The Handbook of Equity Derivatives / Edition 1

by Jack Clark Francis, William W. Toy, J. Gregg Whittaker

Overview

"There are so many ways to use derivatives that I'm almost surprised when someone doesn't use them. Producers and consumers, investors and issuers, hedgers and speculators, governments and financial institutions: almost everyone can use them." - From the Foreword by Fischer Black, Cocreator of the Black-Scholes Model

Never before has there been so much interest in equity derivatives-or so much innovation in structuring these products. As new forms of instruments proliferate, their complexity has grown as well. Even equity derivatives professionals are unlikely to know all the details about every existing structure. With equity derivatives comprising one of the most important components of the capital markets, it's more crucial than ever for every financial professional, specialist and nonspecialist alike, to understand how derivative instruments behave, how they're structured, and how to use them profitably.

Edited by leading thinkers in the field, The Handbook of Equity Derivatives, Revised Edition, assembles dozens of experts from universities and Wall Street to help the reader gain a practical grasp of the growing variety of financial instruments and how they work. Contributions from such respected authorities as Gary Gastineau, Mark Rubinstein,

J. Gregg Whittaker, and Fischer Black outline the full range of the equity derivatives market, from classic warrants, options, and futures to the new and innovative PERCs, equity swaps, and equity-linked bonds.

In nonmathematical language, the book provides a clear introduction to equity derivatives, including the fundamentals and history of options, basic equity structures, and pricing determinants, along with a historical perspective on their evolution. You'll find thorough surveys of:
* The burgeoning field of synthetic structures-OTC options and exotics, equity swaps, SPINs, SIGNs, PENs, MITTs, and SuperShares
* U.S. and foreign derivatives traded on organized exchanges
* Issuer derivative structures, such as warrants, convertibles, PERCs, and unbundled stock units
* The unique tax, legal, accounting, and regulatory features of derivatives
* How to make the most profitable use of the many equity derivative products
* Why some financial instruments succeed-and others fail
* The future of the equity derivative market- place

Whether you're a finance student becoming familiar with the field or a practicing professional seeking better ways to exploit the tremendous potential of equity derivatives for profit, The Handbook of Equity Derivatives, Revised Edition belongs on your bookshelf.

"I heartily endorse The Handbook of Equity Derivatives . . . while the market is continuously inventing new instruments and discarding older ones, the clarity and straightforward nature of the handbook hints at a longevity that will make it useful for many years to come." - Stephen A. Ross Sterling Professor of Economics and Finance, MIT (on the first edition)

The most relied-upon resource on equity derivative instruments, their structure, and diverse global markets- now extensively revised and updated

Once, equity derivatives were exotic instruments relegated to the hands of specialists. Today, they are among the institutional investor's most popular tools for managing risk and uncovering new profit opportunities. Recognized for its authoritative contributors and its accessible, comprehensive coverage of the entire field, The Handbook of Equity Derivatives has become the standard reference on the subject for specialist and nonspecialist alike. Now, this essential resource has been carefully updated and revised to cover the most current innovations in these continually evolving investment vehicles, including:
* Comprehensive coverage of the all-important OTC market
* Basic equity structures and how they work
* Pricing determinants
* PERCs, SPIDERs, and WEBs
* The Black-Scholes model
* The best uses for and profit potential of new derivative products
* Key accounting, tax, and regulatory issues

Product Details

ISBN-13:	9780471326038
Publisher:	Wiley
Publication date:	11/08/1999
Series:	Wiley Series in Financial Engineering , #14
Edition description:	Revised Edition
Pages:	720
Product dimensions:	6.38(w) x 9.53(h) x 1.50(d)

About the Author

JACK CLARK FRANCIS is Professor of Economics and Finance at Bernard M. Baruch College in New York City and author of several well-known college textbooks. He previously served as a Federal Reserve economist and on the finance faculty of the University of Pennsylvania's Wharton School.

WILLIAM W. TOY is a Vice President in the Equity Derivatives Department at Goldman, Sachs & Co. and a codeveloper of the Black-Derman-Toy interest rate options model. He holds a PhD in physics from MIT.

J. GREGG WHITTAKER is a Managing Director and the global head of Credit Derivatives for Chase Securities in New York City and a former senior vice president of S. G. Warburg, where he headed the Equity-Structured Products Division. He earned a PhD in economics from the University of Wisconsin.

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Note: The Figures and\or Tables mentioned in this sample chapter do not appear on the Web

Chapter 1

Options Fundamentals

An option is anything which affords an opportunity of choice. A financial option contract has a more specific focus and application.

TERMINOLOGY

In a financial context, every option is a contract which gives the "holder" (that is, the owner of the option) the right to exchange a certain amount of one thing for a certain amount of another thing. There are primarily two types of options: Calls and Puts. A Call option gives the holder the right to buy (or "call to oneself") a certain amount of an asset or "underlying" for a prespecified price. A Put option gives the holder the right to sell (or "put to someone else") a certain amount of an underlying at a prespecified price.

The "underlying" indicates the asset in question (whether it is an equity, a stock index, a foreign currency, a precious metal, a bond or other interest rate product, a commodity, or even a futures contract). This chapter focuses on the case where the underlying is an individual stock and only later touches on the case of a stock index.

What is the "prespecified price" in every option contract? It is sometimes called the "strike price" (because it is the price at which the contract is "struck" and on which the potential future exchange will be based). If an option holder decides to employ that right to buy (if it is a Call) or right to sell (if it is a Put), the option is said to be "exercised." For this reason, the prespecified price in an option contract is usually called the "exercise price". Typical notation for the strike or exercise price involves "X" (an abbreviation for the "eXercise" price) or "K" (used in baseball scoring to denote a strikeout).

One can buy or sell options. If someone has sold an option and the option holder decides to exercise, the option seller is said to be "assigned" on that sold (or short) option position. This terminology relates back to the exchange convention of "assigning" or matching every option that is exercised with a short option counterparty to satisfy the terms of that contract.

Every option contract has associated with it an "expiration date" (or "expiry") after which the "option" ceases to exist; at some point in time, the opportunity to engage in the specified exchange expires. Because of this characteristic, options have sometimes been referred to as "wasting assets."

