Connectedness and Chaotic Price Movements
Tear the end of my involvement with WorldCom, when I IV was particularly concerned about what the new day would bring, I would sometimes wake up very early, grab a Diet Coke, and check how the stock was faring on the German or English exchanges. As the computer was booting up, I • grew more and more apprehensive. The European response to bad overnight news sometimes prefigured Wall Street’s response, and I dreaded seeing a steeply downward-sloping graph pop up on my screen. More often the European exchanges treaded water on WCOM until trading began in New York. Occasionally I’d be encouraged when the stock was up there, but I soon learned that the small volume sold on overseas exchanges didn’t always mean much.
Whether haunted by a bad investment or not, we’re all connected. No investor is an island (or even a peninsula). Stated mathematically, this means that statistical independence often fails; your actions affect mine. Most accounts of the stock market acknowledge in a general way that we learn from and respond to one another, but a better understanding of the market requires that one’s models reflect the complexity of investors’ interaction. In a sense, the market is the interaction. Stocks R Us. Before discussing some of the consequences of
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this complexity, let me consider three such sources for it: one micro, one macro, and the third mucro (yup, it’s a word).
The micro example involves insider trading, which has always struck me as an odd sort of crime. Few people who aren’t psychopaths daydream about murder or burglary, but many investors, I suspect, fantasize about coming upon inside information and making a bundle from it. The thought of finding myself on a plane next to Bernie Ebbers and Jack Grubman (assuming they flew economy class on commercial airliners) and overhearing their conversation about an impending merger or IPO offering, for example, did cross my mind a few times. Insider trading seems the limit or culmination of what investors and traders do naturally: getting all the information possible and acting on it before others see and understand what they see and understand.
Insider Trading and Subterranean Information Processing
The kind of insider trading I want to consider is relevant to seemingly unexplained price movements. It’s also related to good poker playing, which may explain why the training program of at least one very successful hedge fund has a substantial unit on the game. The strategies associated with poker include learning not only the relevant probabilities but also the bluffing that is a necessary part of the game. Options traders often deal with relatively few other traders, many of whom they recognize, and this gives rise to the opportunity for feints, misdirection, and the exploitation of idiosyncrasies.
The example derives from the notion of common knowledge introduced in chapter 1. Recall that a bit of information is common knowledge among a group of people if they all know it, know that the others know it, know that the others
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know that they know it, and so on. Robert Aumann, who first defined the notion, proved a theorem that can be roughly paraphrased as follows: Two individuals cannot forever agree to disagree. As their beliefs, formed in rational response to different bits of private information, gradually become common knowledge, the beliefs change and eventually coincide.
When private information becomes common knowledge, it induces decisions and actions. As anyone who has overheard teenagers’ gossip with its web of suppositions can attest, this transition to common knowledge sometimes relies on convoluted inferences about others’ beliefs. Sergiu Hart, an economist at Hebrew University and one of a number of people who have built on Aumann’s result, demonstrates this with an example relevant to the stock market. Superficially complicated, it nevertheless requires no particular background besides an ability to decode gossip, hearsay, and rumor and decide what others really think.
Hart asks us to consider a company that must make a decision. In keeping with the WorldCom leitmotif, let’s suppose it to be a small telecommunications company that must decide whether to develop a new handheld device or a cell phone with a novel feature. Assume that the company is equally likely to decide on one or the other of these products, and assume further that whatever decision it makes, the product chosen has a 50 percent chance of being successful, say being bought in huge numbers by another company. Thus there are four equally likely outcomes: Handheld+, Handheld-, Phone+, Phone- (where Handheld+ means the handheld device was chosen for development and it was a success, Handheld- means the handheld was chosen but it turned out to be a failure, and similarly for Phone+ and Phone-).
Let’s say there are two influential investors, Alice and Bob. They both decide that at the current stock price, if the chances of success of this product development are better
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than 50 percent, they should (continue to) buy, and if they’re 50 percent or less they should (continue to) sell.
Furthermore, they are each privy to a different piece of information about the company. Because of her inside contacts, Alice knows which product decision was made, Handheld or Phone, but not whether it was successful or not.
