Options, Risk, and Volatility
/Consider a rather ugly mathematical physicist who goes to the same bar every evening, always takes the second to the last seat, and seems to speak toward the empty seat next to his as if someone were there. The bartender notes this, and on Valentine’s Day when the physicist seems to be especially fervent in his conversation, he asks why he is talking into the air. The physicist scoffs that the bartender doesn’t know anything about quantum mechanics. “There is no such thing as a vacuum. Virtual particles flit in and out of existence, and there is a non-zero probability that a beautiful woman will materialize and, when she does, I want to be here to ask her out.” The bartender is baffled and asks why the physicist doesn’t just ask one of the real women who are in the bar. “You never know. One of them might say yes.” The physicist sneers, “Do you know how unlikely that is?”
Being able to estimate probabilities, especially minuscule ones, is essential when dealing with stock options. I’ll soon describe the language of puts and calls, and we’ll see why the January 2003 calls on WCOM at 15 have as much chance of ending up in the money as Britney Spears has of suddenly materializing before the ugly physicist.
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Options and the Calls of the Wild
Here’s a thought experiment: Two people (or the same person in parallel universes) have roughly similar lives until each undertakes some significant endeavor. The endeavors are equally worthy and equally likely to result in success, but one endeavor ultimately leads to good things for X and his family and friends, and the other leads to bad things for Y and his family and friends. It seems that X and Y should receive roughly comparable evaluations for their decision, but generally they won’t. Unwarranted though it may be, X will be judged kindly and Y harshly. I tell this in part because I’d like to exonerate myself for my investing behavior by claiming status as a faultless Mr. Y, but I don’t qualify.
By late January 2002, WCOM had sunk to about $10 per share, and I was feeling not only dispirited but guilty about losing so much money on it. Losing money in the stock market often induces guilt in those who have lost it, whether they’ve done anything culpable or not. Whatever your views on the randomness of the market, it’s indisputable that chance plays a huge role, so it makes no sense to feel guilty about having called heads when a tails comes up. If this was what I’d done, I could claim to be a Mr. Y: It wouldn’t have been my fault. Alas, as I mentioned, it does make sense to blame yourself for betting recklessly on a particular stock (or on options for it).
There is a term used on Wall Street to describe traders and others who “blow up” (that is, lose a fortune) and as a result become hollow, sepulchral figures. The term is “ghost” and I have developed more empathy for ghosts than I wanted to have. Often they achieve their funereal status by taking unnecessary risks, risks that they could and should have “diversified away.” One perhaps counterintuitive way in which to reduce risk is to buy and sell stock options.
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Many people think of stock options as slot machines, roulette wheels, or dark horse long shots; that is, as pure gambles. Others think of them as absurdly large inducements for people to stay with a company or as rewards for taking a company public. I have no argument with these characterizations, but much of the time an option is more akin to a boring old insurance policy. Just as one buys an insurance policy in case one’s washing machine breaks down, one often buys options in case one’s stock breaks down. They lessen risk, which is the bete noire, bugbear, and bane of investors’ lives and the topic of this chapter.
How options work is best explained with a few numerical examples. (How they’re misused is reserved for the next section.) Assume that you have 1,000 shares of AOL (just to give WCOM a rest), and it is selling at $20 per share. Although you think it’s likely to rise in the long term, you realize there’s a chance that it may fall significantly in the next six months. You could insure against this by buying 1,000 “put” options at an appropriate price. These would give you the right to sell 1,000 shares of AOL for, say, $17.50 for the next six months. If the stock rises or falls less than $2.50, the puts become worthless in six months (just as your washing machine warranty becomes worthless on its expiration if your machine has not broken down by then). Your right to sell shares at $17.50 is not attractive if the price of the stock is more than that. However, if the stock plunges to, say, $10 per share within the six-month period, your right to sell shares at $17.50 is worth at least $7.50 per share. Buying put options is a hedge against a precipitous decline in the price of the underlying stock.
As I was first writing this, only a few paragraphs and a few days after WCOM had fallen to $10, it fell to under $8 per share, and I wished I had bought a boatload of puts on it months before when they were dirt cheap.
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In addition to put options, there are “call” options. Buying them gives you the right to buy a stock at a certain price within a specified period of time. You might be tempted to buy calls when you strongly believe that a stock, say Intel this time (abbreviated INTC), selling at $25 per share, will rise substantially during the next year. Maybe you can’t afford to buy many shares of INTC, but you can afford to buy calls giving you the right to buy shares at, say, $30 during the next year. If the stock falls or rises less than $5 during the next year, the calls become worthless. Your right to buy shares at $30 is not attractive if the price of the stock is less than that. But if the stock rises to, say, $40 per share within the year, each call is worth at least $10. Buying call options is a bet on a substantial rise in the price of the stock. It is also a way to insure that you are not left out when a stock, too expensive to buy outright, begins to take off. (The figures $17.50 and $30 in the AOL and INTC examples above are the “strike” prices of the respective options; this is the price of the stock that determines the point at which the option has intrinsic value or is “in the money.”)
One of the most alluring aspects of buying puts and calls is that your losses are limited to what you have paid for them, but the potential gains are unlimited in the case of calls and very substantial in the case of puts. Because of these huge potential gains, options probably induce a comparably huge amount of fantasy—countless investors thinking something like “the option for INTC with a $30 strike price costs around a dollar, so if the stock goes to $45 in the next year, I’ll make 15 times my investment. And if it goes to $65, I’ll make 35 times my investment.” The attraction for some speculators is not much different from that of a lottery.
Although I’ve often quoted approvingly Voltaire’s quip that lotteries are a tax on stupidity (or at least on innumeracy), yes, I did buy a boatload of now valueless WCOM calls. In
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fact, over the two years of my involvement with the stock, I bought many thousands of January 2003 calls on WCOM at $15.1 thought that whatever problems the company had were temporary and that by 2003 it would right itself and, in the process, me as well. Call me an ugly physicist.
There is, of course, a market in puts and calls, which means that people sell them as well as buy them. Not surprisingly, the payoffs are reversed for sellers of options. If you sell calls for INTC with a strike price of $30 that expires in a year, then you keep your proceeds from the sale of the calls and pay nothing unless the stock moves above $30. If, however, the stock moves to, say, $35, you must supply the buyer of the calls with shares of INTC at $30. Selling calls is thus a bet that the stock will either decline or rise only slightly in a given time period. Likewise, selling puts is a bet that the stock will either rise or decline only slightly.
One common investment strategy is to buy shares of a stock and simultaneously sell calls on them. Say, for example, you buy some shares of INTC stock at $25 per share and sell six-month calls on them with a strike price of $30. If the stock price doesn’t rise to $30, you keep the proceeds from the sale of the calls, but if the stock price does exceed $30, you can sell your own shares to the buyer of the calls, thus limiting the considerable risk in selling calls. This selling of “covered” calls (covered because you own the stock and don’t have to buy it at a high price to satisfy the buyer of the call) is one of many hedges investors can employ to maximize their returns and minimize their risks.
More generally, you can buy and sell the underlying stock and mix and match calls and puts with different expiration dates and strike prices to create a large variety of potential profit and loss outcomes. These combinations go by names like “straddles,” “strangles,” “condors,” and “butterflies,” but whatever strange and contorted animal they’re named for,
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like all insurance policies, they cost money. A surprisingly difficult question in finance has been “How does one place a value on a put or a call?” If you’re insuring your house, some of the determinants of the policy premium are the replacement cost of the house, the length of time the policy is in effect, and the amount of the deductible. The considerations for a stock include these plus others having to do with the rise and fall of stock prices.
Although the practice and theory of insurance have a long history (Lloyd’s of London dates from the late seventeenth century), it wasn’t until 1973 that a way was found to rationally assign costs to options. In that year Fischer Black and Myron Scholes published a formula that, although much refined since, is still the basic valuation tool for options of all sorts. Their work and that of Robert Merton won the Nobel prize for economics in 1997.
Louis Bachelier, whom I mentioned in chapter 4, also devised a formula for options more than one hundred years ago. Bachelier’s formula was developed in connection with his famous 1900 doctoral dissertation in which he was the first to conceive of the stock market as a chance process in which price movements up and down were normally distributed. His work, which utilized the mathematical theory of Brownian motion, was way ahead of its time and hence was largely ignored. His options formula was also prescient, but ultimately misleading. (One reason for its failure is that Bachelier didn’t take account of the effect of compounding on stock returns. Over time this leads to what is called a “lognormal” distribution rather than a normal one.)
The Black-Scholes options formula depends on five parameters: the present price of the stock, the length of time until the option expires, the interest rate, the strike price of the option, and the volatility of the underlying stock. Without getting into the mechanics of the formula, we can see that certain general
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relations among these parameters are commonsensical. For example, a call that expires two years from now has to cost more than one that expires in three months since the later expiration date gives the stock more time to exceed the strike price. Likewise, a call with a strike price a point or two above the present stock price will cost more than one five points above the stock price. And options on a stock whose volatility is high will cost more than options on stocks that barely move from quarter to quarter (just as a short man on a pogo stick is more likely to be able to peek over a nine-foot fence than a tall man who can’t jump). Less intuitive is the fact that the cost of an option also rises with the interest rate, assuming all other parameters remain unchanged.
