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Trading and Capital-Markets Activities Manual

Instrument Profiles: Options   (Continued)
Source: Federal Reserve System 
(The complete Activities Manual (pdf format) can be downloaded from the Federal Reserve's web site)


Options trade both on exchanges and OTC. The vast majority of exchange options are American, while most OTC options tend to be European. Exchange-traded (or simply traded) options are generally standardized as to the underlying asset, expiration dates, and exercise prices. OTC options are generally tailored to meet a customer's specific needs. 

Banks, investment banks, and certain insurance companies are active market makers in OTC options. End-users of options include banks, money managers, hedge funds, insurance companies, corporations, and sovereign institutions. 


In terms of valuation and risk measurement, instruments with option characteristics differ significantly from other assets. In particular, options require an assessment of the probability distribution of possible movements in the relevant market-risk factors. Changes in the expected volatility of an instrument's price will affect the value of the option. Option values not only vary with the degree of expected volatility in the price of the underlying asset, but also vary with the price of the underlying in a decidedly asymmetric way. 

Although the supply and demand for options is what directly determines their market prices, option valuation theory plays a crucial role in informing market participants on both sides of the market. A number of valuation techniques are used by market participants and are described below. 

Approaches to Option Valuation 


The ''standard'' model used to value options is the Black-Scholes option pricing model. Based on a few key assumptions-including that asset prices follow a ''random walk'' (they fluctuate randomly up or down), the risk-free interest rate remains constant, and the option can be exercised only at expiration-the Black-Scholes model can incorporate all the main risk concepts of options and, therefore, provides a useful basis for discussion. In practice, many financial institutions use more sophisticated models, in some cases proprietary models. 

The Black-Scholes formula for the value of a call option depends on five variables: (1) the price of the underlying asset, (2) the time to expiration of the option, (3) the exercise price, (4) the risk-free interest rate (the interest rate on a financial institution deposit or a Treasury bill of the same maturity as the option), and (5) the asset's expected volatility. Of the five variables, only four are known to market participants. The asset price and the deposit or Treasury bill rate of the appropriate maturity can be ascertained from dealers or a public information source. The maturity of the option and the strike price are known from the terms of the option contract. 

Assuming that the price of an asset follows a random walk, Black and Scholes derived their formula for pricing a call option on that asset given the current spot price (St) at time t, the exercise price (X), the option's remaining time to maturity (T), the probability distribution (standard deviation) of the asset price (s), and a constant interest rate r. Specifically, the price C at time t of a call option with a strike price of X which matures at time T is-

where N(d) is the probability that a standardized normally distributed random variable takes on a value less than d, and

The easiest way to understand this formula is as the present value of the expected difference between the future price of the underlying asset and the exercise price, adjusted for the probability of exercise. In other words, it is the expected value of the payoff, discounted to the present at the risk-free rate. The first term in the Black-Scholes equation is the present value of the expected asset price at expiration given that the option finishes in the money. The standard normal term, N(d), is the probability that the option expires in the money; hence, the entire second term, Xe - r(T - t)N(d), is the present value of the exercise price times the probability of exercise. 

The key unknown in the formula is future volatility of the underlying asset price. There are two ways of estimating this price. First, it can be estimated directly from historical data on the asset price, for example, by calculating the standard deviation of daily price changes over some recent period. When calculating volatility using historical prices, different estimates of volatility may be arrived at (and consequently, also different estimates of an option's value), depending on the historical period chosen and other factors. Hence, the historical period used in volatility estimates should be chosen with some care.

Alternatively, volatility can be estimated by using the Black-Scholes formula, together with the market prices of options, to back out the estimate of volatility implicit in the market price of the option, given the four known variables. This is called the implied volatility of the option. Note that the use of implied volatility may not be appropriate for thinly traded options due to the wide variation of options prices in thin markets. 

Some institutions use a combination of both historical and implied volatilities to arrive at an appropriate estimate of expected volatility. Examiners should determine if management and the traders understand the benefits and shortcomings of both the estimated implied volatility and historical methods of calculating volatility, considering that the values derived under either or both methods may be appropriate in certain instances and not appropriate in others. In any case, the method used to estimate volatility should be conservative, independent of individual traders, and not subject to manipulation in risk and profitability calculations. The last point is especially important because volatilities are a critical component for calculating option values for internal control purposes. 