Bob, because of his position with another company, stands to get the “rejects” from a failed phone project, so he knows whether or not the cell phone was chosen for development and failed. That is, Bob knows whether Phone- or not.
Let’s assume that the handheld device was chosen for development. So the true situation is either Handheld+ or Handheld-. Alice therefore knows Handheld, while Bob knows that the decision is not Phone- (else he would have received the rejects).
After the first period (week, day, or hour), Alice sells since Handheld+ and Handheld- are equally likely, and one sells if the probability of success is 50 percent or less. Bob buys since he estimates that the probability for success is 2/3. With Phone- ruled out, the remaining possibilities are Handheld+, Handheld-, and Phone+, and two out of three of them are successes.
After the second period, it is common knowledge that the true situation is not Phone- since otherwise Bob would have sold in the first period. This is not news to Alice, who continues to sell. Bob continues to buy.
After the third period, it is common knowledge that it is not Phone (neither Phone+ nor Phone-) since otherwise Alice would have bought in the second period. Thus it’s either Handheld+ or Handheld-. Both Bob and Alice take the probability of success to be 50 percent, thus both sell, and there is a mini-crash of the stock price. (Selling by both influential investors triggers a general sell-off.)
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Note that at the beginning both Alice and Bob know that the true situation is not Phone-, but this knowledge is mutual, not common. Alice knows that Bob knows it is not Phone-, but Bob does not know that Alice knows this. From his position the true situation might be Phone+, in which case Alice would know Phone but not whether the situation is Phone+ or Phone-.
The example can be varied in a number of ways: there needn’t be merely three periods before a crash, but an arbitrary number; there may be a bubble (sellers suddenly switching to become buyers) instead of a crash; there may be an arbitrarily large number of investors or investor groups; there may be an issue other than buying or selling under deliberation, perhaps a decision whether to employ one stock-picking approach rather than another.
In all these cases the stock’s price can move in response to no external news. Nevertheless, the subterranean information processing leading to common knowledge among the investors eventually leads to precipitous and unexpected movement in the stock’s price. Analysts will express surprise at the crash (or bubble) because “nothing happened.”
The example is also relevant to what I suspect is a relatively common kind of insider trading, in which “partial insiders” are privy to bits of insider information but not to the whole story.
Trading Strategies, Whim, and Ant Behavior
A more macro-level interaction among investors occurs between technical traders and value traders. Also contributing over time to booms and busts, this interaction comes through clearly in computer models of the following commonsense dynamic.
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Let’s suppose that value traders perceive individual stocks or the market as a whole to be strongly undervalued. They start buying and, by doing so, raise prices. As prices increase, a trend develops and technical traders, as is their wont, follow it, increasing prices even further. Soon enough, the market is seen as overvalued by value traders, who begin to sell and thereby slow and then reverse the trend. The trendfollowing technical traders eventually follow suit, and the cycle begins over again. There are, of course, other sources of variation (one being the number of people who are technical traders and value traders at any given time), and the oscillations are irregular.
The bottom line of much of this modeling is that contrarian value traders have a stabilizing effect on the market, whereas technical traders increase volatility. So does computergenerated program trading, which tends to produce buying or selling in lockstep. There are other sorts of interaction among different classes of investors leading to cycles of varying duration, all of which have differential impacts on the others on which they are superimposed.
In addition to these more or less rational interactions among investors I must also note influences inspired by nothing more than whim, where behavior turns on a mucro. I recall many times, for example, reluctantly beginning work on a project when a niggling detail about some utterly irrelevant matter came to mind. It may have concerned the etymology of a word, or the colleague whose paper bag ripped open at a departmental meeting revealing an embarrassing magazine inside, or why caller ID misidentified a friend’s telephone number. These in turn brought to mind the next in a train of associations and musings, which ultimately led me to an entirely different project. My impulsively deciding, while browsing in Borders, to make my first margin call on WCOM is another instance.
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When this capriousness extends to influential analysts, the effect is more pronounced. In November 2002 the New York Times reported on such a case involving Jack Grubman, telecommunications analyst and anxious father. In an email to a friend Grubman allegedly stated that his boss, Sanford Weill, the chairman of Citigroup, helped get Grubman’s children into an exclusive nursery school after he raised his rating of AT&T in 1999. Gretchen Morgenson, the article’s author, further reported that Weill had his own personal reasons for wanting this upgrade. Whether these particular charges are true or not is immaterial. It’s very hard to believe, however, that this sort of influence is rare.