Although there are any number of books and websites on the Black-Scholes formula, it and its variants are more likely to be used by professional traders than by gamblers, who rely on commonsense considerations and gut feel. Viewing options as pure bets, gamblers are generally as interested in carefully pricing them as casino-goers are in the payoff ratios of slot machines.
The Lure of Illegal Leverage
Because of the leverage possible with the purchase, sale, or mere possession of options, they sometimes attract people who aren’t content to merely play the slots but wish to stick their thumbs onto the spinning disks and directly affect the outcomes. One such group of people are CEOs and other management personnel who stand to reap huge amounts of money if they can somehow contrive (by hook, crook, or, too often, by cooking the books) to raise their companies’ stock price. Even if the rise is only temporary, the suddenly valuable call options can “earn” them tens of millions of dollars. This
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is the luxury version of “pump and dump” that has animated much of the recent corporate malfeasance.
(Such malfeasance might make for an interesting novel. On public television one sometimes sees a fantasia in which diverse historical figures are assembled for an imaginary conversation. Think, for example, of Leonardo da Vinci, Thomas Edison, and Benjamin Franklin discussing innovation. Sometimes a contemporary is added to the mix or simply paired with an illustrious precursor—maybe Karl Popper and David Hume, Stephen Hawking and Isaac Newton, or Henry Kissinger and Machiavelli. Recently I tried to think with whom I might pair a present-day ace CEO, investor, or analyst. There are a number of books about the supposed relevance to contemporary business practices of Plato, Aristotle, and other ancient wise men, but the conversation I’d be most interested in would be one between a current wheeler-dealer and some accomplished hoaxer of the past, maybe Dennis Koslowski and P. T. Barnum, or Kenneth Lay and Harry Houdini, or possibly Bernie Ebbers and Elmer Gantry.)
Option leverage works in the opposite direction as well, the options-fueled version of “short and distort.” One particularly abhorrent example may have occurred in connection with the bombing of the World Trade Center. Just after September 11, 2001, there were reports that A1 Qaeda operatives in Europe had bought millions of dollars worth of puts on various stock indices earlier in the month, reasoning that the imminent attacks would lead to a precipitous drop in the value of these indices and a consequent enormous rise in the value of their puts. They may have succeeded, although banking secrecy laws in Switzerland and elsewhere make that unclear.
Much more commonly, people buy puts on a stock and then try to depress its price in less indiscriminately murderous ways. A stockbroker friend of mine tells me, for example, of his fantasy of writing a mystery novel in which speculators
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buy puts on a company whose senior management is absolutely critical to the success of the company. The imaginary speculators then proceed to embarrass, undermine, and ultimately kill the senior management in order to reap the benefit of the soon-to-be valuable puts. The WorldCom chatroom, home to all sorts of utterly baseless rumors, once entertained a brief discussion about the possibility of WorldCom management having been blackmailed into doing all the ill-considered things they did on pain of having some awful secrets revealed. The presumption was that the blackmailers had bought WCOM puts.
Intricacies abound, but the same basic logic governing stock options is at work in the pricing of derivatives. Sharing only the same name as the notion studied in calculus, derivatives are financial instruments whose value is derived from some underlying asset—the stock of a company, commodities like cotton, pork bellies, and natural gas, or almost anything whose value varies significantly over time. They present the same temptation to directly change, affect, or manipulate conditions, and the opportunities for doing so are more varied and would also make for an intriguing business mystery novel.
The leverage involved in trading options and derivatives brings to mind a classic quote from Archimedes, who maintained that given a fulcrum, a long enough lever, and a place to stand, he could move the earth. The world-changing dreams that created the suggestively named WorldCom, Global Crossing, Quantum Group (George Soros’ companies, no stranger to speculation), and others may have been similar in scope. The metaphorical baggage of levers and options is telling.
One can also look at seemingly non-financial situations and discern something like the buying, selling, and manipulating of options. For example, the practice of defraying the medical bills of AIDS patients in exchange for being made the beneficiary of their insurance policies has disappeared
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with the increased longevity of those with AIDS. However, if the deal were modified so that the parties put a time limit on their agreement, it could be considered a standard option sale. The “option buyer” would pay a sum of money, and the patient/option seller would make the buyer the beneficiary for an agreed-upon period of time. If the patient happens not to expire within that time, the “option” does. Maybe another mystery novel here?
Less ghoulish variants of option buying, selling, and manipulating play an important role in everyday life from education and family planning to politics. Political options, better known as campaign contributions to relatively unknown candidates, usually expire worthless after the candidate loses the race. If he or she is elected, however, the “call option” becomes very valuable, enabling the contributor to literally call on the new officeholder. There is no problem with that, but direct manipulation of conditions that might increase the value of the political option is generally called “dirty tricks.”
For all the excesses options sometimes inspire, they are generally a good thing, a valuable lubricant that enables prudent hedgers and adventurous gamblers to form a mutually advantageous market. It’s only when the option holders do something to directly affect the value of the options that the lure of leverage turns lurid.
Short-Selling, Margin Buying, and Familial Finances
An old Wall Street couplet says, “He who sells what isn’t his’n must buy it back or go to prison.” The lines allude to “short-selling,” the selling of stocks one doesn’t own in the hope that the price will decline and one can buy the shares back at a lower price in the future. The practice is very risky
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because the price might rise precipitously in the interim, but many frown upon short-selling for another reason. They consider it hostile or anti-social to bet that a stock will decline. You can bet that your favorite horse wins by a length, not that some other horse breaks its leg. A simple example, however, suggests that short-selling can be a necessary corrective to the sometimes overly optimistic bias of the market.
Imagine that a group of investors has a variety of attitudes to the stock of company X, ranging from a very bearish 1 through a neutral 5 or 6 to a very bullish 10. In general, who is going to buy the stock? It will generally be those whose evaluations are in the 7 to 10 range. Their average valuation will be, let’s assume, 8 or 9. But if those investors in the 1 to 4 range who are quite dubious of the stock were as likely to short sell X as those in the 7 to 10 range were to buy it, then the average valuation might be a more realistic 5 or 6.
Another positive way to look at short-selling is as a way to double the number of stock tips you receive. Tips about a bad stock become as useful as tips about a good one, assuming that you believe any tips. Short-selling is occasionally referred to as “selling on margin,” and it is closely related to “buying on margin,” the practice of buying stock with money borrowed from your broker.
To illustrate the latter, assume you own 5,000 shares of WCOM and it’s selling at $20 per share (ah, remembrance of riches past). Since your investment in WCOM is worth $100,000, you can borrow up to this amount from your broker and, if you’re very bullish on WCOM and a bit reckless, you can use it to buy an additional 5,000 shares on margin, making the total market value of your WCOM holdings $200,000 ($20 x 10,000 shares). Federal regulations require that the amount you owe your broker be no more than 50 percent of the total market value of your holdings. (Percentages vary with the broker, stock, and type of account.) This is
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no problem if the price of WCOM rises to $25 per share, since the $100,000 you owe your broker will then constitute only 40 percent of the $250,000 ($25 x 10,000) market value of your WCOM shares. But consider what happens if the stock fails to $15 per share. The $100,000 you owe now constitutes 67 percent of the $150,000 ($15 x 10,000) market value of your WCOM shares, and you will receive a “margin call” to deposit immediately enough money ($25,000) into your account to bring you back into compliance with the 50 percent requirement. Further declines in the stock price will result in more margin calls.
I’m embarrassed to reiterate that my devotion to WCOM (others may characterize my relationship to the stock in less kindly terms) led me to buy it on margin and to make the margin calls on it as it continued its long, relentless decline. Receiving a margin call (which often takes the literal form of a telephone call) is, I can attest, unnerving and confronts you with a stark choice. Sell your holdings and get out of the game now or quickly scare up some money to stay in it.
My first margin call on WCOM is illustrative. Although the call was rather small, I was leaning toward selling some of my shares rather than depositing yet more money in my account. Unfortunately (in retrospect), I needed a book quickly and decided to go to the Borders store in Center City, Philadelphia, to look for it. While doing so, I came across the phrase “staying in the game” while browsing and realized that staying in the game was what I still wanted to do. I realized too that Schwab was very close to Borders and that I had a check in my pocket.
My wife was with me, and though she knew of my investment in WCOM, at the time she was not aware of its extent nor of the fact that I’d bought on margin. (Readily granting that this doesn’t say much for the transparency of my financial practices, which would not likely be approved by even
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the most lax Familial Securities Commission, I plead guilty to spousal deception.) When she went upstairs, I ducked out of the store and made the margin call. My illicit affair with WCOM continued. Occasionally exciting, it was for the most part anxiety-inducing and pleasureless, not to mention costly.