When may one exercise one's option? It depends. Options may be either European-style or American-style. "European" options can only be exercised at their expiration (on the expiration date), whereas "American" options may be exercised at any time. Note that this "exercise" refers to the buying or selling which the option contract provides. A European Call or Put option (while it cannot be exercised until expiration) can be bought or sold prior to its expiration date. Because of the additional flexibility that they provide, American options must trade for at least the same price as their (otherwise identical) European counterparts; American options often trade for more.

Perhaps the most difficult task associated with acquiring a command of options is mastering all of this terminology. The preceding may have sounded somewhat vague; the following will make it more explicit.

Example

The December 2001 150 (Strike) IBM Call is an option that is listed on the Chicago Board Options Exchange (CBOE). This option gives the holder the right to buy 100 shares of IBM common stock for a price of $150 per share on or before (since it is an American option) its expiration date in December of the year 2001. The actual exchange, if this option were exercised, would involve the option holder receiving 100 shares of IBM common stock and paying $15,000 ($ 150 per share ' 100 shares) following notice of "exercise."

Option settlement may involve the physical underlying stock (even if it is electronically booked) or cash. Since an option on, say, the S& P 500 stock index would make physical delivery of the 500 different stocks difficult, equity index options are typically "cash-settled." That is, the holder receives the cash difference between the underlying index value and the strike price scaled up by a "multiplier" upon exercise or at expiration.

A number of questions that must be answered in order to specify an option contract include:

Are you the option buyer (to be the option holder) or the seller (also called the "writer") of the option?

Is the option a Call option or a Put option? American or European?

What is the strike or exercise price?

When is the option's expiration date?

How will the option "settle" (specify the delivery process)?

What does it cost? That is, what is the option price or option premium?

OVER-THE-COUNTER (OTC) VS. EXCHANGE-TRADED OPTIONS

Two of the primary differences between OTC and exchange-traded options are the way in which the strike prices are referenced and the way in which the option premium is quoted. On exchanges, the strike price is traditionally quoted as a dollar price (in the United States) and the option premium is also quoted in terms of dollars (usually on a per share basis). 5 If the stock is trading at a relatively low price, say S = $15.75, then the listed strike prices on the exchanges will be $2.50 apart (e. g., X = 10.00, 12.50, 15.00, 17.50, and 20.00). If the stock is trading around S = $80.00, the listed strikes will be $5.00 apart (e. g., X = 70.00, 75.00, 80.00, 85.00, and 90.00).

In the over-the-counter market, one often hears reference made to the 95 strike or the 110 strike. These strike prices are being quoted in percentage terms relative to the current stock price. In other words, if a stock is trading for S = $20.00, the 95 Put refers to a Put with a strike price of X = $19.00 and the 110 Call refers to a Call with a strike price of X = $22.00. Moreover, OTC options may have their premiums quoted in percentage terms; a premium of 430 may refer to an option trading for 430 basis points (or 4.30 percent) of the face value of the stock underlying the contract. OTC options can, of course, be quoted in terms of dollars as well.

OTC contracts can be tailored in any way the client wishes with respect to strike, notional, expiration, delivery (or even, increasingly, with respect to additional nonstandard or "exotic" characteristics such as knockout barriers or spot price averaging). In an effort to accommodate nonstandard requests, exchanges have made attempts to offer "flexible" contracts or "FLEX" options that differ with respect to expiration date, strike price, or contract size from their standard counterparts; this often requires the support of designated marketmakers who have agreed to price such nonstandard contracts in a timely fashion when inquiries are made.

Exchange-traded options typically require "margining". That means a "good faith" deposit of cash (and possibly securities) must be posted to the exchange/ clearinghouse. This margin is then adjusted for daily option price changes (in an effort to limit the counterparty risk taken on by the exchange). Although OTC options are not necessarily marked-to-market, there is counter-party risk to the two sides of an OTC option trade.

Some firms trade actively both over-the-counter and on the options exchanges; they step in and "arbitrage" any significant differences between option prices in these markets-- so prices at the two markets are usually "in line."

The commonly referenced advantages and disadvantages of each type of contract (e. g., margining, marking-to-market, counterparty risk, liquidity, standardization, tailored contracts, transparency, anonymity, etc.) are no different for options than for Forward versus Futures contracts.

FORWARDS VS. OPTIONS

An option is best understood by considering a related contract: a Forward. A "Forward" contract is an agreement between two parties to make an exchange of an underlying asset for an agreed upon price (that is, an exchange of one asset for another at a previously contracted rate) at some point in the future. This definition should sound familiar.

A "long" Forward contract involves an agreement to buy the underlying on some future date for a quoted Forward Price. A "short" Forward contract is an agreement to sell the underlying on that future date.

The primary difference between a European Call option and a long Forward contract centers on the nature of the agreement between the two counterparties. With a Forward contract, there is the unmistakable assent that both parties agree to the anticipated exchange as an obligation. With an Option, the acquiring party recognizes ownership as a right to a future exchange (fully expecting that the party which sold the option will perform if the option holder decides to exercise that option).

A table of prices and the associated instrument values (at expiration) makes clear the difference between an Option and a Forward. The long one-year Forward contract with a Forward price of F = $42.00 is compared in Table 1.1 to the one-year European Call with a strike price of X = $42.00.

This table can be summarized, and provide some intuition, by graphing the values associated with these two instruments on the same axes (see Figure 1.1). If the underlying price were to rise above the price specified in the Forward contract (the Forward price), that Forward would take on a positive value.Similarly, if the underlying stock price settled above the strike price at expiration, the option would take on a positive value.

If F = $42.00 is the correct market Forward price, no additional premium or charge is necessary for someone to enter into this contract; this is illustrated in Figure 1.1. Clearly, when one buys (or sells) a Forward, that person assumes both upside and downside risk. When one acquires an option, there is an asymmetry. With a Call option, there is only upside; for that reason, one must pay a premium to acquire an option. This premium payment was not shown in Figure 1.1.

OPTION GRAPHS

Option graphs provide an excellent framework for understanding Calls and Puts. Consider the 42.00 Strike Call pictured in Figure 1.1. One could ask, "What value would that option have if it were held till expiration?" If the stock (spot) price (denoted by "S") was below X = $42.00 at expiration, the option would have no value. For every dollar the spot price exceeds $42.00 at expiration, the option will increase in value a dollar. At expiration, every option should trade for its parity (or "intrinsic value").

For a Call option, parity = Max [( S D X), 0]

and for a Put option, parity = Max [( X D S), 0]

where Max [a, b] indicates the larger of a or b.

Arbitrage ensures that every option trade at parity at expiration.