Other Closed-Form Models 

Since the publication of Black-Scholes, other widely-used formula-based valuation techniques have been developed for use by market makers to value European options as well as options on interest-bearing assets. These techniques include the Hull and White model and the Black, Derman, and Toy (BDT) model. These models are often described as no-arbitrage models and are designed so that the model is, or can be made, consistent with the current term structure of interest rates. Other models, such as the Cox, Ingersoll, and Ross (CIR) model, apply other disciplines to the term structure but allow prices to evolve in a way that need not be consistent with today's term structure of interest rates. 

Binomial Model 

An alternative technique used to value options is the binomial model. It is termed ''binomial'' because it is constructed as a ''tree'' of successive event points in which each branch has two possible events: the asset price either rises or falls. The amount of the rise or fall at each event point depends on the volatility of the underlying asset price. Each path of the variable-from the valuation date through each event point until expiration-then leads to an ultimate profit or loss for the option holder. The value of the option is then the ''average'' present value of these various ultimate outcomes. 

The binomial approach is attractive because it is capable of pricing a wider variety of options than Black-Scholes. For example, a binomial model can allow for a different value function to be applied at different points in time or for options with multiple exercise dates. The binomial model is used by some to value options because it is perceived to be a more reasonable representation of observed prices in particular markets. It is also used to check the accuracy of modifications to the Black-Scholes model. (The Ho-Lee model of interest-rate options, for example, is an elaboration of a binomial model.) In addition, although it requires more computing time than the Black-Scholes model, the binomial model can be more easily adapted for computer use than other still more rigorous techniques. Under the same restrictive assumptions described above, the binomial model and the Black-Scholes formula will produce identical option values. 

Monte Carlo Simulations 

A final approach to valuing options is simply to value them using a large sample of randomly drawn potential future movements in the asset price, and calculate the average or expected value of the option. The random draws are based on the expected volatility of the asset price so that a sufficiently large sample will (by the Law of Large Numbers) accurately portray the expected value of the option, considering the entire probability distribution of the asset price. The advantage of this technique is that it allows for different value functions under different conditions, particularly if the value of an instrument at a point in time depends in part on past movements in market-risk factors. Thus, for example, the value of a collateralized mortgage obligation security at a point in time will depend in part on the level of rate-motivated mortgage prepayments that have taken place in the past, making Monte Carlo simulation the valuation technique market participants prefer. Because of the time and computer resources required, this technique is generally reserved for the most complex option valuation problems. 

Sensitivity of Market Risk for Options 

Given the complexity of the market risk arising from options, and the different models of option valuation, a set of terms has evolved in the market and in academic literature that now serves as a common language for discussing options risk. The key terms (loosely known as ''the greeks'') are described below. Each term is linked to one of the key variables needed to price an option, as described earlier; however, there is no ''greek'' for the exercise price. 

Delta and Gamma 

Delta and gamma both describe the sensitivity of the option price with respect to changes in the price of the underlying asset. The delta of an option is the degree to which the option's value will be affected by a (small) change in the price of the underlying instrument. As such, the estimate of an instrument's delta can be used to determine the appropriate option hedge ratio for an unhedged position in that instrument. 

Gamma refers to the degree to which the option's delta will change as the instrument's price changes. The existence of gamma risk means that the use of delta hedging techniques is less effective against large changes in the price of the underlying instrument. While a delta-hedged short option position is protected against small changes in the price of the underlying asset, large price changes in either direction will produce losses (though of smaller magnitude than would have occurred had the price moved against a naked written option). 


The vega of an option, or a portfolio of options, is the sensitivity of the option value to changes in the market's expectations for the volatility of the underlying instrument. An option value is heavily dependent upon the expected price volatility of the underlying instrument over the life of the option. If expected volatility increases, for example, there is a greater probability that an option may become in the money (profitable for the holder to exercise); thus the vega is typically positive. As noted above, market participants rely on implied rather than historical volatility in this type of analysis and measurement. 