I took some comfort from the fact that my margin buying distantly mirrored that of WorldCom’s Bernie Ebbers, who borrowed approximately $400 million to buy WCOM shares. (More recent allegations have put his borrowings at closer to $1 billion, some of it for personal reasons unrelated to WorldCom. Enron’s Ken Lay, by contrast, borrowed only $10 to $20 million.) When he couldn’t make the ballooning margin calls, the board of directors extended him a very low interest loan that was one factor leading to further investor unrest, massive sell-offs, and more trips to Borders for me.
Relatively few individuals short-sell or buy on margin, but the practice is very common among hedge funds—private, lightly regulated investment portfolios managed by people who employ virtually every financial tool known to man. They can short-sell, buy on margin, use various other sorts of leverage, or engage in complicated arbitrage (the near simultaneous buying and selling of the same stock, bond, commodity, or anything else, in order to profit from tiny price discrepancies). They’re called “hedge funds” because many of them try to minimize the risks of wealthy investors. Others fail to hedge their bets at all.
A prime example of the latter is the collapse in 1998 of Long-Term Capital Management, a hedge fund, two of whose founding partners, Robert Merton and Myron Scholes, were the aforementioned Nobel prize winners who, together with Fischer Black, derived the celebrated formula for pricing options. Despite the presence of such seminal thinkers on the board of LTCM, the debacle roiled the world’s financial markets and, had not emergency measures been enacted, might
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have seriously damaged them. (Then again, there is a laissez- faire argument for letting the fund fail.)
I admit I take a certain self-serving pleasure from this story since my own escapades pale by comparison. It’s not clear, however, that the LTCM collapse was the fault of the Nobel laureates and their models. Many believe it was a consequence of a “perfect storm” in the markets, a vanishingly unlikely confluence of chance events. (The claim that Merton and Scholes were not implicated is nevertheless a bit disingenuous, since many invested in LTCM precisely because the fund was touting them and their models.)
The specific problems encountered by LTCM concerned a lack of liquidity in world markets, and this was exacerbated by the disguised dependence of a number of factors that were assumed to be independent. Consider, for illustration’s sake, the likelihood that 3,000 specific people will die in New York on any given day. Provided that there is no connection among them, this is an impossibly minuscule number—a small probability raised to the 3,000th power. If most of the people work in a pair of buildings, however, the independence assumption that allows us to multiply probabilities fails. The 3,000 deaths are still extraordinarily unlikely, but not impossibly minuscule. Of course, the probabilities associated with possible LTCM scenarios were nowhere near as small and, according to some, could and should have been anticipated.
Are Insider Trading and Stock Manipulation So Bad?
It’s natural to take a moralistic stance toward the corporate fraud and excess that have dominated business news the last couple of years. Certainly that attitude has not been completely absent from this book. An elementary probability puzA
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zle and its extensions suggest, however, that some arguments against insider trading and stock manipulation are rather weak. Moral outrage, rather than actual harm to investors, seems to be the primary source of many people’s revulsion toward these practices.
Let me start with the original puzzle. Which of the following two situations would you prefer to be in? In the first one you’re given a fair coin to flip and are told that you will receive $1,000 if it lands heads and lose $1,000 if it lands tails. In the second you’re given a very biased coin to flip and must decide whether to bet on heads or tails. If it lands the way you predict you win $1,000 and, if not, you lose $1,000. Although most people prefer to flip the fair coin, your chances of winning are 1/2 in both situations, since you’re as likely to pick the biased coin’s good side as its bad side.
Consider now a similar pair of situations. In the first one you are told you must pick a ball at random from an urn containing 10 green balls and 10 red balls. If you pick a green one, you win $1,000, whereas if you pick a red one, you lose $1,000. In the second, someone you thoroughly distrust places an indeterminate number of green and red balls in the urn. You must decide whether to bet on green or red and then choose a ball at random. If you choose the color you bet on, you win $1,000 and, if not, you lose $1,000. Again, your chances of winning are 1/2 in both situations.
Finally, consider a third pair of similar situations. In the first one you buy a stock that is being sold in a perfectly efficient market and your earnings are $1,000 if it rises the next day and -$1,000 if it falls. (Assume that in the short run it moves up with probability 1/2 and down with the same probability.) In the second there is insider trading and manipulation and the stock is very likely to rise or fall the next day as a result of these illegal actions. You must decide whether to buy or sell the stock. If you guess correctly, your earnings are
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$1,000 and, if not, -$1,000. Once again your chances of winning are 1/2 in both situations. (They may even be slightly higher in the second situation since you might have knowledge of the insiders’ motivations.)
In each of these pairs, the unfairness of the second situation is only apparent. You have the same chance of winning that you do in the first situation. I do not by any means defend insider trading and stock manipulation, which are wrong for many other reasons, but I do suggest that they are, in a sense, simply two among many unpredictable factors affecting the price of a stock.
I suspect that more than a few cases of insider trading and stock manipulation result in the miscreant guessing wrong about how the market will respond to his illegal actions. This must be depressing for the perpetrators (and funny for everyone else).
Expected Value, Not Value Expected
What can we anticipate? What should we expect? What’s the likely high, low, and average value? Whether the quantity in question is height, weather, or personal income, extremes are more likely to make it into the headlines than are more informative averages. “Who makes the most money,” for example, is generally more attention-grabbing than “what is the average income” (although both terms are always suspect because—surprise—like companies, people lie about how much money they make).
Even more informative than averages, however, are distributions. What, for example, is the distribution of all incomes and how spread out are they about the average? If the average income in a community is $100,000, this might reflect the fact that almost everyone makes somewhere between $80,000
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and $120,000, or it might mean that a big majority earns less than $30,000 and shops at Kmart, whose spokesperson, the (too) maligned Martha Stewart, also lives in town and brings the average up to $100,000. “Expected value” and “standard deviation” are two mathematical notions that help clarify these issues.
An expected value is a special sort of average. Specifically, the expected value of a quantity is the average of its values, but weighted according to their probabilities. If, for example, based on analysts’ recommendations, our own assessment, a mathematical model, or some other source of information, we assume that 1/2 of the time a stock will have a 6 percent rate of return, that 1/3 of the time it will have a -2 percent rate of return, and that the remaining 1/6 of the time it will have a 28 percent rate of return, then, on average, the stock’s rate of return over any given six periods will be 6 percent three times, -2 percent twice, and 28 percent once. The expected value of its return is simply this probabilistically weighted average— (6% + 6% + 6% + (-2%) + (-2%) + 28%)/6, or 7%.
Rather than averaging directly, one generally obtains the expected value of a quantity by multiplying its possible values by their probabilities and then adding up these products. Thus .06 x 1/2 + (-.02) x 1/3 + .28 x 1/6 = .07, or 7%, the expected value of the above stock’s return. Note that the term “mean” and the Greek letter p (mu) are used interchangeably with “expected value,” so 7% is also the mean return, p.
The notion of expected value clarifies a minor investing mystery. An analyst may simultaneously and without contradiction believe that a stock is very likely to do well but that, on average, it’s a loser. Perhaps she estimates that the stock will rise 1 percent in the next month with probability 95 percent and that it will fall 60 percent in the same time period with probability 5 percent. (The probabilities might come, for example, from an appraisal of the likely outcome of
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an impending court decision.) The expected value of its price change is thus (.01 x .95) + (-.60) x .05), which equals -.021 or an expected loss of 2.1%. The lesson is that the expected value, -2.1%, is not the value expected, which is 1%.
The same probabilities and price changes can also be used to illustrate two complementary trading strategies, one that usually results in small gains but sometimes in big losses, and one that usually results in small losses but sometimes in big gains. An investor who’s willing to take a risk to regularly make some “easy money” might sell puts on the above stock, puts that expire in a month and whose strike price is a little under the present price. In effect, he’s betting that the stock won’t decline in the next month. Ninety-five percent of the time he’ll be right, and he’ll keep the put premiums and make a little money. Correspondingly, the buyer of the puts will lose a little money (the put premiums) 95 percent of the time. Assuming the probabilities are accurate, however, when the stock declines, it declines by 60 percent, and so the puts (the right to sell the stock at a little under the original price) become very valuable 5 percent of the time. The buyer of the puts then makes a lot of money and the seller loses a lot.
Investors can play the same game on a larger scale by buying and selling puts on the S&P 500, for example, rather than on any particular stock. The key to playing is coming up with reasonable probabilities for the possible returns, numbers about which people are as likely to differ as they are in their preferences for the above two strategies. Two exemplars of these two types of investor are Victor Niederhoffer, a well- known futures trader and author of The Education of a Speculator, who lost a fortune by selling puts a few years ago, and Nassim Taleb, another trader and the author of Fooled by Randomness, who makes his living by buying them.