There are primarily four types of option graphs: (1) the individual option value graphs at expiration (long Call, long Put, short Call, short Put); (2) the profit/ loss (P/ L) or breakeven graphs for these four basic building blocks (which incorporate the option premium into the diagram); (3) the option spread or combination graphs; and (4) the option value graphs prior to expiration. A long Call "value" graph at expiration is shown in Figure 1.1.

A long 42.00 strike Put value graph at expiration (see Figure 1.2) is equivalent to a one-sided short Forward position. If one is long the Put (owns the Put), then, if exercised, one would be selling the underlying asset for the Strike Price. This long Put corresponds to an asymmetric short position in the underlying. In terms of option value at expiration, the worst case for a long Put holder is zero value; the option may expire unexercised. This Put will have no value at expiration if the Spot Price exceeds the Strike Price; it would be better to sell the stock in the stock market rather than to exercise the option at the lower Strike Price.

If one sells options, one is transferring to someone else the "upside" or positive value that options potentially provide. Ignoring the option premium (and independent of any other positions in your portfolio), short options appear only as downside. A short Call loses value for the Call seller on a dollar-per-dollar basis once the spot price exceeds the strike price. Consider the value graph of the short 42.00 Call (Figure 1.3).

A short Put loses value similarly, as the spot price falls beneath the strike price. Consider the value graph of the short 42.00 Put (Figure 1.4).

BREAKEVEN ANALYSIS

Since options provide only upside to their owners, they sell for a positive premium. Conversely, if one sells options, one must be compensated.

If the 42.00 Call costs 5.00, then we must recognize that the net value which the option provides should be reduced by the option premium paid. This results in the net payoff from that option position. Mechanically, the option value graph is lowered (shifted down) by the amount of the premium. This can be seen in the P/ L (profit/ loss) graph shown in Figure 1.5.

Figure 1.5 is also referred to as a "breakeven" graph because it helps identify the expiration spot price at which the option buyer "recovers" the option premium. 8 In this case, the "breakeven price" is S* = 47.00.

If the 42.00 Put costs 5.00, then the graph (shifted downward by the premium) results in a "breakeven" of S* = $37.00 (see Figure 1.6).

Would the 42 Put be exercised if, at expiration, the spot price was at S = 40.00? At first glance, one might conjecture that it would not be exercised since that would result in a loss (of 3.00). But, the 42 Put has value if S = 40.00. On reflection, it is better to exercise and recoup 2.00 of value (losing 3.00 net) than to let an option that has value expire unexercised and forfeit the full premium (losing 5.00).

If one sold the 45.00 Call option for a premium of $1.75, then the breakeven would occur at S* = 46.75 (see Figure 1.7).

If one sold the 40.00 Put for 2.25, its breakeven would be S* = 37.75 (see Figure 1.8).

OPTION SPREADS

Familiarity with simple option graphs and their breakevens facilitates the analysis of more complex options positions or "option spreads". These typically involve two or more options incorporated into a single strategy, which, therefore, can be combined in a single diagram.

One approach to arriving at a single combined options graph is to

1. identify the individual options' value graphs;
2. combine the "pieces" and obtain a single value graph; and then
3. adjust that graph for the net option premium (shifting the graph up if premium was received, shifting it down if a premium was paid).

Let's examine three of the more common option combinations and consider why one would want to trade that option spread.

If one buys a Call and buys the same strike (and expiration) Put, one is said to be long a "Straddle". If one bought the 42.00 Call for 5.00 and also bought the 42.00 Put for 5.00, then the combined P/ L graph would appear as shown in Figure 1.9.

Clearly the worst possible outcome would occur for the buyer of this Straddle if the spot price at expiration ended up at S = 42.00. In this case, both options end up without value and the option purchaser would be out the entire premium of 10.00 per share. This position gains in value as spot moves away from S = 42.00 in either direction. Why might someone buy a Straddle? If one expects a large movement in the underlying price, but is unsure of which way it might move, a Straddle seems like a reasonable buy. Straddles are a popular strategy around elections and other important events that have a strong likelihood of moving the markets. In this example, the breakeven prices are S* = 32.00 and S* = 52.00.

Another combination is known as a short "Strangle". In this strategy, a Call is sold and a Put (with a different Strike but the same expiration) is also sold (see Figure 1.10). Note that if these options had the same Strike Price, this would have been simply a short Straddle. If the 40 Put is sold for 2.25 and the 45 Call is sold for 1.75, then the combined graph will have breakevens at S* = 36.00 and S* = 49.00. If there is little activity in the markets, it is difficult to make money trading the underlying assets, but selling options could generate positive returns (if the spot price at expiration ends up anywhere between the two breakevens).

One additional representative of these combined option strategies is the long "Call Spread". This would involve buying a more valuable (lower strike) Call and selling a less valuable (higher strike) Call. If one bought the 40 Call for 5.50 and sold the 45 Call for 1.75 the combined graph is in Figure 1.11. Note the 3.75 (= 5.50 D 1.75) "shift" down based on the net option premium. Since both options have zero value, if the spot is below the lower strike price of X = 40.00, the loss in that range will be 3.75. Above the higher strike price of X = 45.00, any further gains from the 40 Call are offset by losses associated with the short 45 Call. Therefore, the position "flattens out". With a breakeven price at S* = 43.75, the investor would have to decide on the appeal of this spread based on the perceived risk-reward tradeoff.

The options markets seem to enjoy making up monikers for option spreads. These names include Butterflies, Boxes, Calendars, Condors, Reversals, Christmas Trees, Seagulls, Strips, Straps, Stupids, Conversions, and Wrangles.

SYNTHETICS

A synthetic material is a constructed fabric; in other words, it was "fabricated" artificially (perhaps out of plastic). One can also fabricate synthetic underlying positions and synthetic options positions.

If one buys the 42 Call and sells the 42 Put, then one has "fabricated" a Synthetic Forward position with a "Forward" price of F = 42.00 (see Figure 1.12. )

If the spot price ends above the strike price of 42, then the Call will be exercised, the Put will be valueless, and one will purchase the underlying for the strike. If the spot price ends up below the strike, the Call will go unexercised, one will be "assigned" on the short Put; if the owner of the Put wishes to sell, the Put seller is compelled to buy (at the strike price of 42). Regardless of where spot is at expiration, one way or another, the long called- short Put combination will induce a purchase of the asset in the future for the strike price. This is essentially a long Forward contract (since the purchase will take place), but it is synthetic since it was fabricated from options. This synthetic Forward can be summarized by the following formula:

+F = + C - P

where F, C , and P denote the prices of a forward, a call, and a put on the same underlying.