Such episodes strongly suggest to me that there will never be a precise science of finance or economics. Buying and selling must surely partake of a similar iffiness, at least sometimes. Butterfly Economics, by the British economic theorist Paul Ormerod, faults these disciplines for not sufficiently taking into account the commonsense fact that people, whether knowledgeable or not, influence each other.
People do not, as chapter 2 demonstrated, have a set of fixed preferences on which they coolly and rationally base their economic decisions. The assumption that investors are sensitive only to price and a few ratios simplifies the mathematical models, but it is not always true to our experience of fads, fashions, and people’s everyday monkey-see, monkey-do behavior.
Ormerod tells of an experiment involving not monkeys but ants that provides a useful metaphor. Two identical piles of food are set up at equal distances from a large nest of ants. Each pile is automatically replenished and the ants have no reason to prefer one to the other. Entomologists tell us that once an ant has found food, it usually returns to the same source. Upon returning to the nest, however, it physically stimulates other ants, who might be frequenting the other pile, to follow it to the first pile.
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So where do the ants go? It might be speculated that either they would split into two roughly even groups or perhaps a large majority would arbitrarily settle on one or the other pile. Their actual behavior is counterintuitive. The number of ants going to each pile fluctuates wildly and doesn’t ever settle down. A graph of these fluctuations looks suspiciously like a graph of the stock market.
And in a way, the ants are like stock traders (or people deciding whether or not to make a margin call). Upon leaving the nest, each ant must make a decision: Go to the pile visited last time, be influenced by another to switch piles, or switch piles of its own volition. This slight openness to the influence of other ants is enough to insure the complicated and volatile fluctuations in the number of ants visiting the two sites.
An astonishingly simple formal model of such influence is provided by Stephen Wolfram in his book A New Kind of Science. Imagine a colossally high brick wall wherein each brick rests on parts of two bricks below it and, except for the top row, has parts of two bricks above it. Imagine further that the top row has some red bricks and some green ones. The coloring of the bricks in the top row determines the coloring of the bricks in the second row as follows. Pick a brick in the second row and check the colors of the two bricks above it in the first row. If exactly one of these bricks is green, then the brick in the second row is colored green. If both or neither are green, then the brick is colored red. Do this for every brick in the second row.
The coloring of the bricks in the second row determines the coloring of the bricks in the third row in the same way, and in general, the coloring of the bricks in any row determines the coloring of the bricks in the row below it in the same way. That’s it.
Now if we interpret a row of bricks as a collection of investors at any given instant, green ones for buyers and red ones
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for sellers, then the change from moment to moment of investor sentiment is reflected in the changing color composition of the succeeding rows of bricks. If we let P be the difference between the number of green bricks and the number of red bricks, then P is a rough analogue of a stock’s price. Graphing it, we see that it oscillates up and down in a way that looks random.
The model can be made more realistic, but it is significant that even this bare-bones version, like the ant behavior, evinces a kind of internally generated random noise. This suggests that part of the oscillation of stock prices is also internally generated and is not a response to anything besides investors’ reactions to each other. The theme of Wolfram’s book, borne out here, is that complex behavior can result from very simple rules of interaction.
Chaos and Unpredictability
What is the relative importance of private information, investor trading strategies, and pure whim in predicting the market? What is the relative importance of conventional economic news (interest rates, budget deficits, accounting scandals, and trade balances), popular culture fads (in sports, movies, fashions), and germane political and military events (terrorism, elections, war) too disparate even to categorize? If we were to carefully define the problem, predicting the market with any precision is probably what mathematicians call a universal problem, meaning that a complete solution to it would lead immediately to solutions for a large class of other problems. It is, in other words, as hard a problem in social prediction as there is.