For a more pedestrian illustration, consider an insurance company. From past experience, it has good reason to believe
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that each year, on average, one out of every 10,000 of its homeowners’ policies will result in a claim of $400,000, one out of 1,000 policies will result in a claim of $60,000, one out of 50 will result in a claim of $4,000, and the remainder will result in a claim of $0. The insurance company would like to know what its average payout will be per policy written. The answer is the expected value, which in this case is ($400,000 x 1/10,000) + ($60,000 x 1/1,000) + ($4,000 x 1/50) + ($0 x 9,979/10,000) = $40 + $60 + $80 + $0 = $180. The premium the insurance company charges the homeowners will no doubt be at least $181.
Combining the techniques of probability theory with the definition of expected value allows for the calculation of more interesting quantities. The rules for the World Series of baseball, for example, stipulate that the series ends when one team wins four games. The rules further stipulate that team A plays in its home stadium for games 1 and 2 and however many of games 6 and 7 are necessary, whereas team B plays in its home stadium for games 3, 4, and, if necessary, game 5. If the teams are evenly matched, you might be interested in the expected number of games that will be played in each team’s stadium. Skipping the calculation, I’ll simply note that team A can expect to play 2.9375 games and team B 2.875 games in their respective home stadiums.
Almost any situation in which one can calculate (or reasonably estimate) the probabilities of the values of a quantity allows us to determine the expected value of that quantity. An example more tractable than the baseball problem concerns the decision whether to park in a lot or illegally on the street. If you park in a lot, the rate is $10 or $14, depending upon whether you stay for less than an hour, the probability of which you estimate to be 25 percent. You may, however, decide to park illegally on the street and have reason to believe that 20 percent of the time you will receive a simple parking
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ticket for $30, 5 percent of the time you will receive an obstruction of traffic citation for $100, and 75 percent of the time you will get off for free.
The expected value of parking in the lot is ($10 x .25) + ($14 x .75), which equals $13. The expected value of parking on the street is ($100 x .05) + ($30 x .20) + ($0 x .75), which equals $11. For those to whom this is not already Greek, we might say that pL, the mean costs of parking in the lot, and ps, the mean cost of parking on the street, are $13 and $11, respectively.
Even though parking in the street is cheaper on average (assuming money was your only consideration), the variability of what you’ll have to pay there is much greater than it is with the lot. This brings us to the notion of standard deviation and stock risk.
What’s Normal? Not Six Sigma
Risk in general is frightening, and the fear it engenders explains part of the appeal of quantifying it. Naming bogeymen tends to tame them, and chance is one of the most terrifying bogeyman around, at least for adults.
So how might one get at the notion of risk mathematically? Let’s start with “variance,” one of several mathematical terms for variability. Any chance-dependent quantity varies and deviates from its mean or average; it’s sometimes more than the average, sometimes less. The actual temperature, for example, is sometimes warmer than the mean temperature, sometimes cooler. These deviations from the mean constitute risk and are what we want to quantify. They can be positive or negative, just as the actual temperature minus the mean temperature can be positive or negative, and hence they tend to cancel out. If we square them, however, the deviations are all positive,
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and we come to the definition: the variance of a chance- dependent quantity is the expected value of all its squared deviations from the mean. Before I numerically illustrate this, note the etymological/psychological association of risk with “deviation from the mean.” This is a testament, I suspect, to our fear not only of risk but of anything unusual, peculiar, or deviant.
Be that as it may, let’s switch from temperature back to our parking scenario. Recall that the mean cost of parking in the lot is $13, and so ($10 - $13)2 and ($14 - $13)2, which equal $9 and $1, respectively, are the squares of the deviations of the two possible costs from the mean. They don’t occur equally frequently, however. The first occurs with probability 25%, and the second occurs with probability 75%, and so the variance, the expected value of these numbers, is ($9 x .25) + ($1 x .75), or $3. More commonly used in statistical applications in finance and elsewhere is the square root of the variance, which is usually symbolized by the Greek letter or (sigma). Termed the “standard deviation,” it is in this case the square root of $3, or approximately $1.73. The standard deviation is (not exactly, but can be thought of as) the average deviation from the mean, and it is the most common mathematical measure of risk.
Forget the numerical examples if you like, but remember that, for any quantity, the larger the standard deviation, the more spread out its possible values are about the mean; the smaller it is, the more tightly the possible values cluster around the mean. Thus, if you read that in Japan the standard deviation of personal incomes is much less than it is in the United States, you should infer that Japanese incomes vary considerably less than U.S. incomes.
Returning to the street, you may wonder what the variance and standard deviation are of your parking costs there. The mean cost of parking in the street is $11, and the squares of
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the deviations of the three possible costs from the mean are ($100 - $11)2, ($30 - $11)2, and ($0 - $11)2, or $7,921, $361, and $121, respectively. The first occurs with probability 5%, the second with probability 20%, and the third with probability 75%, and so the variance, the expected values of these numbers, is ($7,921 x .05) + ($361 x .20) + ($0 x .75), or $468.25. The square root of this gives us the standard deviation of $21.64, more than twelve times the standard deviation of parking in the lot.
Despite this blizzard of numbers, I reiterate that all we have done is quantify the obvious fact that the possible outcomes of parking on the street are much more varied and unpredictable than those of parking in the lot. Even though the average cost of parking in the street ($11) is less than that of parking in the lot ($13), most would prefer to incur less risk and would therefore park in the lot for prudential reasons, if not moral ones.
This brings us to the market’s use of standard deviation (sigma) to measure a stock’s volatility. Let’s use the same approach to calculate the variance of the returns for our stock that yields a rate of 6% about 1/2 the time, -2% about 1/3 of the time, and 28% the remaining 1/6 of the time. The mean or expected value of its returns is 7%, and so the squares of the deviations from the mean are (.06 - .07)2, (-.02 - .07)2, and (.28 - .07)2 or .0001, .0081, and .0441, respectively. These occur with probabilities 1/2, 1/3, and 1/6, and so the variance, the expected value of the squares of these deviations from the mean, is (.0001 x 1/2) + (.0081 x 1/3) + (.0441 x 1/6), which is .01. The square root of .01 is .10 or 10%, and this is the standard deviation of the returns for this stock.
The Greek lesson again: The expected value of a quantity is its (probabilistically weighted) average and is symbolized by the letter p (mu), and the standard deviation of a quantity is a measure of its variability and is symbolized by the letter
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<t (sigma). If the quantity in question is the rate of return on a stock price, its volatility is generally taken to be the standard deviation.
If there are only two or three possible values a quantity might assume, the standard deviation is not that helpful a notion. It becomes very useful, however, when a quantity can assume many different values and these values, as they often do, have an approximately normal bell-shaped distribution—high in the middle and tapering off on the sides. In this case, the expected value is the high point of the distribution. Moreover, approximately 2/3 of the values (68 percent) lie within one standard deviation of the expected value, and 95 percent of the values lie within two standard deviations of the expected value.
Before we go on, let’s list a few of the quantities that have a normal distribution: age-specific heights and weights, natural gas consumption in a city for any given winter day, water use between 2 A.M. and 3 A.M. in a given city, thicknesses of a particular machined part coming off an assembly line, I.Q.s (whatever it is that they measure), the number of admissions to a large hospital on any given day, distances of darts from a bull’s-eye, leaf sizes, nose sizes, the number of raisins in boxes of breakfast cereal, and possible rates of return for a stock. If we were to graph any of these quantities, we would obtain bell-shaped curves whose values are clustered about the mean.
Take as an example the number of raisins in a large box of cereal. If the expected number of raisins is 142 and the standard deviation is 8, then the high point of the bell-shaped graph would be at 142. About two-thirds of the boxes would contain between 134 and 150 raisins, and 95 percent of the boxes would contain between 126 and 158 raisins.
Or consider the rate of return of a conservative stock. If the possible rates are normally distributed with an expected value of 5.4 percent and a volatility (standard deviation, that is) of
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only 3.2 percent, then about two-thirds of the time, the rate of return will be between 2.2 percent and 8.6 percent, and 95 percent of the time the rate will be between -1 percent and
11.8 percent. You might prefer this stock to a more risky one with the same expected value but a volatility of, say, 20.2 percent. About two-thirds of the time, the rate of return of this more volatile stock will be between -14.8 percent and 25.6 percent, and 95 percent of the time it will be between -35 percent and 45.8 percent.
In all cases, the more standard deviations from the expected value, the more unusual the result. This fact helps account for the many popular books on management and quality control having the words “six sigma” in their titles. The covers of many of these books suggest that by following their precepts, you can attain results that are six standard deviations above the norm, leading, for example, to a minuscule number of product defects. A six-sigma performance is, in fact, so unlikely that the tables in most statistics texts don’t even include values for it. If you look into the books on management, however, you learn that Sigma is usually capitalized and means something other than sigma, the standard deviation of a chance-dependent quantity. A new oxymoron: minor capital offense.