A synthetic option position (as opposed to a synthetic underlying position) is similar in the sense that it is manufactured using components other than the contract that is meant to be fabricated. To construct a long synthetic Call, one might start by buying stock (replicating the "long" exposure which a Call provides), but to limit the stock's downside, one would have to buy a Put. The position could be summarized:

+C = + S + P

Note that this is simply a rearrangement of the earlier formula.

In the equity world, although the payoff diagrams may look identical for the synthetic and real positions, they can differ in terms of dividend treatment, carry costs, voting rights, tax treatment, and so on.

FORWARD PRICING

Earlier, we looked at the 42.00 strike Call-- which might have been written on, say, Toys R Us stock. We then compared the one-year 42.00 Call to the 42.00 Long Forward. Where might this Forward price of F = $42.00 have come from? Good question. One might ask, "What would be the price quoted today for a transaction on Toys R Us?" To be specific, if Toys R Us is trading in the spot market (today) for S = $40.00 per share, at what current price do you think someone would be willing to enter into an agreement to buy or sell Toys R Us in one year's time?

The answer might seem to depend on the individual investor. But, as will be seen, arbitrage will drive the Forward price to one very specific numerical value that is independent of investor sentiment and expectations.

Assuming Toys R Us does not pay a dividend (as has historically been the case) and interest rates are 5 percent, the one-year Toys R Us Forward should be valued at $42.00. This is calculated using the following relationship:

F = S + Srt

where S is the spot price, r is the (annual) interest rate, and t is the time (in years). What does it mean to say that this is the "correct" Forward Price?

If a Forward price of F = $45.00 was quoted in the market, arbitrageurs (who have no inherent interest in owning or speculating in Toys R Us stock) would sell the stock Forward (because F is too high relative to the preceding "fair value" calculation), buy that stock in the spot market (borrowing the purchase price at the prevailing interest rate), and then reverse that trade in the future to lock in an arbitrage profit (in this case of $3.00). Similarly, if the Forward price was quoted at, say, F = $38.00, arbitrageurs would buy the stock Forward, sell it spot (depositing the proceeds in the bank and, therefore, receiving interest on the sale), and, again, reverse the trade in the future, ensuring a profit (this time, of $4.00).

A rational investor who wishes to own Toys R Us stock "long term" should be indifferent between purchasing (going long) the stock today for S = $40.00 or agreeing to purchase the stock in one year at the "fair value" Forward price of $42.00. Because of the costs (and possibly benefits) associated with buying and carrying stock, the prices of stock at two different points in time will usually not be the same. One should be indifferent, however, between paying a lower spot price today and agreeing to pay a higher price in the future (the Forward price) which reflects all the costs of carrying the stock (for a year) net of any pecuniary benefits which that stock might confer. Since we assumed Toys R Us pays no dividend and it is costly to carry, one should be willing to pay a higher price in the future for the opportunity to acquire it later and avoid incurring these "carry" costs.

The primary cost associated with "carrying" stock is the interest on the spot price (whether the money which is handed over to purchase the stock is borrowed from the bank or withdrawn from an interest-bearing bank account). The primary benefits associated with owning and carrying stocks are the dividends.

A similar argument could be made for a dividend-paying stock like IBM. If IBM stock was trading for S = 150.00, interest rates were 5 percent, and IBM was fully expected to pay a dividend of $1.00 over the next year, then the IBM one-year Forward would be priced around F = $156.50. Again, arbitrage would drive the Forward back to this value (with spot at S = $150.00 and interest rates at r = 5 percent).

Although Forwards and Futures are frequently quoted for currencies, commodities, and certain interest rate instruments, we do not usually think of Forwards or Futures on individual stocks. Forwards on stocks trade infrequently; Futures on individual stocks are not even listed in the U. S. 10 For this reason, even those working in equities do not typically think in terms of equity Forward pricing. Nevertheless, understanding Forward value takes us halfway to understanding options.

OFF-MARKET FORWARDS

A case has been made that the one-year Toys R Us Forward should be priced at F = 42.00. What if you were a marketmaker in this stock's Forward contract and a customer told you they wanted to buy Toys R Us one year in the future for a price of F = 35? Your response might be, "So do I." If the client is sufficiently persistent or important, you might eventually say, "Okay, I'll sell you Toys R Us stock in the future for a price of F = 35.00 per share," but that surely wouldn't be the end of the transaction. Obviously the Forward price is 7.00 too low (relative to where it should be). You, as the seller of Toys R Us Forward, would need to be compensated with a "premium" of 7.00 (a premium of 7.00 in one year's time, that is-- since the 42.00 and 35.00 are prices for exchanges one year in the future). If you are to receive a payment today, all you need to receive is the present value of 7.00 (which could then be invested and which would grow to 7.00 in one year, making you "whole" on your "off-market" Forward trade). Similarly, agreeing to buy IBM forward for $180 should result in your receiving the present value of 23.50.

We have said that options are like one-sided Forward contracts. Since the strike price on any option may be above or below the current Forward price for that time horizon, most options are, in a sense, (one-sided) off-market Forwards.

FORWARD PRICES VERSUS SPOT PRICES IN THE FUTURE

Where might the spot price of Toys R Us or IBM be in one year's time? We have no idea today, but anyone willing to trade above or below the Forward price is willing to provide a riskless profit to an arbitrageur.

If there was a systematic deviation of the one-year Forward price (today) from the spot price one year later (in the future), speculators would attempt to profit from this "relationship". Assuming the one-year Forward price was ALWAYS exactly $3.50 higher than the resultant spot price in one year, what trades would one enter into? One could "arbitrage" this persistent discrepancy by selling the Forward, waiting a year to buy spot, and then delivering against your short Forward contract. For as long as this situation persisted, one would ALWAYS make exactly $3.50. What if this relationship were true only 95 percent of the time? Some people would attempt to take advantage of this phenomenon. What do you think would happen if it was true 75 percent of the time? What about 60 percent of the time?

Although no one knows where any given stock price will be trading at any given point in the future, it is safe to say that there is some probability that the future spot price will be higher than the Forward price and some probability that it will be lower. Exactly how much higher or lower might be unclear and the associated probabilities are not obvious. But, in a very real sense, the Forward price should be located at the "center" of the probability distribution associated with the possible future spot prices.

