it refers to – the counterparty gets the corresponding obligation. The underlying asset is usually a financial instrument such as a stock, a commodity, a bond, a currency pair, a futures contract or an index but in principle, options can be created for almost any asset. The price of an option is contingent on the difference between the value of the referenced asset and the reference price in addition to a premium, which depends largely on the time left for the contract to expire.

An option that gives the right to buy the underlying asset at a stipulated price is known as call option while the option that gives the right to sell at a stipulated price is known as put option. The stipulated price at which the underlying asset may be traded in the option contract is known as the strike price. Most options have an expiry date but may be activated on or before that date, a process that is known as exercising an option. The option is voided (becomes worthless) if not exercised.

The option writer (the seller) collects the premium from the buyer and must act on the promise and deliver the underlying asset or its cash equivalent, if the other party exercises the option. It works the other way round in case of put option.

An option is a tradable derivative and can be sold to another party before it expires. Standardized options are traded in any exchange that allows derivative trading. There are also over-the-counter options that are customized and improvised for specific buyer needs. Such options are normally created by investment banks.

Contract Stipulations

Every option contract, standardized or customized, is a contract that gives corresponding rights and obligation to the two parties to the contract. The terms of the contract are specified in a term sheet. Some options contracts may be more complicated than others, but most of them stipulate the following.

- if it is a call option, it provides the option holder the right to buy, while a put option conveys the right to sell
- the name and quantity of the underlying asset
- the strike price at which the transaction will occur in case the option is exercised
- expiration date, the date after which the option is voided
- the terms of settlement of the contract, for example whether the settlement would involve delivery of the asset or cash equal to the value of the asset
- the amount that the option writer pays to the option holder

Types of Options

Options can be classified on the basis of various criteria. Some of the common types of options include:

Exchange Traded Options

These are standardized listed options contracts specifying the quantity of the underlying asset and the terms on which the option will be traded and settled and traded on a recognized options exchange. Exchange trade options are settled through the exchange or a clearing house nominated by the exchange, which also stands guarantee to the fulfillment of contractual obligations. Most exchanges offer the following options:

- stock options
- index options
- options of futures contract
- interest rate options including bond options
- callable bull/bear contract

Over-the-Counter Options

Over-the-counter options, commonly known as OTC options are dealer options, meaning that the contract is between two parties. OTC options are not listed on an exchange and are not standardized contracts – the terms may be customized to meet individual needs. Since it is a contract between two private parties, it is presumed that at least one of them is an adequately capitalized entity of good repute. Interest rate options, exchange rate options (currency options) and options on swaps are some of the common options traded over the counter.

Other Types

These include stock options given to employees of corporations as performance incentives and/or as part of a salary package. Many financial contracts include options of different types; in real estate industry these are used for assembling large parcels of land and prepayment of loan option is often included in mortgage contracts.

Option Styles

Option style is a general term that denotes the class into which the options falls, typically defined by the dates for exercising the option. These include:

- American option is an option that can be exercised anytime on or before the expiry date
- European option is exercisable only on expiration
- Barrier option is characterized by a condition that it can be exercised only after the price of the underlying asset crosses a certain level or ’barrier’
- Bermudan option can be exercised only on a specific dates on or before the option expires
- Exotic option is the term applied to a broad category of options that are marked by complex features
- Vanilla option is another term for a normal option that is not an exotic option

Option Valuation

Valuation of options may be done with simple quantitative techniques or based on sophisticated models. The Black-Scholes model is the most basic model that uses stochastic calculus and is based on the concept of risk neutral pricing. Sophisticated models are implemented using a variety of numerical techniques and attempt to simulate the volatility smile. Generally, standard models of option valuation depend largely on factors such as:

- The current market price of the asset the option refers to
- The relationship between the strike price and the current market price that is whether it is an in-the-money, at-the money or out-of-the money option
- The cost, interest or dividends, of holding a position in the option
- Expiry time along with whether there are any restrictions on when the option can be exercised
- Estimation of the future volatility of the underlying assets over the duration of the option

Advanced model may take into account observed patterns, implied volatility and the ever changing stochastic interest rates. Some of the commonly used popular valuation techniques for evaluating option contracts are as follows.