Certainly, too little notice is taken of the complicated connections among these variables, even the more clearly defined
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economic ones. Interest rates, for example, have an impact on unemployment rates, which in turn influence revenues; budget deficits affect trade deficits, which sway interest rates and exchange rates; corporate fraud influences consumer confidence, which may depress the stock market and alter other indices; natural business cycles of various periods are superimposed on one another; an increase in some quantity or index positively (or negatively) feeds back on another, reinforcing or weakening it and being reinforced or weakened in turn.
Few of these associations are accurately described by a straight-line graph and so they bring to a mathematician’s mind the subject of nonlinear dynamics, more popularly known as chaos theory. The subject doesn’t deal with anarchist treatises or surrealist manifestoes but with the behavior of so-called nonlinear systems. For our purposes these may be thought of as any collection of parts whose interactions and connections are described by nonlinear rules or equations. That is to say, the equations’ variables may be multiplied together, raised to powers, and so on. As a consequence the system’s parts are not necessarily linked in a proportional manner as they are, for example, in a bathroom scale or a thermometer; doubling the magnitude of one part will not double that of another—nor will outputs be proportional to inputs. Not surprisingly, trying to predict the precise long-term behavior of such systems is often futile.
Let me, in place of a technical definition of such nonlinear systems, describe instead a particular physical instance of one. Picture before you a billiards table. Imagine that approximately twenty-five round obstacles are securely fastened to its surface in some haphazard arrangement. You hire the best pool player you can find and ask him to place the ball at a particular spot on the table and take a shot toward one of the round obstacles. After he’s done so, his challenge is to make exactly the same shot from the same spot with another ball.
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Even if his angle on this second shot is off by the merest fraction of a degree, the trajectories of these two balls will very soon diverge considerably. An infinitesimal difference in the angle of impact will be magnified by successive hits of the obstacles. Soon one of the balls will hit an obstacle that the other misses entirely, at which point all similarity between the two trajectories ends.
The sensitivity of the billiard balls’ paths to minuscule variations in their initial angles is characteristic of nonlinear systems. The divergence of the billiard balls is not unlike the disproportionate effect of seemingly inconsequential events, the missed planes, serendipitous meetings, and odd mistakes and links that shape and reshape our lives.
This sensitive dependence of nonlinear systems on even tiny differences in initial conditions is, I repeat, relevant to various aspects of the stock market in general, in particular its sometimes wildly disproportionate responses to seemingly small stimuli such as companies’ falling a penny short of earnings estimates. Sometimes, of course, the differences are more substantial. Witness the notoriously large discrepancies between government economic figures on the size of budget surpluses and corporate accounting statements of earnings and the “real” numbers.
Aspects of investor behavior too can no doubt be better modeled by a nonlinear system than a linear one. This is so despite the fact that linear systems and models are much more robust, with small differences in initial conditions leading only to small differences in final outcomes. They’re also easier to predict mathematically, and this is why they’re so often employed whether their application is appropriate or not. The chestnut about the economist looking for his lost car keys under the street lamp comes to mind. “You probably lost them near the car,” his companion remonstrates, to which the economist responds, “I know, but the light is better over here.”
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The “butterfly effect” is the term often used for the sensitive dependence of nonlinear systems, a characteristic that has been noted in phenomena ranging from fluid flow and heart fibrillations to epilepsy and price fluctuations. The name comes from the idea that a butterfly flapping its wings someplace in South America might be sufficient to change future weather systems, helping to bring about, say, a tornado in Oklahoma that would otherwise not have occurred. It also explains why long-range precise prediction of nonlinear systems isn’t generally possible. This non-predictability is the result not of randomness but of complexity too great to fathom.
Yet another reason to suspect that parts of the market may be better modeled by nonlinear systems is that such systems’ “trajectories” often follow a fractal course. The trajectories of these systems, of which the stock price movements may be considered a proxy, turn out to be aperiodic and unpredictable and, when examined closely, evince even more intricacy. Still closer inspection of the system’s trajectories reveals yet smaller vortices and complications of the same general kind.
In general, fractals are curves, surfaces, or higher dimensional objects that contain more, but similar, complexity the closer one looks. A shoreline, to cite a classic example, has a characteristic jagged shape at whatever scale we draw it; that is, whether we use satellite photos to sketch the whole coast, map it on a fine scale by walking along some small section of it, or examine a few inches of it through a magnifying glass. The surface of the mountain looks roughly the same whether seen from a height of 200 feet by a giant or close up by an insect. The branching of a tree appears the same to us as it does to birds, or even to worms or fungi in the idealized limiting case of infinite branching.