DIGRESSION: EXPECTED VALUE

If one rolls a standard six-sided dice, how many dots would one expect to see as a result of any given roll? Assuming the dice is fair (all the sides have an equal probability of appearing), the choice of any one number may seem as good as any other. There is, however, a "best" guess here. That best guess is 3 1 /2. Now there is no way you can realize 3 1 /2 dots on any given toss, but, "on average", there will be 3 1 /2 dots per roll. Considering all of the possible out-comes, weighting them by their respective probabilities, and then summing these numbers give an expected value of 3 1 /2.

( 1 /6)( 1) + ( 1 /6)( 2) + ( 1 /6)( 3) + ( 1 /6)( 4) + ( 1 /6)( 5) + ( 1 /6)( 6) = 3 1 /2

What this really means is that if one, say, rolled a dice a million times and summed all of the dots appearing on all of these tosses, one would "expect" to see a total of about 3 1 /2 million dots. This probability distribution (the way in which the likelihood of occurrence is spread over the various outcomes) can be represented as shown in Figure 1.13

If one rolls two dice simultaneously, what is one's best guess for the total number of dots showing on the two top surfaces? The possible outcomes range from 2 (snake eyes) to 12 (box cars). By enumeration (of the 6 ' 6 = 36 possible outcomes), the most likely outcome is "7" [obtainable in six different ways: (1,6), (6,1), (2,5), (5,2), (3,4), and (4,3)]. Our best "guess" for the outcome of the roll is "7," but do we really think 7 dots will appear? No-- the odds are 30 /36 = 5 /6 that something other than a "7" will occur. The probability distribution associated with this experiment is seen in Figure 1.14.

OPTION VALUATION

What do dice have to do with options? Both have to do with uncertain outcomes and risk. In both cases, one must have some idea as to the possible outcomes and the associated probabilities. In a very real sense, the Forward price of a stock is similar to the "7" above. Although the Forward price provides little predictive power as to where the future spot price will be, it is in a sense our best "guess." It must, therefore, be at the center of the probability distribution of spot prices in the future. This is integral to what is referred to as "Risk Neutral Option Valuation".

The probability distribution for future stock prices will depend on a number of things: the Forward price (itself based on the current spot price plus the "carry"), the time horizon under consideration, and the volatility of the stock price. An example for the Toys R Us Stock case is presented in Figure 1.15.

Obviously, the simple discrete probability distribution presented is simplistic, but it can be used to demonstrate the principles of theoretical option valuation.

There are two ways in which an option's value can be computed:

1. 1. An option can be thought of as a one-sided Forward contract or, stated slightly differently, as a Forward plus "insurance" (insurance that the option provides in the event the holder would choose not to engage in the forward transaction). An option value, then, would consist of Forward value and "insurance value."
2. 2. An option can also be thought of as a contract that may have a positive value or payoff at expiration; this view suggests approaching option valuation as an expected value calculation.

Using the distribution set forth in Figure 1.15, one can ask,

"What is the value of the 25 Call?" (see Figure 1.16). The first approach shows that the 25 (off-market) Forward would have a value (in the future) of 17.00 (because this strike price is 17.00 below the Forward). Note, however, that this option is "better" than the 25 Forward because there is a possibility that the spot price will be at S = 16.00 at expiration (in which case, the 25 Call would be preferred to the Forward). For this reason, the 25 Call should be worth more than the 17.00. How much more? It depends on the value of the "insurance".

With the Forward, one might end up buying for 25.00 a stock that is trading in the market for S = 16.00. In that case, one "loses" 9.00. The Call provides protection against that loss that considers both the size of the loss and its likelihood of occurrence. Since the 9.00 loss occurs with a 10 percent probability, the "fair value" of the "insurance" is (9.00) ' (. 10) = .90. The full option value (in the future) then would be 17.00 + .90 = 17.90. Since option premium is paid up-front, discounting by 5 percent for one year gives a present value for the 25 Call of 17.05.

Alternatively, one could simply look at all the possible payoffs at expiration associated with this option and weigh them by their probabilities (see Figure 1.16). In this case, the (future) value of the 25 Call would be:

(4) ' (. 20) + (17) ' (. 40) + (30) ' (. 20) + (43) ' (. 10) = 17.90.

Again, today's present value of the option's value is 17.05. Both approaches generate the same option value.

The value of the 25 Put (using the expected value approach) would derive solely from the (outside) chance that the spot price at expiration is below 25.00; in this example, this means if S = 16.00. This would imply that the 25 Put's value will simply be (9) ' (. 10) = .90 (in the future) or .86 (today). It is interesting that the "insurance value" in the Call should equal the "opportunity value" in the Put. In both cases, the "insurance" or "opportunity" value derived from the fact that the spot price in the future might not be equal to the Forward price. To avoid the cumbersome use of terms like "insurance value" and "opportunity value", both may be referred to as "volatility value" since this captures the fact that stock prices fluctuate and may rise or fall in the future (relative to the current Forward price).

Valuing the 42.00 Call (see Figure 1.17) using the "Forward Value plus Volatility Value" approach (and suppressing the present valuing), one finds:

Forward Value =

0

Insurance Value = (26) ' (. 10) + (13) ' (. 20) = 5.20

So the 42 Call would be worth the present value of 5.20 or 4.95

.

Using the "Expected Value Approach" (see Figure 1.17), one finds the (future value of the) 42 Call to be worth:

(13) ' (. 20) + (26) ' (. 10) = 5.20

The 42 Put is also worth the present value of 5.20, or 4.95

.

Finally, calculating the values of the one-year 60 strike options (see Figure 1.18), we find

60 C = PV (. 80) = .76 and

60 P = PV (18.00 + .80) = 17.90.

Note once again the equivalence of the options' volatility values; this is not just a coincidence. This idea will be revisited.

In short, considering the first approach (Option Value = the present value of [Forward Value + Volatility Value]), we can summarize the idea behind Forward Value:

for a Call option, Forward Value = Max [( F D X), 0]

and for a Put option, Forward Value = Max [( X D F), 0]

Although the use of a simple probability distribution may seem unrealistic, there is an assumed probability distribution inside every theoretical option valuation model which is employed in the calculation of option values along the lines described here.