Black-Scholes

Fischer Black and Myron Scholes derived a differential equation that must be satisfied by the price of a derivative based on a non-dividend paying stock. Developed from the earlier work done by Louis Bachelier and later by Edward O. Thorp, the Black-Scholes mathematical model of a financial market produces a close-form solution for a European option’s theoretical price using the technique of building a risk neutral portfolio replicating the returns of holding an option. It also generates quantities representing the sensitivities of price or hedge parameters of an option required for managing risk of holding options. The Black-Scholes model used innovative ideas that eventually resulted in Scholes and Merton getting the associated Prize for Achievement in Economics (also known as the Nobel Prize of Economics) endowed by Sweden’s central bank Sveriges Riksbank. However, the application of the model is difficult to handle in real-life options trading because of the assumptions of continuous (or no) dividends and a constant volatility and interest rate. Regardless, the Black-Scholes model remains one of the most important methods as it throws results within a reasonable range.

Stochastic Volatility Models

Stochastic volatility models assume that asset volatility is neither constant nor deterministic but follows a random process. This came to light in market crash of 1987 when it was observed that implied volatility for options of lower strike prices were typically higher than for higher strike prices. The Heston model is a principal stochastic volatility model that can be solved in closed-form and enjoys a distinct advantage over stochastic models that employ complex numerical methods.

Model Implementation

There is a variety of techniques used to take mathematical pricing models for implementing the chosen valuation model.

Analytic

In some cases closed form solutions such as the Black-Scholes and the Black model can be developed by taking the mathematical model and using analytical methods. The results, as well as the hedge parameters, arrived at are easily computable. While the Roll-Geske-Whaley model applies well to an American call with one dividend, it fails to provide closed form solutions for other American options. Barone-Adesi and Whaley and others are included in the approximation formulas.

Binomial Tree Pricing Model

John Cox, Stephen Ross and Mark Rubinstein developed the binomial options pricing model’s original version by following the Black and Scholes derivation. It models changes in the theoretical value of an option for discrete time intervals over the life of an option. Just as in the Black-Scholes model, on construction of a risk neutral portfolio of option and stock, a simple formula is used to arrive at the price of the option at each node in the tree. The value thus arrived, is an approximation to the desired degree of precision, of the theoretical value thrown by the Black-Scholes model. Since the binomial tree model is more flexible as it is able to correctly model the discrete future dividends, it is considered to be more accurate than the Black-Scholes model. Similar to the binomial model is the trinomial tree that allows for an up, down or stable path. Considered more accurate especially when fewer time-steps are modeled, it is a complex numerical model and thus used less frequently.

Monte Carlo Models

Many classes of options are inherently complex instruments, which makes traditional valuation methods difficult to manage. The Monte Carlo approach often proves to be more useful in such cases. Monte Carlo models use simulation to produce random price paths of the referenced asset, each of which results in a payoff for the option, which is an entirely different approach than solving differential equations of motions that describe the value of the option in relation to the price of the underlying asset. The average of the payoffs for the option can be discounted to give an expectation value for the option. Although the model is more flexible, these models are more intricate especially when used for simulating American style options.

Finite Difference Models

The equations used in option valuation models are often shown as partial differential equations, for example the Black-Scholes equation. Once shown in this form, a finite difference model can be developed for obtaining the valuation. There are a number of implementations of the finite difference methods for obtaining option valuations. These include implicit finite difference, explicit finite difference and the Crank-Nicholson method. A simplified application of the explicit finite difference method can be expressed as a trinomial tree option pricing model. The finite difference models are mathematically more sophisticated but are useful where changes are assumed over time in model inputs that are not easily manageable such as dividend yield, risk free rate or volatility or a combination of these.

Other Models

Finite element methods are among the other numerical implementations that have been used for valuing options. Various short rate models have also been developed for valuation of bond options, interest rate derivatives and swap options. These methods allow for lattice-based, closed-form and simulation-based modeling.

Risks

Trading in options, just like trading in all securities, implicates the risk of the value of the option changing over time. While the return from a traditional security varies linearly with its value, the return from holding an option varies non-linearly with the value of the asset it refers to. There are also certain other factors that affect the return from holding an option. Hence, understanding risks of holding options is more intricate and difficult to predict.

