Contents

§ Valuation overview [ edit ]

Options valuation is a topic of ongoing research in academic and practical finance. In 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 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 expected value of that difference at expiration.

Although options valuation has been studied at least since the nineteenth century, the contemporary approach is based on the Black–Scholes model which was first published in 1973. [ 1 ] [ 2 ]

Options contracts have been known for many centuries, however both trading activity and academic interest increased when, as from 1973, options were issued with standardized terms and traded through a guaranteed clearing house at the Chicago Board Options Exchange. 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. [ 3 ] [ 4 ]

§ Contract specifications [ edit ]

Every 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: [ 5 ]

• whether the option holder has the right to buy (a call option ) or the right to sell (a put option )
• the quantity and class of the underlying asset(s) (e.g. 100 shares of XYZ Co. B stock)
• the strike price. also known as the exercise price, which is the price at which the underlying transaction will occur upon exercise
• 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

§ Types [ edit ]

Options can be classified in a few ways.

§ According to the option rights [ edit ]

§ According to the underlying assets [ edit ]

§ According to the trading markets [ edit ]

§ Other option types [ edit ]

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.

§ Option styles [ edit ]

Naming conventions are used to help identify properties common to many different types of options. These include:

• European option – an option that may only be exercised on expiration .
• American option – an option that may be exercised on any trading day on or before expiry.
• 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 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 option – An all-or-nothing option that pays the full amount if the underlying security meets the defined condition on expiration otherwise it expires worthless.
• Exotic option – any of a broad category of options that may include complex financial structures. [ 8 ]
• Vanilla option – any option that is not exotic.

§ Valuation models [ edit ]

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 Black–Scholes model. More sophisticated models are used to model the volatility smile. These models are implemented using a variety of numerical techniques. [ 9 ] In general, standard option valuation models depend on the following factors:

• The current market price of the underlying security,
• the strike price 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, and
• 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 [ edit ]

Following early work by Louis Bachelier and later work by Robert C. Merton. 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 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. [ 10 ] 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 in Economics), [ 11 ] 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. [ 12 ]

§ Stochastic volatility models [ edit ]

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. [ 13 ] One principal advantage of the Heston model is that it can be solved in closed-form, while other stochastic volatility models require complex numerical methods. [ 13 ]

§ Model implementation [ edit ]

Once a valuation model has been chosen, there are a number of different techniques used to take the mathematical models to implement the models.

§ Analytic techniques [ edit ]

In some cases, one can take the mathematical model and using analytical methods develop closed form solutions such as Black–Scholes 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 options. closed form solutions are not available; approximations here include Barone-Adesi and Whaley. Bjerksund and Stensland and others.

§ Binomial tree pricing model [ edit ]

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. [ 14 ] [ 15 ] 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 options can be modeled as well as European ones. Binomial models are widely used by professional option traders. The Trinomial tree is a similar model, allowing for an up, down or stable path; although considered more accurate, particularly when fewer time-steps are modelled, it is less commonly used as its implementation is more complex.

§ Monte Carlo models [ edit ]

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 to generate random price paths of the underlying asset, each of which results in a payoff for the option. The average of these payoffs can be discounted to yield an expectation value for the option. [ 16 ] Note though, that despite its flexibility, using simulation for American styled options is somewhat more complex than for lattice based models.

§ Finite difference models [ edit ]

The equations used to model the option are often expressed as partial differential equations (see for example Black–Scholes equation ). Once expressed in this form, a finite difference 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-Nicholson 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 [ edit ]

Other numerical implementations which have been used to value options include finite element methods. Additionally, various short rate models have been developed for the valuation of interest rate derivatives. bond options and swaptions. These, similarly, allow for closed-form, lattice-based, and simulation-based modelling, with corresponding advantages and considerations.

§ Risks [ edit ]

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 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 Itō’s lemma as:

where the Greeks , , and are the standard hedge parameters calculated from an option valuation model, such as Black–Scholes. and , and 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, , and , 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 of shares in the underlying, a trader can form a delta neutral portfolio that is hedged from loss for small changes in the underlying’s price. The corresponding price sensitivity formula for this portfolio is:

§ Example [ edit ]

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 , , , 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, whereby the trader had also sold 44 shares of XYZ stock as a hedge, the net loss under the same scenario would be ($15.86).

§ Pin risk [ edit ]

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 worthless. 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 [ edit ]

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.

§ Trading [ edit ]

The most common way to trade options is via standardized options contracts that are listed by various futures and options exchanges. [ 17 ] 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:

• fulfillment of the contract is backed by the credit of the exchange, which typically has the highest rating (AAA),
• counterparties remain anonymous,
• enforcement of market regulation to ensure fairness and transparency, and
• maintenance of orderly markets, especially during fast trading conditions.

Over-the-counter options contracts are not traded on exchanges, but instead between two independent parties. Ordinarily, at least one of the counterparties is a well-capitalized 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 other’s clearing and settlement procedures.

With few exceptions, [ 18 ] there are no secondary markets for employee stock options. These must either be exercised by the original grantee or allowed to expire worthless.

§ The basic trades of traded stock options (American style) [ edit ]

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. [ 19 ]