As the mathematician Benoit Mandelbrot, the discoverer of fractals, has famously written, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” These and many other shapes in nature are near fractals, having
characteristic zigzags, push-pulls, bump-dents at almost every size scale, greater magnification yielding similar but ever more complicated convolutions.
And the bottom line, or, in this case, the bottom fractal, for stocks? By starting with the basic up-down-up and down-up- down patterns of a stock’s possible movements, continually replacing each of these patterns’ three segments with smaller versions of one of the basic patterns chosen at random, and then altering the spikiness of the patterns to reflect changes in the stock’s volatility, Mandelbrot has constructed what he calls multifractal “forgeries.” The forgeries are patterns of price movement whose general look is indistinguishable from that of real stock price movements. In contrast, more conventional assumptions about price movements, say those of a strict random-walk theorist, lead to patterns that are noticeably different from real price movements.
These multifractal patterns are so far merely descriptive, not predictive of specific price changes. In their modesty, as well as in their mathematical sophistication, they differ from the Elliott waves mentioned in chapter 3.
Even this does not prove that chaos (in the mathematical sense) reigns in (part of) the market, but it is clearly a bit more than suggestive. The occasional surges of extreme volatility that have always been a part of the market are not as nicely accounted for by traditional approaches to finance, approaches Mandelbrot compares to “theories of sea waves that forbid their swells to exceed six feet.”
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Extreme Price Movements,
Power Laws, and the Web
Humans are a social species, which means we’re all connected to each other, some in more ways than others. This is especially so in financial matters. Every investor responds not only
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to relatively objective economic considerations, but also in varying degrees to the pronouncements of national and world leaders (not least of those Mr. Greenspan), consumer confidence, analysts’ ratings (bah), general and business media reports and their associated spin, investment newsletters, the behavior of funds and large institutions, the sentiments of friends, colleagues, and of course the much-derided brother- in-law.
The linkage of changes in stock prices to the varieties of investor responses and interactions suggests to me that communication networks, degrees of connectivity, and so-called small world phenomena (“Oh, you must know my uncle Waldo’s third wife’s botox specialist”) can shine a light on the workings of Wall Street.
First the conventional story. Movements in a stock or index over small units of time are usually slightly positive or slightly negative, less frequently very positive or very negative. A large fraction of the time, the price will rise or fall between 0 percent and 1 percent; a smaller fraction of the time, it will rise or fall between 1 percent and 2 percent; a very small fraction of the time will the movement be more than, say, 10 percent up or down. In general, the movements are well described by a normal bell-shaped curve. The most likely change for a small unit of time is probably a minuscule jot above zero, reflecting the market’s long-term (and recently invisible) upward bias, but the fact remains that extremely large price movements, whether positive or negative, are rare.
It’s been clear for some time, however (that is, since Mandelbrot made it clear), that extreme movements are not as rare as the normal curve would predict. If you measure commodity price changes, for example, in each of a large number of small time units and make from these measurements a histogram, you will notice that the graph is roughly normal near its middle. The distribution of these price movements, howA
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ever, seems to have “fatter tails” than the normal distribution, suggesting that crashes and bubbles in a stock, an index, or the entire market are less unlikely than many would like to admit. There is, in fact, some evidence that very large movements in stock prices are best described by a so-called power law (whose definition I’ll get to shortly) rather than the tails of the normal curve.
An oblique approach to such evidence is via the notions of connectivity and networks. Everyone’s heard people exclaim about how amazed they were to run into someone they knew so far from home. (What I find amazing is how they can be continually amazed at this sort of thing.) Most have heard too of the alleged six degrees of separation between any two people in this country. (Actually, under reasonable assumptions each of us is connected to everyone else by an average of two links, although we’re not likely to know who the two intermediate parties are.) Another popular variant of the notion concerns the number of movie links between film actors, say between Marlon Brando and Christina Ricci or between Kevin Bacon and anyone else. If A and B appeared together in X, and B and C appeared together in Y, then A is linked to C via these two movies.