The theta of an option, or a portfolio of options, is the measure of how much an option position's value changes as the option moves closer to its expiration date (simply with the passage of time). The more time remaining to expiration, the more time for the option to become profitable to the holder. As time to expiration declines, option values tend to decline. 


The rho of an option, or a portfolio of options, is the measure of how much an option's value changes in response to a change in short-term interest rates. The impact of rho risk is more significant for longer-term or in-the-money options. 


Financial institutions using options may choose from basically three hedging approaches: 

1. hedging on a ''perfectly matched'' basis, 
2. hedging on a ''matched-book'' basis, and 
3. hedging on a portfolio basis. 

Hedging on a Perfectly Matched Basis 

Some financial institutions prefer to trade and hedge options on a perfectly matched basis. In this instance, the financial institution arranges an option transaction only if another offsetting option transaction with exactly the same specifications (that is to say, the same underlying asset, amount, origination date, and maturity date) is simultaneously available. The trade-off in trading options on a perfectly matched basis is that the financial institution may miss opportunities to enter into deals while it is waiting to find the perfect match. However, many risks are reduced or eliminated when options and other instruments are traded on a perfectly matched basis. In any event, the financial institution continues to assume credit risk when hedging on a perfectly matched basis. 

Hedging on a Matched-Book Basis 

As a practical matter, managing a portfolio of perfectly matched transactions is seldom possible because of the difficulty in finding two customers with perfectly offsetting needs. Less than perfectly matched hedging, called the matched-book hedging, attempts to approximate the perfectly matched approach. In matched book hedging, all or most of the terms of the offsetting transactions are close but not exactly the same, or transactions are booked ''temporarily'' without an offsetting transaction. For example, a financial institution may enter into an option transaction with a customer even if an offsetting OTC option transaction with similar terms is not available. The financial institution may temporarily hedge the risk associated with that option by using futures and exchange-traded options, or forward contracts. When an appropriate offsetting transaction becomes available, the temporary hedge is unwound. In reality, it may be some time before an offsetting transaction occurs, and it may never occur. Typically, institutions that run a matched book establish position limits on the amount of residual exposure permitted. By offering transactions on a matched-book basis, financial institutions are able to assist their customers without waiting for a counterparty with simultaneous, offsetting needs to appear. Hedging on a Portfolio Basis More sophisticated institutions usually find it more practical to hedge their exposure on a portfolio basis when trading options (and other traded instruments) in more liquid markets, such as those for interest rates and foreign exchange. Portfolio hedging does not attempt to match each transaction with an offsetting transaction, but rather attempts to minimize and control the residual price exposure of the entire portfolio. 

Risk-management or hedging models determine the amount of exposure remaining in the portfolio after taking into consideration offsetting transactions currently in the book. Offsetting transactions using futures, swaps, exchange traded options, the underlying asset, or other transactions are then entered into to reduce the portfolio's residual risk to a level acceptable to the institution. Portfolio hedging permits financial institutions to act more effectively as market makers for options and other traded instruments, entering into transactions as requested by customers. It is also more efficient and less costly than running a matched book since there is less need to exactly match the particulars of a transaction with an offsetting position. 


Credit Risk 

One of the key risks in an option transaction is the risk that the counterparty will default on its obligation to perform.2 Accordingly, credit risk arises when financial institutions purchase options, not when they write (sell) options. For example, when a financial institution sells a put or call option, it receives a premium for assuming the risk that it may have to perform if the option moves in the money and the buyer chooses to exercise. On the other hand, when a financial institution purchases a put or call option, it is exposed to the possibility that the counterparty may not perform if the option moves in the money. 

When estimating the credit risk associated with an option contract, some institutions calculate credit risk under a worst-case scenario. To develop this scenario, financial institutions typically rely on statistical analysis. In essence, the financial institution attempts to project, within a certain confidence level, how far, in dollar terms, the option can move in the money. This amount represents the ''maximum potential loss exposure'' if the counterparty (option seller) defaults on the option contract and the financial institution is required to replace the transaction in the market. For a discussion of other ways financial institutions measure credit risk, see section 2020.1, ''Counterparty Credit and Pre-settlement Risk.'' 