Generally, the change in option value can be derived from itō’s lemma as:

where the Greeks , , and are the risk sensitivities or standard hedge parameters calculated from option valuation models such as Black–Scholes, and , and are unit changes in the underlying asset’s price, volatility and time respectively.

By calculating its hedge parameters and then approximating the expected change in the model inputs, , and (provided the changes are small), the implicit risk of holding an option can be estimated. It is a technique that is effective in understanding and managing the risks associated with standard options. As an example, a trader can create a hedged delta neutral portfolio that is protected from loss due to small changes in the price of the underlying asset by offsetting a holding in an option with the quantity of shares in the underlying assets. The formula corresponding to price sensitivity for this portfolio is:

Example

- A call option with expiration in 99 days is struck for 100 shares of ABC stock at $50 with current market price of ABC stock at $48
- The future realized volatility over the life of the option is approximated at 25%, which gives the value of the option as $1.89.
- , , and , the hedge parameters are 0.439, 0.0631, 9.6, and −0.022 respectively

Now, assuming that the price of ABC stock goes up to $48.5 and the future realized volatility falls to 23.5%, the hedge parameters can be applied to the new model inputs as under to get the estimated value of the option.

In this example, the value of the option increases to $1.9514 (that is $1.89 + $0.0614). Since the option is for 100 shares, it results in a profit of $ 6.14 (0.0614 X 100). However, for a delta neutral portfolio in which the trader has sold 44 shares of ABC stock for hedging the option, in the same example, the trader would lose $15.86 after considering the loss made on buying 44 shares as the price of the ABC stock has gone up by 0.50 up.

Pin Risk

Pin risk is a situation where the market price of the underlying asset at the time of expiry of an option contract is the same or very close to the strike price. In such a situation, the option writer (seller) does not know whether the option will be allowed to expire worthless or whether the buyer will exercise the option. In such a case, despite best efforts of avoiding it, the trader is left with an unwanted residual position the day after the expiry date.

Counterparty Risk

Most people tend to ignore this risk while trading in derivatives. However, howsoever small it may be the risk is always there of the counterparty refusing to sell or buy the underlying security as per the terms of the option contract. While this risk is eliminated (at least minimized) if trading is done on the platform of a financially strong broker, even the strongest intermediary may not be able to withstand the pressure of defaults in a panic situation or in a crashing market.

Option Trading

Most traders prefer to trade in standardized options offered for trade by recognized exchanges dealing in futures and options. These options are listed and prices are displayed on the ticker tape. Publishing of live rates allows independent traders to initiate trades after engaging in price discovery. The futures and options exchange acts an intermediary to both parties involved in a transaction, a process that provides certain distinct advantages.

- Contracts are guaranteed by the exchange. This is also the reason why it is recommended that an exchange with the highest credit rating should be chosen.
- Traders deal through the exchange and not directly with the counterparty, which remains anonymous.
- Exchanges must adhere to regulations, which ensures transparency and fair trading practices.
- Orderliness is maintained under all circumstances. This proves beneficial in times of rapid trading activity in a particular option or the underlying asset.

Over-the-counter trading in options contracts is done directly between two parties without an intermediary to monitor the transaction. Generally, at least one of the two parties is adequately capitalized institution. Since the contracts in this case are not standardized, the terms of each option can be customized to suit individual preferences. Over-the-counter options trading is neither regulated nor advertised to the market, which necessitates that counterparties are sure of each other’s creditability and accept procedures for clearing and settlement of trades.

Basic Trades in American Style Stock Options

In USA, an option contract normally represents 100 shares of an underlying asset or stock. The following are the basic trades (from the point of view of a trader) that can be initiated. Speculators may combine these trades with other positions and hedge their bets.

Long Call

A trader who is bullish on a stock and believes that its price will move up can buy the right to purchase the stock (a call option) instead of buying the stock. This way the trader would have no obligation to buy the stock but still have the right to buy the stock till the date of expiry of the contract. For creating a long position, the trader pays a premium over the current market price and makes a profit if the stock price on expiry date moves up by more than the premium paid. In case the price closes below the exercise price, the trader lets the options expire worthless and take a loss equal to the amount of premium paid. A call option comprising of 100 shares of a specific stock costs only a fraction of the amount required to buy 100 shares of the same stock. The leverage allows the trader to control a much larger number of shares with the same amount.