MORE OPTION TERMINOLOGY

The discussion of Forward Value above raises a question. What does it mean for an option to be in-the-money, at-the-money, or out-of-the-money? Consider a one-year European 41.00 strike Call with the stock price at 40.00 and the Forward at 42.00. This option has no (parity) value relative to the current spot price, but it does have value relative to the Forward (and this will enter into the option's valuation calculation). This option is out-of-the-money spot but in-the- money forward.

Because forwards on individual stocks are not commonly traded, the standard use of in-, at-, and out-of-the-money with equity options typically refers to the relationship between the spot price and the strike price. Regardless of whether someone is asking for the at-the-money one-week or at-the-money two-year option on IBM, the strike price will be set equal to the current stock price. Clearly, this obfuscates the notion of Forward valuation.

It is the exchange convention to quote strike prices at fixed intervals. Thus, the actively traded options on an exchange might be the 10.00 strike, the 12.50 strike, and the 15.00 strike, with the stock trading at S = 12.75, marketmakers would refer to the 12.50 strike as the at-the-money option, even though spot and strike are not exactly equal.

This convention of comparing the strike price to the spot price (to determine in-, at-, and out-of-the-money) is not the norm with other underlyings. If someone working in foreign exchange asked for an at-the-money three-month "Dollar| Yen Call", that option would have its strike price set equal to the three-month Forward price for Dollar| Yen.

OPTION VALUE

Previously, four types of option graphs were identified. Prior to expiration, options have value above and beyond their parity or intrinsic value (seen in the expiration graphs as the "hockey sticks"). Often this additional value is referred to as "time value"; this consists primarily of what we have labeled "volatility value". The graph of a typical Call value prior to expiration is seen in Figure 1.19.

PUT- CALL PARITY

If one were to buy the 42 Call and sell the 42 Put discussed earlier at their theoretical values (4.95 each), one would construct a synthetic Forward with F = 42. How much should one pay to get long the 42 Forward? Nothing. That's the "right" price. The option premiums bear this out; they cancel each other. If one could (net) be paid to go long the Forward synthetically at 42 and could then sell the real Forward at 42, one would lock in an arbitrage profit. Similarly, if one could sell the 42 Forward synthetically for a credit (that is, taking in net option premium) and buy the real Forward, one would lock in a profit.

Let's look at the 25 Strike options. If one buys the 25 Call for 17.05 and sells the 25 Put at .86, one pays 16.19 to be long at 25. This looks attractive since one is long the 25 Strike synthetic Forward, one only paid 16.19, and one could sell the real Forward for 42.00. Without taking into account the time value of money, it looks like one is long Forward (net) at 41.19. But recognizing that the option premium is paid today, this translates to being long Forward at F = 25 (in the future) with an additional cost (in the future) of 17.00 (the future value of 16.19). How much would someone wishing to go long the off-market Forward at F = 25 have to pay today? One should pay the present value of 17.00, which is 16.19.

Similarly, if one were to buy the 60 Call (for .76) and sell the 60 Put (at 17.90), one would be long the 60 synthetic Forward and in possession of the net option premium (17.14). How much would someone need to be paid to go long the off-market Forward at S = 60? The correct up-front payment would be 17.14 (the present value of the Forward Value of 18.00).

Put- Call Parity can be understood in two different ways. For European options on non-dividend-paying stocks, as we have shown earlier, the Call and the same-Strike (same expiration) Put will have the same volatility value. Stated another way, same-strike European Calls and Puts will differ by the (present value of the) Forward Value. This can be summarized in the formula:

C - P - S + PV( X) = 0

where PV(.) indicates "the present value of". Or, perhaps more insightful,

C - P - PV( F - X) = 0.

OPTIONVALUATION MODELS: THE BLACK- SCHOLES FORMULA

The year 1973 was a landmark year for options. It was the year that options were first traded on an exchange (the Chicago Board Options Exchange or CBOE) in the United States. It was also the year that Fischer Black and Myron Scholes saw the publication of their seminal paper on option valuation. Prior to 1973, options were traded "over the counter" by members of the American Put and Call Brokers Association. They were priced like any commodity (supply and demand, seat-of-the-pants, finger in the air). The Black- Scholes Option Pricing Formula gave derivatives marketmakers a tool for the valuation of the options they traded. More so, they provided insights into the risk measurement and hedging parameters necessary for the risk management of their portfolios.

Although some attempts had been made to devise a scientific or "rational" model for option valuation going all the way back to Louis Bachelier (a French mathematician who wrote his dissertation on options in 1900), the "correct" specification of the probability distribution and the "right" interest rate to employ slowed progress on this problem. Black and Scholes, by way of an ingenious arbitrage argument, "solved" the problem (followed closely by Robert C. Merton and others who went beyond their path-breaking work).

The Black- Scholes Equation specifies an arbitrage-free relationship between three of the primary option risk measures. When solved, this equation is known as the Black- Scholes Formula.

The Black- Scholes Formula for a European Call (on a non-dividend-paying stock) follows.

C = S N( d1) - X e -rt N( d2)

where S = the stock (spot) price

X = the strike or exercise price

t = time till expiration (in years)

r = the annual risk-free rate of interest

s = the annual volatility of the stock price

where N(.) refers to the standard cumulative Normal distribution function. This formula and its counterpart for Puts (derived under very severe assumptions) apply only to the valuation of European options on non-dividend-paying stocks. Later researchers generalized this formula to handle other underlyings. According to the formula, the at-the-money spot three-month (t = 1 /4) Call option on a stock with S = 50.00 = X, no dividends, a volatility of s = 25 percent, and interest rates of r = 5 percent has a value of 50 C = 2.80.

The Black- Scholes Formula has been lauded as the most successful financial (or even economic) model ever. Empirically, Black- Scholes prices are repeatedly confirmed in the market. In a fascinating paper, "Living Up To The Model," Fischer Black noted, though, that, "Because the formula is so popular, because so many traders and investors use it, option prices tend to fit the model even when they shouldn't."

A few years after the Black- Scholes paper appeared, John Cox, Stephen Ross, and Mark Rubinstein published their discrete-time option valuation approach which is commonly referred to as the Binomial Model. More transparent than the Black- Scholes approach, requiring substantially less mathematical hardware to understand, and more general in that it can be employed to value a wider class of options, the Binomial Model (or some variation of it) provides one of the fundamental approaches to option valuation today. Theoretical option valuation is summarized in Figure 1.20.