Whether they are defects, nose sizes, raisins, or water use in a city, almost all normally distributed quantities can be thought of as the average or sum of many factors (genetic, physical, social, or financial). This is not an accident: The so- called Central Limit Theorem states that averages and sums of a sufficient number of chance-dependent quantities are always normally distributed.
As we’ll see in chapter 8, however, not everyone believes that stocks’ rates of return are normally distributed.
Being able to estimate probabilities, especially minuscule ones, is essential when dealing with stock options. I’ll soon describe the language of puts and calls, and we’ll see why the January 2003 calls on WCOM at 15 have as much chance of ending up in the money as Britney Spears has of suddenly materializing before the ugly physicist.
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Options and the Calls of the Wild
Here’s a thought experiment: Two people (or the same person in parallel universes) have roughly similar lives until each undertakes some significant endeavor. The endeavors are equally worthy and equally likely to result in success, but one endeavor ultimately leads to good things for X and his family and friends, and the other leads to bad things for Y and his family and friends. It seems that X and Y should receive roughly comparable evaluations for their decision, but generally they won’t. Unwarranted though it may be, X will be judged kindly and Y harshly. I tell this in part because I’d like to exonerate myself for my investing behavior by claiming status as a faultless Mr. Y, but I don’t qualify.
By late January 2002, WCOM had sunk to about $10 per share, and I was feeling not only dispirited but guilty about losing so much money on it. Losing money in the stock market often induces guilt in those who have lost it, whether they’ve done anything culpable or not. Whatever your views on the randomness of the market, it’s indisputable that chance plays a huge role, so it makes no sense to feel guilty about having called heads when a tails comes up. If this was what I’d done, I could claim to be a Mr. Y: It wouldn’t have been my fault. Alas, as I mentioned, it does make sense to blame yourself for betting recklessly on a particular stock (or on options for it).
There is a term used on Wall Street to describe traders and others who “blow up” (that is, lose a fortune) and as a result become hollow, sepulchral figures. The term is “ghost” and I have developed more empathy for ghosts than I wanted to have. Often they achieve their funereal status by taking unnecessary risks, risks that they could and should have “diversified away.” One perhaps counterintuitive way in which to reduce risk is to buy and sell stock options.
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Many people think of stock options as slot machines, roulette wheels, or dark horse long shots; that is, as pure gambles. Others think of them as absurdly large inducements for people to stay with a company or as rewards for taking a company public. I have no argument with these characterizations, but much of the time an option is more akin to a boring old insurance policy. Just as one buys an insurance policy in case one’s washing machine breaks down, one often buys options in case one’s stock breaks down. They lessen risk, which is the bete noire, bugbear, and bane of investors’ lives and the topic of this chapter.
How options work is best explained with a few numerical examples. (How they’re misused is reserved for the next section.) Assume that you have 1,000 shares of AOL (just to give WCOM a rest), and it is selling at $20 per share. Although you think it’s likely to rise in the long term, you realize there’s a chance that it may fall significantly in the next six months. You could insure against this by buying 1,000 “put” options at an appropriate price. These would give you the right to sell 1,000 shares of AOL for, say, $17.50 for the next six months. If the stock rises or falls less than $2.50, the puts become worthless in six months (just as your washing machine warranty becomes worthless on its expiration if your machine has not broken down by then). Your right to sell shares at $17.50 is not attractive if the price of the stock is more than that. However, if the stock plunges to, say, $10 per share within the six-month period, your right to sell shares at $17.50 is worth at least $7.50 per share. Buying put options is a hedge against a precipitous decline in the price of the underlying stock.
As I was first writing this, only a few paragraphs and a few days after WCOM had fallen to $10, it fell to under $8 per share, and I wished I had bought a boatload of puts on it months before when they were dirt cheap.
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In addition to put options, there are “call” options. Buying them gives you the right to buy a stock at a certain price within a specified period of time. You might be tempted to buy calls when you strongly believe that a stock, say Intel this time (abbreviated INTC), selling at $25 per share, will rise substantially during the next year. Maybe you can’t afford to buy many shares of INTC, but you can afford to buy calls giving you the right to buy shares at, say, $30 during the next year. If the stock falls or rises less than $5 during the next year, the calls become worthless. Your right to buy shares at $30 is not attractive if the price of the stock is less than that. But if the stock rises to, say, $40 per share within the year, each call is worth at least $10. Buying call options is a bet on a substantial rise in the price of the stock. It is also a way to insure that you are not left out when a stock, too expensive to buy outright, begins to take off. (The figures $17.50 and $30 in the AOL and INTC examples above are the “strike” prices of the respective options; this is the price of the stock that determines the point at which the option has intrinsic value or is “in the money.”)
One of the most alluring aspects of buying puts and calls is that your losses are limited to what you have paid for them, but the potential gains are unlimited in the case of calls and very substantial in the case of puts. Because of these huge potential gains, options probably induce a comparably huge amount of fantasy—countless investors thinking something like “the option for INTC with a $30 strike price costs around a dollar, so if the stock goes to $45 in the next year, I’ll make 15 times my investment. And if it goes to $65, I’ll make 35 times my investment.” The attraction for some speculators is not much different from that of a lottery.
Although I’ve often quoted approvingly Voltaire’s quip that lotteries are a tax on stupidity (or at least on innumeracy), yes, I did buy a boatload of now valueless WCOM calls. In
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fact, over the two years of my involvement with the stock, I bought many thousands of January 2003 calls on WCOM at $15.1 thought that whatever problems the company had were temporary and that by 2003 it would right itself and, in the process, me as well. Call me an ugly physicist.
There is, of course, a market in puts and calls, which means that people sell them as well as buy them. Not surprisingly, the payoffs are reversed for sellers of options. If you sell calls for INTC with a strike price of $30 that expires in a year, then you keep your proceeds from the sale of the calls and pay nothing unless the stock moves above $30. If, however, the stock moves to, say, $35, you must supply the buyer of the calls with shares of INTC at $30. Selling calls is thus a bet that the stock will either decline or rise only slightly in a given time period. Likewise, selling puts is a bet that the stock will either rise or decline only slightly.
One common investment strategy is to buy shares of a stock and simultaneously sell calls on them. Say, for example, you buy some shares of INTC stock at $25 per share and sell six-month calls on them with a strike price of $30. If the stock price doesn’t rise to $30, you keep the proceeds from the sale of the calls, but if the stock price does exceed $30, you can sell your own shares to the buyer of the calls, thus limiting the considerable risk in selling calls. This selling of “covered” calls (covered because you own the stock and don’t have to buy it at a high price to satisfy the buyer of the call) is one of many hedges investors can employ to maximize their returns and minimize their risks.
More generally, you can buy and sell the underlying stock and mix and match calls and puts with different expiration dates and strike prices to create a large variety of potential profit and loss outcomes. These combinations go by names like “straddles,” “strangles,” “condors,” and “butterflies,” but whatever strange and contorted animal they’re named for,
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like all insurance policies, they cost money. A surprisingly difficult question in finance has been “How does one place a value on a put or a call?” If you’re insuring your house, some of the determinants of the policy premium are the replacement cost of the house, the length of time the policy is in effect, and the amount of the deductible. The considerations for a stock include these plus others having to do with the rise and fall of stock prices.
Although the practice and theory of insurance have a long history (Lloyd’s of London dates from the late seventeenth century), it wasn’t until 1973 that a way was found to rationally assign costs to options. In that year Fischer Black and Myron Scholes published a formula that, although much refined since, is still the basic valuation tool for options of all sorts. Their work and that of Robert Merton won the Nobel prize for economics in 1997.
Louis Bachelier, whom I mentioned in chapter 4, also devised a formula for options more than one hundred years ago. Bachelier’s formula was developed in connection with his famous 1900 doctoral dissertation in which he was the first to conceive of the stock market as a chance process in which price movements up and down were normally distributed. His work, which utilized the mathematical theory of Brownian motion, was way ahead of its time and hence was largely ignored. His options formula was also prescient, but ultimately misleading. (One reason for its failure is that Bachelier didn’t take account of the effect of compounding on stock returns. Over time this leads to what is called a “lognormal” distribution rather than a normal one.)
The Black-Scholes options formula depends on five parameters: the present price of the stock, the length of time until the option expires, the interest rate, the strike price of the option, and the volatility of the underlying stock. Without getting into the mechanics of the formula, we can see that certain general
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relations among these parameters are commonsensical. For example, a call that expires two years from now has to cost more than one that expires in three months since the later expiration date gives the stock more time to exceed the strike price. Likewise, a call with a strike price a point or two above the present stock price will cost more than one five points above the stock price. And options on a stock whose volatility is high will cost more than options on stocks that barely move from quarter to quarter (just as a short man on a pogo stick is more likely to be able to peek over a nine-foot fence than a tall man who can’t jump). Less intuitive is the fact that the cost of an option also rises with the interest rate, assuming all other parameters remain unchanged.