 2. This discussion of credit risk is relevant for over-the-counter products. Exchange-traded options are guaranteed by a clearing organization and have minimal credit risk.

Settlement Risk 

The importance of settlement risk may vary materially among countries, depending on the settlement procedures used. In the United States, for example, transactions are typically settled on a net payment basis, with payment being made to only one party to the contract. The beneficiary of the payment incurs the credit risk that the counterparty will not make payment and will default, but does not face the greater settlement risk that a one-sided exchange of securities will occur. Examiners should determine what settlement procedures are used by the markets in which the financial institution participates and should determine what procedures the financial institution takes to minimize any settlement risk. For further discussion of settlement-risk issues, see section 2020.1, ''Counterparty Credit and Pre-settlement Risk.'' 

Liquidity Risk 

The financial institution's ability to offset or cancel outstanding options contracts is an important consideration in evaluating the usefulness and safety and soundness of its options activities. OTC options contracts are often illiquid since they can only be canceled by agreement of the counterparty. If the counterparty refuses to cancel an open contract, the financial institution must either find another party with which to enter an offsetting contract or go to one of the exchanges to execute a similar, but offsetting, contract. On the other hand, if a counterparty defaults and the financial institution is unable to enter into an offsetting contract because of market illiquidity, then the default will expose the financial institution to unexpected market risk. 

Exchanges also do not ensure liquidity. First, not all financial contracts listed on exchanges are heavily traded. While some contracts have greater trading volume than the underlying cash markets, others trade infrequently. In addition, even with actively traded futures and options contracts, the bulk of trading occurs in the first or second expiration month. Thus, to be able to offset open contracts quickly as needs change, the financial institution must take positions in the earlier expiration months when the bulk of trading occurs. 

Some exchange-traded contracts limit how far prices can move on any given day. When the market has moved ''limit up'' or ''limit down'' for the day, trading ceases until the next day. These limits cause illiquidity in certain instances. Hedging contracts with such limited price movement potential may not adequately protect the holders against large changes in the value of underlying asset prices. Examiners should review the financial institution's policies and procedures to determine whether the financial institution recognizes problems that these limits could create (for example, ineffective hedges). This review should also determine whether the financial institution has contingency plans for dealing with such situations. 


The accounting treatment for option contracts is determined by the Financial Accounting Standards Board's Statement of Financial Accounting Standards (SFAS) No. 133, ''Accounting for Derivatives and Hedging Activities.'' (See section 2120.1, ''Accounting,'' for further discussion.) 

Purchased Options 

The purchaser of an option has the right, but not the obligation, to purchase a fixed amount of the underlying instrument according to the terms of the option contract. If a purchased option is held as a trading asset or otherwise does not qualify for hedge accounting, it should be marked to market. Options that qualify for hedge accounting should record unrealized gains and losses in the appropriate period to match the recognition of the revenue or expense item of the hedged item. The premium paid on options qualifying as hedges generally are amortized over the life of the option. 

Written Options 

The writer of an option is obligated to perform according to the terms of the option contract. Written options are generally presumed to be speculative and, therefore, should be marked to market through the income statement. 


The credit-equivalent amount of an option contract is calculated by summing- 

1. the mark-to-market value (positive values only) of the contract and 
2. an estimate of the potential future credit exposure over the remaining life of each contract. 

The conversion factors are listed below.

If a bank has multiple contracts with a counterparty and a qualifying bilateral contract with the counterparty, the bank may establish its current and potential credit exposures as net credit exposures. (See section 2110.1, ''Capital Adequacy.'') 


Options are not considered investment securities under 12 USC 24 (seventh). However, the use of these contracts is considered to be an activity incidental to banking within safe and sound banking principles. 


Hull, John C. Options, Futures, and Other Derivative Securities. 2d ed. New York: Prentice Hall, 1993. 
McCann, Karen. Options: Beginning and Intermediate Issues. Federal Reserve Bank of Chicago, November 1994. 
McMillan, Lawrence G. Options as a Strategic Investment. 2d ed. New York: Institute of Finance, 1986.


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