Armed with theoretical values and the risk measures necessary to manage their portfolios, banks and trading firms attempt to buy options a bit below theoretical value and sell them a bit above theoretical value.

Textbooks on options and derivatives often refer to the options markets as zero-sum institutions in which every winner is offset by an equal and opposite loser. This is not true. Options traders who hedge their spot price risk generally take on volatility exposure; clients generally wish the asymmetric, unidirectional price risk which options provide (either to hedge an existing exposure or to implement their market view). Can an end-user"profit" on an option trade and the option dealer "profit" on the other side of that same trade simultaneously? The answer is clearly, "Yes!"

The reason for this has to do with the fact that option values are based on the assumption that the spot price in the future will equal the Forward price; investors, hedgers, and speculators often know that options look attractive or expensive based on their forecasts or beliefs about the future spot price in the real world.

Most option marketmakers, when asked what they do, would respond, "I trade `volatility' ." Now, they don't really trade volatility; they trade Calls and Puts, but because they hedge their spot risk, they are left with (and consider themselves trading) volatility value. If the investor is right about spot price direction and the option trader is right about volatility, both sides can simultaneously benefit. Options are not treasure troves, however; both sides could lose if they are wrong.

If options were "zero-sum," far fewer people would use them. Let's turn now and examine some of the more common equity option strategies used today.

OPTION STRATEGIES FOR EQUITY RISK MANAGEMENT

Today, options are used extensively as integral components in the risk management of equity portfolios.

A Protective Put is one of the more common option strategies. It involves purchasing a (usually out-of-the-money) Put as a hedge for a long stock position. Although premium must be paid for this option, the Put provides downside protection should the stock price fall. One may ask why someone would hold stock that might fall. There are instances in which someone possesses restricted stock (which cannot be sold); the Protective Put is a very reasonable and conservative play in this context.

The graph of a Protective Put strategy and that of a long stock position are compared in Figure 1.21.

To an investor, this strategy preserves upside exposure and eliminates downside risk. Does that description sound familiar? Indeed, a Protective Put strategy is simply a long synthetic Call.

A strategy that is extremely common in the equity world, but seemingly riskier, is the Buy- Write. A Buy-Write or Covered Call or Covered Write strategy involves buying (or being long) a stock and selling (or "writing") a (usually out-of-the-money) Call option against that stock. If the stock price rises, one retains one's appreciating stock exposure up to the Call's strike price (and, in addition, one has taken in option premium). If the stock price remains unchanged or falls, this strategy will dominate a simple long stock position (again, because this strategy generated upfront option premium) (see Figure 1.22). The frequently stated rationale for a Buy-Write strategy is "yield enhancement." The worst case is that the Call one has sold ends up deeply in-the-money at expiration (but since one owns the stock already, it is simply delivered for the strike price). Often, a Buy-Write "program" involves selling Calls on a rolling basis (e. g., every month or every three months). Usually, the strike prices chosen are just outside of the range of the expected appreciation of the stock.

Perhaps the "best" all-around option strategy is the Collar strategy. 12 A Collar consists of a long Put and a short Call (usually both options are out-of-the-money); it is typically thought of in conjunction with a long stock position (see Figure 1.23). The Put provides downside protection for the stock position; the combined position facilitates participation in the appreciation of the stock up to the strike price on the short Call; and the "premium" from the sold Call can be used to finance the Put purchase [thus no premium needs to trade hands-- that is, it can be (and often is constructed to be) a zero upfront cost option strategy]. It addresses the three goals of risk aversion, profit maximization, and option-premium resistance.

Looking at the option graph of a Collar (Figure 1.23) explains its name. The exposure is limited to a range or "collared in"; the stock's downside has been curtailed at the expense of some of the stock's upside.

In practice, a customer might ask for the Collar that provides 95 percent downside protection; for the strategy to be "premium neutral" (that is, for there to be zero net option premium), the Call that must be sold (e. g., the 115 percent Call) can be identified. Similarly, the user might decide that the stock will not appreciate (between now and the contemplated expiration) by more than 15 percent (suggesting the 115 Call would be a reasonable option to "sell"); the downside protection which the Call "premium" could acquire might then be determined (that is, the 115 Call can finance the 95 Put). These sorts of discussions are generally referred to as "strike selection" and, along with a discussion of time horizon/ expiration, are a necessary component of every option strategy.

OTHER EQUITY OPTION ISSUES

There are issues that arise with equity options (generally regulated in the United States by the Securities and Exchange Commission) that do not arise with options on non-equity related underlyings.

What if a stock splits? The norm for, say, a two-for-one stock split would involve multiplying the notional number of shares associated with the option contract by two and halving the strike or exercise price. Although stock splits can be handled precisely in the OTC market, rounding sometimes occurs with exchange-traded contracts.

Other circumstances arise from mergers and acquisitions, rights issues, and dividends paid with stock. In the past, it was not uncommon for OTC options to have adjustments made to the strike price for cash dividends. These were known as "dividend-protected." For example, in the case of a Call, the strike price was lowered by the amount of the dividend (the amount by which the stock price was expected to fall). Although this is no longer the norm, this convention is still applied sporadically (as in the case of payouts from REITs). Any option investor or marketmaker must be aware of the specifics of the contract being traded.

CONCLUSION

The equity options market is one of the most mature of the options markets. The goal of this chapter was to give some institutional insights into the equity options market, lay the foundation of option terminology (which, admittedly, can be confusing), identify the differences between the exchange-traded markets and the OTC market, explain option graphs and their uses, introduce some simple option spreads, deliver some insights into theoretical option valuation (with a heavy reliance on Forward valuation), and catalogue a few of the most commonly employed equity option strategies.

Much of this material was developed at O'Connor and Associates in Chicago, evolved through Swiss Bank Corporation's Education Department, and continues to be used at Warburg Dillon Read (the Investment Banking Division of UBS AG) as an effective approach for educating burgeoning capital market participants.