Although there are any number of books and websites on the Black-Scholes formula, it and its variants are more likely to be used by professional traders than by gamblers, who rely on commonsense considerations and gut feel. Viewing options as pure bets, gamblers are generally as interested in carefully pricing them as casino-goers are in the payoff ratios of slot machines.
The Lure of Illegal Leverage
Because of the leverage possible with the purchase, sale, or mere possession of options, they sometimes attract people who aren’t content to merely play the slots but wish to stick their thumbs onto the spinning disks and directly affect the outcomes. One such group of people are CEOs and other management personnel who stand to reap huge amounts of money if they can somehow contrive (by hook, crook, or, too often, by cooking the books) to raise their companies’ stock price. Even if the rise is only temporary, the suddenly valuable call options can “earn” them tens of millions of dollars. This
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is the luxury version of “pump and dump” that has animated much of the recent corporate malfeasance.
(Such malfeasance might make for an interesting novel. On public television one sometimes sees a fantasia in which diverse historical figures are assembled for an imaginary conversation. Think, for example, of Leonardo da Vinci, Thomas Edison, and Benjamin Franklin discussing innovation. Sometimes a contemporary is added to the mix or simply paired with an illustrious precursor—maybe Karl Popper and David Hume, Stephen Hawking and Isaac Newton, or Henry Kissinger and Machiavelli. Recently I tried to think with whom I might pair a present-day ace CEO, investor, or analyst. There are a number of books about the supposed relevance to contemporary business practices of Plato, Aristotle, and other ancient wise men, but the conversation I’d be most interested in would be one between a current wheeler-dealer and some accomplished hoaxer of the past, maybe Dennis Koslowski and P. T. Barnum, or Kenneth Lay and Harry Houdini, or possibly Bernie Ebbers and Elmer Gantry.)
Option leverage works in the opposite direction as well, the options-fueled version of “short and distort.” One particularly abhorrent example may have occurred in connection with the bombing of the World Trade Center. Just after September 11, 2001, there were reports that A1 Qaeda operatives in Europe had bought millions of dollars worth of puts on various stock indices earlier in the month, reasoning that the imminent attacks would lead to a precipitous drop in the value of these indices and a consequent enormous rise in the value of their puts. They may have succeeded, although banking secrecy laws in Switzerland and elsewhere make that unclear.
Much more commonly, people buy puts on a stock and then try to depress its price in less indiscriminately murderous ways. A stockbroker friend of mine tells me, for example, of his fantasy of writing a mystery novel in which speculators
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buy puts on a company whose senior management is absolutely critical to the success of the company. The imaginary speculators then proceed to embarrass, undermine, and ultimately kill the senior management in order to reap the benefit of the soon-to-be valuable puts. The WorldCom chatroom, home to all sorts of utterly baseless rumors, once entertained a brief discussion about the possibility of WorldCom management having been blackmailed into doing all the ill-considered things they did on pain of having some awful secrets revealed. The presumption was that the blackmailers had bought WCOM puts.
Intricacies abound, but the same basic logic governing stock options is at work in the pricing of derivatives. Sharing only the same name as the notion studied in calculus, derivatives are financial instruments whose value is derived from some underlying asset—the stock of a company, commodities like cotton, pork bellies, and natural gas, or almost anything whose value varies significantly over time. They present the same temptation to directly change, affect, or manipulate conditions, and the opportunities for doing so are more varied and would also make for an intriguing business mystery novel.
The leverage involved in trading options and derivatives brings to mind a classic quote from Archimedes, who maintained that given a fulcrum, a long enough lever, and a place to stand, he could move the earth. The world-changing dreams that created the suggestively named WorldCom, Global Crossing, Quantum Group (George Soros’ companies, no stranger to speculation), and others may have been similar in scope. The metaphorical baggage of levers and options is telling.
One can also look at seemingly non-financial situations and discern something like the buying, selling, and manipulating of options. For example, the practice of defraying the medical bills of AIDS patients in exchange for being made the beneficiary of their insurance policies has disappeared
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with the increased longevity of those with AIDS. However, if the deal were modified so that the parties put a time limit on their agreement, it could be considered a standard option sale. The “option buyer” would pay a sum of money, and the patient/option seller would make the buyer the beneficiary for an agreed-upon period of time. If the patient happens not to expire within that time, the “option” does. Maybe another mystery novel here?
Less ghoulish variants of option buying, selling, and manipulating play an important role in everyday life from education and family planning to politics. Political options, better known as campaign contributions to relatively unknown candidates, usually expire worthless after the candidate loses the race. If he or she is elected, however, the “call option” becomes very valuable, enabling the contributor to literally call on the new officeholder. There is no problem with that, but direct manipulation of conditions that might increase the value of the political option is generally called “dirty tricks.”
For all the excesses options sometimes inspire, they are generally a good thing, a valuable lubricant that enables prudent hedgers and adventurous gamblers to form a mutually advantageous market. It’s only when the option holders do something to directly affect the value of the options that the lure of leverage turns lurid.
Short-Selling, Margin Buying, and Familial Finances
An old Wall Street couplet says, “He who sells what isn’t his’n must buy it back or go to prison.” The lines allude to “short-selling,” the selling of stocks one doesn’t own in the hope that the price will decline and one can buy the shares back at a lower price in the future. The practice is very risky
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because the price might rise precipitously in the interim, but many frown upon short-selling for another reason. They consider it hostile or anti-social to bet that a stock will decline. You can bet that your favorite horse wins by a length, not that some other horse breaks its leg. A simple example, however, suggests that short-selling can be a necessary corrective to the sometimes overly optimistic bias of the market.
Imagine that a group of investors has a variety of attitudes to the stock of company X, ranging from a very bearish 1 through a neutral 5 or 6 to a very bullish 10. In general, who is going to buy the stock? It will generally be those whose evaluations are in the 7 to 10 range. Their average valuation will be, let’s assume, 8 or 9. But if those investors in the 1 to 4 range who are quite dubious of the stock were as likely to short sell X as those in the 7 to 10 range were to buy it, then the average valuation might be a more realistic 5 or 6.
Another positive way to look at short-selling is as a way to double the number of stock tips you receive. Tips about a bad stock become as useful as tips about a good one, assuming that you believe any tips. Short-selling is occasionally referred to as “selling on margin,” and it is closely related to “buying on margin,” the practice of buying stock with money borrowed from your broker.
To illustrate the latter, assume you own 5,000 shares of WCOM and it’s selling at $20 per share (ah, remembrance of riches past). Since your investment in WCOM is worth $100,000, you can borrow up to this amount from your broker and, if you’re very bullish on WCOM and a bit reckless, you can use it to buy an additional 5,000 shares on margin, making the total market value of your WCOM holdings $200,000 ($20 x 10,000 shares). Federal regulations require that the amount you owe your broker be no more than 50 percent of the total market value of your holdings. (Percentages vary with the broker, stock, and type of account.) This is
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no problem if the price of WCOM rises to $25 per share, since the $100,000 you owe your broker will then constitute only 40 percent of the $250,000 ($25 x 10,000) market value of your WCOM shares. But consider what happens if the stock fails to $15 per share. The $100,000 you owe now constitutes 67 percent of the $150,000 ($15 x 10,000) market value of your WCOM shares, and you will receive a “margin call” to deposit immediately enough money ($25,000) into your account to bring you back into compliance with the 50 percent requirement. Further declines in the stock price will result in more margin calls.
I’m embarrassed to reiterate that my devotion to WCOM (others may characterize my relationship to the stock in less kindly terms) led me to buy it on margin and to make the margin calls on it as it continued its long, relentless decline. Receiving a margin call (which often takes the literal form of a telephone call) is, I can attest, unnerving and confronts you with a stark choice. Sell your holdings and get out of the game now or quickly scare up some money to stay in it.
My first margin call on WCOM is illustrative. Although the call was rather small, I was leaning toward selling some of my shares rather than depositing yet more money in my account. Unfortunately (in retrospect), I needed a book quickly and decided to go to the Borders store in Center City, Philadelphia, to look for it. While doing so, I came across the phrase “staying in the game” while browsing and realized that staying in the game was what I still wanted to do. I realized too that Schwab was very close to Borders and that I had a check in my pocket.
My wife was with me, and though she knew of my investment in WCOM, at the time she was not aware of its extent nor of the fact that I’d bought on margin. (Readily granting that this doesn’t say much for the transparency of my financial practices, which would not likely be approved by even
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the most lax Familial Securities Commission, I plead guilty to spousal deception.) When she went upstairs, I ducked out of the store and made the margin call. My illicit affair with WCOM continued. Occasionally exciting, it was for the most part anxiety-inducing and pleasureless, not to mention costly.