Although they may not know of Kevin Bacon and his movies, most mathematicians are familiar with Paul Erdos and his theorems. Erdos, a prolific and peripatetic Hungarian mathematician, wrote hundreds of papers in a variety of mathematical areas during his long life. Many of these had co-authors, who are therefore said to have Erdos number 1. Mathematicians who have written a joint paper with someone with Erdds number 1 are said to have Erdos number 2, and so on.
Ideas about such informal networks lead naturally to the network of all networks, the Internet, and to ways to analyze its structure, shape, and “diameter.” How, for example, are
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the Internet’s nearly 1 billion web pages connected? What constitutes a good search strategy? How many links does the average web page contain? What is the distribution of document sizes? Are there many with, say, more than 1,000 links? And, perhaps most intriguingly, how many clicks on average does it take to get from one of two randomly selected documents to another?
A couple of years ago, Albert-Laszlo Barabasi, a physics professor at Notre Dame, and two associates, Reka Albert and Hawoong Jeong, published results that strongly suggest that the web is growing and that its documents are linking in a rather collective way that accounts for, among other things, the unexpectedly large number of very popular documents. The increasing number of web pages and the “flocking effect” of many pages pointing to the same popular addresses, causing proportionally more pages to do the same thing, is what leads to a power law.
Barabasi, Albert, and Jeong showed that the probability that a document has k links is roughly proportional to 1/k3— or inversely proportional to the third power of k. (I’ve rounded off; the model actually predicts an exponent of 2.9.) This means, for example, that there are approximately one- eighth as many documents with twenty links as there are documents with ten links since 1/203 is one-eighth of 1/103. Thus the number of documents with k links declines quickly as k increases, but nowhere near as quickly as a normal bellshaped distribution would predict. This is why the power law distribution has a fatter tail (more instances of very large values of k) than does the normal distribution.
The power laws (sometimes called scaling laws, sometimes Pareto laws) that characterize the web also seem to characterize many other complex systems that organize themselves into a state of skittish responsiveness. The physicist Per Bak, who has made an extensive study of them, described in his book
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How Nature Works, claims that such l/km laws (for various exponents m) are typical of many biological, geological, musical, and economic processes, and that they tend to arise in a wide variety of complex systems. Traffic jams, to cite a different domain and seemingly unrelated dynamic, also seem to obey a power law, with jams involving k cars occurring with a probability roughly proportional to l/km for an appropriate m.
There is even a power law in linguistics. In English, for example, the word “the” appears most frequently and is said to have rank order 1; the words “of,” “and,” and “to” rank 2, 3, and 4, respectively. “Chrysanthemum” has a much higher rank order. Zipf’s Law relates the frequency of a word to its rank order k and states that a word’s frequency in a written text is proportional to 1/k1; that is, inversely proportional to the first power of k. (Again, I’ve rounded off; the power of k is close to, but not exactly 1.) Thus a relatively unusual word whose rank order is 10,000 will still appear with a frequency proportional to 1/10,000, rather than essentially not at all as would be the case if word frequencies were described by the tail of a normal distribution. The size of cities also follows a power law with k close to 1, the kth largest city having a population proportional to 1/k.
One of the most intriguing consequences of the Barabasi- Albert-Jeong model is that because of the power law distribution of links to and from documents on web sites (the nodes of the network), the diameter of the web is only nineteen clicks. By this they mean that you can travel from one arbitrarily selected web page to any other in approximately nineteen clicks, far fewer than had been conjectured. On the other hand, comparing nineteen with the much smaller number of links between arbitrarily selected people, we may wonder why the diameter is as big as it is. The answer is that the average web page contains only seven links, whereas the average person knows hundreds of people.
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Even though the web is expected to grow by a power of 10 over the next few years, its diameter will likely grow by only a couple of clicks, from nineteen to twenty-one. The growth and preferential linking assumptions above indicate that the web’s diameter D is governed by a logarithmic law; D is a bit more than 2 log(N), where N is the number of documents, presently about 1 billion.
If the Barabasi model is valid (and more work needs to be done), the web is not as unmanageable and untraversable as it often seems. Its documents are much more closely interconnected than they would be if the probability that a document has k links were described by a normal distribution.