APPENDIX: THE VALUATION OF STOCK INDEX FUTURES OPTIONS

Valuation of European Options on Stock Index Futures is similar to valuation of individual equity options with one or two adjustments. After having stressed the importance of Forward pricing for option valuation, the"carry" or"basis" can be suppressed in this instance since the net carry on the index will have been incorporated into the price of the Stock Index Future itself. It is not necessary to incorporate this "drift" into what is otherwise the standard Black- Scholes Formula. The European Call formula for a Stock Index Future appears as follows:

C = [F N( d1) - X N (d2)] e -rt

where F = the Futures price

X = the strike or exercise price

t = time till expiration (in years)

r = the annual risk-free rate of interest

s = the annual volatility of the Futures price

where N(.) refers to the standard cumulative Normal distribution function. Caution should be exercised at the application of a formula, however. First, many of the exchange-listed options on stock indexes in the U. S. have American-style exercise (so this formula is not applicable). Second, the units used for F and X are index points, not dollars; these must be converted to a dollar (or currency) figure by the proper application of the contracts "multiplier." Third, although a volatility forecast may be undertaken for a stock index future (just as it may for an individual equity), one should be aware that there may be a relationship between interest rates and that index (as might exist, for example, for bank stocks). This would either bolster or diminish the value of the option (for purposes of valuation), but begs the question of modeling the stock index-interest rate correlation. Finally, the uniform application of a single volatility to options of every strike and expiration would generally result in many "incorrect" values. This phenomena, referred to as the volatility "smile," is typically observed with options on individual stocks as well and reflects an understanding, on the part of the participants in these markets, that the model (and its assumed probability distribution) is simply not precisely "correct."

Nevertheless, we could apply the formula to value the six-month European 1250 Call on a Stock Index Future where the Future is trading at F = 1200, interest rates are 5 percent, and the index has a volatility of s = 12 percent. This gives us an option value of 20.58 (which, if the multiplier were $100 per point, would imply a cash premium of $2,058 for this option).

Although many options on stock indexes are cash-settled, options on stock index futures are typically futures-settled (i. e., if you exercise an in-the-money Call, you will "buy" or get "long" a stock index future, receiving the "parity" in cash through your margin account). This has implications for futures-settled option exercise. Exercise of an in-the-money Call option on an individual stock will result in the option holder making a cash payout (money flows out as a result of the exercise); exercise of an in-the-money future-settled option results in a cash influx.

REFERENCES

Bernstein, Peter. Capital Ideas: The Improbable Origins of Modern WallStreet. New York: Free Press, 1992.

Black, Fischer and Myron Scholes. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81 (1973): 637- 654.

Black, Fischer. "The Pricing of Commodity Contracts." Journal of Financial Economics 3 (1976): 167- 179.

Black, Fischer. "The Holes in Black-Scholes." Risk 1 (1988): 30- 33.

Black, Fischer. "How We Came Up With the Option Formula." Journal of Portfolio Management (1989): 4- 8.

Black, Fischer. "Living Up To The Model." Risk 3 (1990): 11- 13.

Cox, John, Stephen Ross, and Mark Rubinstein. "Option Pricing: A Simplified Approach." Journal of Financial Economics 7 (1979): 229- 263.

Cox, John, and Mark Rubinstein. Options Markets. Englewood Cliffs, NJ: Prentice-Hall, 1985.

Gemmill, Gordon. Options Pricing. New York: McGraw-Hill, 1993.

Hull, John C. Options, Futures, and Other Derivatives (3rd ed.). New York; Prentice-Hall, 1997.

Jarrow, Robert, and Andrew Rudd. Option Pricing. Homewood, IL: Dow-Jones Irwin, 1983.

Merton, Robert C. "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4 (1973): 141- 183.

Natenburg, Sheldon. Option Volatility & Pricing (2d ed.). Chicago, IL: Probus, 1994.

Satty, Glenn and Joe Troccolo. Options: The Fundamentals. Basel, Switzerland: Swiss Bank Corporation, 1994.

Stoll, Hans." The Relationship Between Put and Call Option Prices." Journal of Finance 24 (1969): 801- 824.

Sullivan, Edward J., and Timothy M. Weithers. "Louis Bachelier: The Father of Modern Option Pricing Theory." Journal of Economic Education 22 (1991): 165- 171.

Sullivan, Edward J. and Timothy M. Weithers. "The History and Development of the Option Pricing Formula." in Research In the History of Economic Thought and Methodology, Warren J. Samuels and Jeff Biddle (eds.), pp. 31- 43. Greenwich, CT: JAI Press, 1994.

Table of Contents

Options Fundamentals (T. Weithers).

A Historical Perspective on Equity Derivatives (J. Hill & B. Dunn).

PRODUCTS TRADED ON ORGANIZED EXCHANGES IN THE UNITED STATES.

U.S. Options Exchanges (J. Junkus).

U.S. Stock Index Futures Markets (T. Petzel).

Index Participation Units (E. Kirzner).

Exchange-Traded Equity Funds-Genesis, Growth, and Outlook (G. Gastineau & C. Weber).

International Equity Derivatives (J. Junkus).

ISSUER DERIVATIVE STRUCTURES.

Warrants (E. Harty).

Japanese Warrants (B. Sucher & J. Knight).

Convertibles (A. Aithal).

Buy-Write Securities (T. McManus).

The Enigma of Tax-Driven Security Design: The Unbundled Stock Units and Other Experiences (R. Sah & N. Vasavada).

SYNTHETIC STRUCTURES.

Applications of OTC Options and Other Structured Products (M. Zurack).

Americus Trust's Primes and Scores: Precursors to Leaps, Percs, Elks, Yeelds and Chips, and Supertrust (E. Harty).

Public Equity-Linked Debt (W. Toy & M. Ryan).

Equity Swaps: Structures, Uses, and Pricing (J. Marshall & R. Yuyuenyongwatana).

Nikkei 225 Put Warrants (M. Ryan & R. Granovsky).

Synthetic Warrants (T. Lindberg).

Supershares (M. Rubinstein).

Structured Equity Derivative Products (J. Whittaker).

A Practitioner's Guide To Structuring Listed Equity Derivative Securities (J. Braddock & B. Krause).

ACCOUNTING, TAX, AND REGULATORY ISSUES.

Accounting Considerations for Equity Derivatives (M. Joseph).

Legal Issues Concerning Equity Derivative Instruments (P. Gottlieb & J. Gatsos).

The U.S. Taxation of Equity Derivative Instruments (E. Kleinbard).

Developments in U.S. Derivatives Regulation (T. Russo & M. Vinciguerra).

SEARCHING FOR MEANINGFUL PATTERNS.

Why Financial Instruments Fail or Succeed (J. Francis).

What Lies Ahead? (G. Gastineau & L. Margolis).

The Theory of Security Design (A. Boot, et al.).

Index.