Long Put

A long put is the opposite of a long call. A trader who believes that the price of a stock will go down can buy the right to sell (a long put) the stock at a predetermined price on the expiry date. In this case too, the trader has the right to sell but is not under any obligation to sell. If the price of the stock decreases by more than the premium paid, the trader takes home the difference as profit. In the price is more than the exercise price on the date of expiry, the trader loses the premium paid at the time of the buying the long put.

Short Call

When a trader believes that the price of a stock will decrease, he may choose to sell or ‘write’ a call. This is known as short selling a stock. In this case, the trader has the obligation to sell the stock to the buyer should the buyer want to exercise the option. The trader, in this case, makes a profit in the amount of the premium should the price of the stock decrease on expiry. The trader makes a loss if the stock price increases over the strike price by more than the premium paid. The potential of loss is unlimited in this case.

Short Put

A trader may ‘write’ or sell a put option if he believes that the price of the stock will increase. The trader in this case has the obligation to buy the stock from the buyer of the put should the buyer so desire and exercise the option. The put ‘writer’ makes a profit in the amount of the premium should the price of the stock rise above the exercise price on expiration. The trader takes a loss if the price of the stock is below the strike price by more than the premium amount. The potential of loss, in this case, is up to the full value of the stock.

Option Strategies

Traders can strategize by combining the above four basic option trades using different exercise prices and varying expirations and the two basic kinds of stock trades (buy and sell). A simple strategy normally comprises of only a few trades; the more complicated ones however comprise of several interlinked trades.

Strategies are primarily plans for managing a specific risk profile to the price movements in the underlying asset. A butterfly spread, for example involves going long on one X1 call and going short on two X2 calls and going long on one X3 call. This allows the traders to make a profit if the price of the stock is near the middle strike price, X2 with little exposure to a large loss.

Iron condor is similar to a butterfly spread except that in this case the strike prices are for short options. This strategy has a greater likelihood of profit but with a lower net outlay as compared to the butterfly spread.

A trader may sell a straddle, which is selling a call and a put at the same exercise price. This has a greater potential of profit as compared to a butterfly spread should the price of the stock end up at near the exercise price at expiry. However, the potential of loss is also greater.

A strangle is similar to a straddle except for that in this case the call and put have different strike rates. This not only reduces the amount of net outlay but also reduces the risk of loss.

A covered call is a well-known and very commonly used strategy. The trader buys a stock or holds a previously purchased stock and sells a call. Should the price of the stock rise above the exercise price, the buyer will exercise the call and the trader gets a fixed profit. On the other hand, should the price of the stock fall, the call will not be exercised and the premium received on selling the call will offset the loss incurred by the trader. On the whole, the payoffs in this case are almost the same as the payoffs from selling a put. This is known as the put-call parity, a relationship that offers insights for financial theory.

History of Options

Options have a long history and have been in use since long.

The world’s first option buyer is supposedly the ancient Greek mathematician and philosopher Thales of Miletus. He predicted that the olive harvest would be exceptionally good and bought the right to use a number of olive presses in the off-season. His prediction proved correct and he exercised his option and rented out presses at a price much higher than he paid for the option.

In real estate, brokers and traders have been using call options to accumulate large parcels of land. For example, a developer pays for the right to buy, but not the obligation, land from separate owners of adjacent plots. The developer may not buy the plots till the time he is able to buy all the plots in the parcel. Similarly, in the theatre and film industry, producers or directors may buy the right, but not the obligation, to dramatize a script or a book. In banking, a credit line provided by the bank gives the borrower the right, but not the obligation, to borrow within a specific time period.

In the securities market, convertible bonds give the buyer the option to convert them into common stock or be called back at the issuer’s option. In a mortgage contract, the borrower often has the option of early repayment of the loan.

Puts and ‘refusal’ (calls) were well-known trading instruments in London during the 1690s. Specialized dealers offered privileges on shares (both put and call options) over the counter in the 19^{th} century in America. The expiry date was generally three months from the date of the contract with the exercise price fixed at the rounded-off market price on the day or week the option was bought.