I took some comfort from the fact that my margin buying distantly mirrored that of WorldCom’s Bernie Ebbers, who borrowed approximately $400 million to buy WCOM shares. (More recent allegations have put his borrowings at closer to $1 billion, some of it for personal reasons unrelated to WorldCom. Enron’s Ken Lay, by contrast, borrowed only $10 to $20 million.) When he couldn’t make the ballooning margin calls, the board of directors extended him a very low interest loan that was one factor leading to further investor unrest, massive sell-offs, and more trips to Borders for me.
Relatively few individuals short-sell or buy on margin, but the practice is very common among hedge funds—private, lightly regulated investment portfolios managed by people who employ virtually every financial tool known to man. They can short-sell, buy on margin, use various other sorts of leverage, or engage in complicated arbitrage (the near simultaneous buying and selling of the same stock, bond, commodity, or anything else, in order to profit from tiny price discrepancies). They’re called “hedge funds” because many of them try to minimize the risks of wealthy investors. Others fail to hedge their bets at all.
A prime example of the latter is the collapse in 1998 of Long-Term Capital Management, a hedge fund, two of whose founding partners, Robert Merton and Myron Scholes, were the aforementioned Nobel prize winners who, together with Fischer Black, derived the celebrated formula for pricing options. Despite the presence of such seminal thinkers on the board of LTCM, the debacle roiled the world’s financial markets and, had not emergency measures been enacted, might
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have seriously damaged them. (Then again, there is a laissez- faire argument for letting the fund fail.)
I admit I take a certain self-serving pleasure from this story since my own escapades pale by comparison. It’s not clear, however, that the LTCM collapse was the fault of the Nobel laureates and their models. Many believe it was a consequence of a “perfect storm” in the markets, a vanishingly unlikely confluence of chance events. (The claim that Merton and Scholes were not implicated is nevertheless a bit disingenuous, since many invested in LTCM precisely because the fund was touting them and their models.)
The specific problems encountered by LTCM concerned a lack of liquidity in world markets, and this was exacerbated by the disguised dependence of a number of factors that were assumed to be independent. Consider, for illustration’s sake, the likelihood that 3,000 specific people will die in New York on any given day. Provided that there is no connection among them, this is an impossibly minuscule number—a small probability raised to the 3,000th power. If most of the people work in a pair of buildings, however, the independence assumption that allows us to multiply probabilities fails. The 3,000 deaths are still extraordinarily unlikely, but not impossibly minuscule. Of course, the probabilities associated with possible LTCM scenarios were nowhere near as small and, according to some, could and should have been anticipated.
Are Insider Trading and Stock Manipulation So Bad?
It’s natural to take a moralistic stance toward the corporate fraud and excess that have dominated business news the last couple of years. Certainly that attitude has not been completely absent from this book. An elementary probability puzA
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zle and its extensions suggest, however, that some arguments against insider trading and stock manipulation are rather weak. Moral outrage, rather than actual harm to investors, seems to be the primary source of many people’s revulsion toward these practices.
Let me start with the original puzzle. Which of the following two situations would you prefer to be in? In the first one you’re given a fair coin to flip and are told that you will receive $1,000 if it lands heads and lose $1,000 if it lands tails. In the second you’re given a very biased coin to flip and must decide whether to bet on heads or tails. If it lands the way you predict you win $1,000 and, if not, you lose $1,000. Although most people prefer to flip the fair coin, your chances of winning are 1/2 in both situations, since you’re as likely to pick the biased coin’s good side as its bad side.
Consider now a similar pair of situations. In the first one you are told you must pick a ball at random from an urn containing 10 green balls and 10 red balls. If you pick a green one, you win $1,000, whereas if you pick a red one, you lose $1,000. In the second, someone you thoroughly distrust places an indeterminate number of green and red balls in the urn. You must decide whether to bet on green or red and then choose a ball at random. If you choose the color you bet on, you win $1,000 and, if not, you lose $1,000. Again, your chances of winning are 1/2 in both situations.
Finally, consider a third pair of similar situations. In the first one you buy a stock that is being sold in a perfectly efficient market and your earnings are $1,000 if it rises the next day and -$1,000 if it falls. (Assume that in the short run it moves up with probability 1/2 and down with the same probability.) In the second there is insider trading and manipulation and the stock is very likely to rise or fall the next day as a result of these illegal actions. You must decide whether to buy or sell the stock. If you guess correctly, your earnings are
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$1,000 and, if not, -$1,000. Once again your chances of winning are 1/2 in both situations. (They may even be slightly higher in the second situation since you might have knowledge of the insiders’ motivations.)
In each of these pairs, the unfairness of the second situation is only apparent. You have the same chance of winning that you do in the first situation. I do not by any means defend insider trading and stock manipulation, which are wrong for many other reasons, but I do suggest that they are, in a sense, simply two among many unpredictable factors affecting the price of a stock.
I suspect that more than a few cases of insider trading and stock manipulation result in the miscreant guessing wrong about how the market will respond to his illegal actions. This must be depressing for the perpetrators (and funny for everyone else).
Expected Value, Not Value Expected
What can we anticipate? What should we expect? What’s the likely high, low, and average value? Whether the quantity in question is height, weather, or personal income, extremes are more likely to make it into the headlines than are more informative averages. “Who makes the most money,” for example, is generally more attention-grabbing than “what is the average income” (although both terms are always suspect because—surprise—like companies, people lie about how much money they make).
Even more informative than averages, however, are distributions. What, for example, is the distribution of all incomes and how spread out are they about the average? If the average income in a community is $100,000, this might reflect the fact that almost everyone makes somewhere between $80,000
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and $120,000, or it might mean that a big majority earns less than $30,000 and shops at Kmart, whose spokesperson, the (too) maligned Martha Stewart, also lives in town and brings the average up to $100,000. “Expected value” and “standard deviation” are two mathematical notions that help clarify these issues.
An expected value is a special sort of average. Specifically, the expected value of a quantity is the average of its values, but weighted according to their probabilities. If, for example, based on analysts’ recommendations, our own assessment, a mathematical model, or some other source of information, we assume that 1/2 of the time a stock will have a 6 percent rate of return, that 1/3 of the time it will have a -2 percent rate of return, and that the remaining 1/6 of the time it will have a 28 percent rate of return, then, on average, the stock’s rate of return over any given six periods will be 6 percent three times, -2 percent twice, and 28 percent once. The expected value of its return is simply this probabilistically weighted average— (6% + 6% + 6% + (-2%) + (-2%) + 28%)/6, or 7%.
Rather than averaging directly, one generally obtains the expected value of a quantity by multiplying its possible values by their probabilities and then adding up these products. Thus .06 x 1/2 + (-.02) x 1/3 + .28 x 1/6 = .07, or 7%, the expected value of the above stock’s return. Note that the term “mean” and the Greek letter p (mu) are used interchangeably with “expected value,” so 7% is also the mean return, p.
The notion of expected value clarifies a minor investing mystery. An analyst may simultaneously and without contradiction believe that a stock is very likely to do well but that, on average, it’s a loser. Perhaps she estimates that the stock will rise 1 percent in the next month with probability 95 percent and that it will fall 60 percent in the same time period with probability 5 percent. (The probabilities might come, for example, from an appraisal of the likely outcome of
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an impending court decision.) The expected value of its price change is thus (.01 x .95) + (-.60) x .05), which equals -.021 or an expected loss of 2.1%. The lesson is that the expected value, -2.1%, is not the value expected, which is 1%.
The same probabilities and price changes can also be used to illustrate two complementary trading strategies, one that usually results in small gains but sometimes in big losses, and one that usually results in small losses but sometimes in big gains. An investor who’s willing to take a risk to regularly make some “easy money” might sell puts on the above stock, puts that expire in a month and whose strike price is a little under the present price. In effect, he’s betting that the stock won’t decline in the next month. Ninety-five percent of the time he’ll be right, and he’ll keep the put premiums and make a little money. Correspondingly, the buyer of the puts will lose a little money (the put premiums) 95 percent of the time. Assuming the probabilities are accurate, however, when the stock declines, it declines by 60 percent, and so the puts (the right to sell the stock at a little under the original price) become very valuable 5 percent of the time. The buyer of the puts then makes a lot of money and the seller loses a lot.
Investors can play the same game on a larger scale by buying and selling puts on the S&P 500, for example, rather than on any particular stock. The key to playing is coming up with reasonable probabilities for the possible returns, numbers about which people are as likely to differ as they are in their preferences for the above two strategies. Two exemplars of these two types of investor are Victor Niederhoffer, a well- known futures trader and author of The Education of a Speculator, who lost a fortune by selling puts a few years ago, and Nassim Taleb, another trader and the author of Fooled by Randomness, who makes his living by buying them.
For a more pedestrian illustration, consider an insurance company. From past experience, it has good reason to believe
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that each year, on average, one out of every 10,000 of its homeowners’ policies will result in a claim of $400,000, one out of 1,000 policies will result in a claim of $60,000, one out of 50 will result in a claim of $4,000, and the remainder will result in a claim of $0. The insurance company would like to know what its average payout will be per policy written. The answer is the expected value, which in this case is ($400,000 x 1/10,000) + ($60,000 x 1/1,000) + ($4,000 x 1/50) + ($0 x 9,979/10,000) = $40 + $60 + $80 + $0 = $180. The premium the insurance company charges the homeowners will no doubt be at least $181.
Combining the techniques of probability theory with the definition of expected value allows for the calculation of more interesting quantities. The rules for the World Series of baseball, for example, stipulate that the series ends when one team wins four games. The rules further stipulate that team A plays in its home stadium for games 1 and 2 and however many of games 6 and 7 are necessary, whereas team B plays in its home stadium for games 3, 4, and, if necessary, game 5. If the teams are evenly matched, you might be interested in the expected number of games that will be played in each team’s stadium. Skipping the calculation, I’ll simply note that team A can expect to play 2.9375 games and team B 2.875 games in their respective home stadiums.
Almost any situation in which one can calculate (or reasonably estimate) the probabilities of the values of a quantity allows us to determine the expected value of that quantity. An example more tractable than the baseball problem concerns the decision whether to park in a lot or illegally on the street. If you park in a lot, the rate is $10 or $14, depending upon whether you stay for less than an hour, the probability of which you estimate to be 25 percent. You may, however, decide to park illegally on the street and have reason to believe that 20 percent of the time you will receive a simple parking
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ticket for $30, 5 percent of the time you will receive an obstruction of traffic citation for $100, and 75 percent of the time you will get off for free.
The expected value of parking in the lot is ($10 x .25) + ($14 x .75), which equals $13. The expected value of parking on the street is ($100 x .05) + ($30 x .20) + ($0 x .75), which equals $11. For those to whom this is not already Greek, we might say that pL, the mean costs of parking in the lot, and ps, the mean cost of parking on the street, are $13 and $11, respectively.
Even though parking in the street is cheaper on average (assuming money was your only consideration), the variability of what you’ll have to pay there is much greater than it is with the lot. This brings us to the notion of standard deviation and stock risk.
What’s Normal? Not Six Sigma
Risk in general is frightening, and the fear it engenders explains part of the appeal of quantifying it. Naming bogeymen tends to tame them, and chance is one of the most terrifying bogeyman around, at least for adults.
So how might one get at the notion of risk mathematically? Let’s start with “variance,” one of several mathematical terms for variability. Any chance-dependent quantity varies and deviates from its mean or average; it’s sometimes more than the average, sometimes less. The actual temperature, for example, is sometimes warmer than the mean temperature, sometimes cooler. These deviations from the mean constitute risk and are what we want to quantify. They can be positive or negative, just as the actual temperature minus the mean temperature can be positive or negative, and hence they tend to cancel out. If we square them, however, the deviations are all positive,
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and we come to the definition: the variance of a chance- dependent quantity is the expected value of all its squared deviations from the mean. Before I numerically illustrate this, note the etymological/psychological association of risk with “deviation from the mean.” This is a testament, I suspect, to our fear not only of risk but of anything unusual, peculiar, or deviant.
Be that as it may, let’s switch from temperature back to our parking scenario. Recall that the mean cost of parking in the lot is $13, and so ($10 - $13)2 and ($14 - $13)2, which equal $9 and $1, respectively, are the squares of the deviations of the two possible costs from the mean. They don’t occur equally frequently, however. The first occurs with probability 25%, and the second occurs with probability 75%, and so the variance, the expected value of these numbers, is ($9 x .25) + ($1 x .75), or $3. More commonly used in statistical applications in finance and elsewhere is the square root of the variance, which is usually symbolized by the Greek letter or (sigma). Termed the “standard deviation,” it is in this case the square root of $3, or approximately $1.73. The standard deviation is (not exactly, but can be thought of as) the average deviation from the mean, and it is the most common mathematical measure of risk.
Forget the numerical examples if you like, but remember that, for any quantity, the larger the standard deviation, the more spread out its possible values are about the mean; the smaller it is, the more tightly the possible values cluster around the mean. Thus, if you read that in Japan the standard deviation of personal incomes is much less than it is in the United States, you should infer that Japanese incomes vary considerably less than U.S. incomes.
Returning to the street, you may wonder what the variance and standard deviation are of your parking costs there. The mean cost of parking in the street is $11, and the squares of
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the deviations of the three possible costs from the mean are ($100 - $11)2, ($30 - $11)2, and ($0 - $11)2, or $7,921, $361, and $121, respectively. The first occurs with probability 5%, the second with probability 20%, and the third with probability 75%, and so the variance, the expected values of these numbers, is ($7,921 x .05) + ($361 x .20) + ($0 x .75), or $468.25. The square root of this gives us the standard deviation of $21.64, more than twelve times the standard deviation of parking in the lot.
Despite this blizzard of numbers, I reiterate that all we have done is quantify the obvious fact that the possible outcomes of parking on the street are much more varied and unpredictable than those of parking in the lot. Even though the average cost of parking in the street ($11) is less than that of parking in the lot ($13), most would prefer to incur less risk and would therefore park in the lot for prudential reasons, if not moral ones.
This brings us to the market’s use of standard deviation (sigma) to measure a stock’s volatility. Let’s use the same approach to calculate the variance of the returns for our stock that yields a rate of 6% about 1/2 the time, -2% about 1/3 of the time, and 28% the remaining 1/6 of the time. The mean or expected value of its returns is 7%, and so the squares of the deviations from the mean are (.06 - .07)2, (-.02 - .07)2, and (.28 - .07)2 or .0001, .0081, and .0441, respectively. These occur with probabilities 1/2, 1/3, and 1/6, and so the variance, the expected value of the squares of these deviations from the mean, is (.0001 x 1/2) + (.0081 x 1/3) + (.0441 x 1/6), which is .01. The square root of .01 is .10 or 10%, and this is the standard deviation of the returns for this stock.
The Greek lesson again: The expected value of a quantity is its (probabilistically weighted) average and is symbolized by the letter p (mu), and the standard deviation of a quantity is a measure of its variability and is symbolized by the letter
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<t (sigma). If the quantity in question is the rate of return on a stock price, its volatility is generally taken to be the standard deviation.
If there are only two or three possible values a quantity might assume, the standard deviation is not that helpful a notion. It becomes very useful, however, when a quantity can assume many different values and these values, as they often do, have an approximately normal bell-shaped distribution—high in the middle and tapering off on the sides. In this case, the expected value is the high point of the distribution. Moreover, approximately 2/3 of the values (68 percent) lie within one standard deviation of the expected value, and 95 percent of the values lie within two standard deviations of the expected value.
Before we go on, let’s list a few of the quantities that have a normal distribution: age-specific heights and weights, natural gas consumption in a city for any given winter day, water use between 2 A.M. and 3 A.M. in a given city, thicknesses of a particular machined part coming off an assembly line, I.Q.s (whatever it is that they measure), the number of admissions to a large hospital on any given day, distances of darts from a bull’s-eye, leaf sizes, nose sizes, the number of raisins in boxes of breakfast cereal, and possible rates of return for a stock. If we were to graph any of these quantities, we would obtain bell-shaped curves whose values are clustered about the mean.
Take as an example the number of raisins in a large box of cereal. If the expected number of raisins is 142 and the standard deviation is 8, then the high point of the bell-shaped graph would be at 142. About two-thirds of the boxes would contain between 134 and 150 raisins, and 95 percent of the boxes would contain between 126 and 158 raisins.
Or consider the rate of return of a conservative stock. If the possible rates are normally distributed with an expected value of 5.4 percent and a volatility (standard deviation, that is) of
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only 3.2 percent, then about two-thirds of the time, the rate of return will be between 2.2 percent and 8.6 percent, and 95 percent of the time the rate will be between -1 percent and
11.8 percent. You might prefer this stock to a more risky one with the same expected value but a volatility of, say, 20.2 percent. About two-thirds of the time, the rate of return of this more volatile stock will be between -14.8 percent and 25.6 percent, and 95 percent of the time it will be between -35 percent and 45.8 percent.
In all cases, the more standard deviations from the expected value, the more unusual the result. This fact helps account for the many popular books on management and quality control having the words “six sigma” in their titles. The covers of many of these books suggest that by following their precepts, you can attain results that are six standard deviations above the norm, leading, for example, to a minuscule number of product defects. A six-sigma performance is, in fact, so unlikely that the tables in most statistics texts don’t even include values for it. If you look into the books on management, however, you learn that Sigma is usually capitalized and means something other than sigma, the standard deviation of a chance-dependent quantity. A new oxymoron: minor capital offense.
Whether they are defects, nose sizes, raisins, or water use in a city, almost all normally distributed quantities can be thought of as the average or sum of many factors (genetic, physical, social, or financial). This is not an accident: The so- called Central Limit Theorem states that averages and sums of a sufficient number of chance-dependent quantities are always normally distributed.
As we’ll see in chapter 8, however, not everyone believes that stocks’ rates of return are normally distributed.
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