What is the relevance of power laws, networks, and diameters to extreme price movements? Investors, companies, mutual funds, brokerages, analysts, and media outlets are connected via a large, vaguely defined network, whose nodes exert influence on the nodes to which they’re connected. This network is likely to be more tightly connected and to contain more very popular (and hence very influential) nodes than people realize. Most of the time this makes no difference and price movements, resulting from the sum of a myriad of investors’ independent zigs and zags, are governed by the normal distribution.
But when the volume of trades is very high, the trades are strongly influenced by relatively few popular nodes—mutual funds, for example, or analysts or media outlets—becoming aligned in their sentiments, and this alignment can create extreme price movements. (WCOM often led the Nasdaq in volume during its slide.) That there exist a few very popular, very connected nodes is, I reiterate, a consequence of the fact that a power law and not the normal distribution governs their frequency. A contagious alignment of this handful of very popular, very connected, very influential nodes will occur
more frequently than people expect, as will, therefore, extreme price movements.
Other examples suggest that the exponent m in market power laws, l/km, may be something other than 3, but the point stands. The trading network is sometimes more herdlike and volatile in its behavior than standard pictures of it acknowledge. The crash of 1929, the decline of 1987, and the recent dot-com meltdown should perhaps not be seen as inexplicable aberrations (or as “just deserts”) but as natural consequences of network dynamics.
Clearly much work remains to be done to understand why power laws are so pervasive. What is needed, I think, is something like the central limit theorem in statistics, which explains why the normal curve arises in so many different contexts. Power laws provide an explanation, albeit not an airtight one, for the frequency of bubbles and crashes and the so-called volatility clustering that seem to characterize real markets. They also reinforce the impression that the market is a different sort of beast than that usually studied by social scientists or, perhaps, that social scientists have been studying these beasts in the wrong way.
I should note that my interest in networks and connectivity is not unrelated to my initial interest in WorldCom, which owned not only MCI, but, as I’ve mentioned twice already, UUNet, “the backbone of the Internet.” Obsessions fade slowly.
Economic Disparities and Media Disproportions
WorldCom may have been based in Mississippi, but Bernie Ebbers, who affected an unpretentious, down-home style,
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wielded political and economic influence foreign to the average Mississippian and the average WorldCom employee. For this he may serve as a synecdoche for the following.
More than a mathematical pun suggests that power laws may have relevance to economic, media, and political power as well as to the stock market. Along various social dimensions, the dynamics underlying power laws might allow for the development of more centers of concentration than we might otherwise expect. This might lead to larger, more powerful economic, media, and political elites and consequent great disparities. Whether or not this is the case, and whether or not great disparities are necessary for complex societies to function, such disparities certainly reign in modern America. Relatively few people, for example, own a hugely disproportionate share of the wealth, and relatively few people attract a hugely disproportionate share of media attention.
The United Nations issued a report a couple of years ago saying that the net worth of the three richest families in the world—the Gates family, the sultan of Brunei, and the Walton family—was greater than the combined gross domestic product of the forty-three poorest nations on Earth. The U.N. statement is misleading in an apples-and-oranges sort of way, but despite the periodic additions, subtractions, and reshufflings of the Forbes 400 and the fortunes of underdeveloped countries, some appropriately modified conclusion no doubt still holds.
(On the other hand, the distribution of wealth in some of the poorest nations—where almost everybody is poverty- stricken—is no doubt more uniform than it is here, indicating that relative equality is no solution to the problem of poverty. I suspect that significant, but not outrageous, disparities of wealth are probably more conducive to wealth creation than is relative uniformity, provided the society meets some minimal conditions: It’s based on law, offers some educational and
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other opportunities, and allows for a modicum of private property.)
The dynamic whereby the rich get richer is nowhere more apparent than in the pharmaceutical industry, in which companies understandably spend far, far more money researching lifestyle drugs for the affluent than life-saving drugs for the hundreds of millions of the world’s poor people. Instead of trying to come up with treatments for malaria, diarrhea, tuberculosis, and acute lower-respiratory diseases, resources go into treatments for wrinkles, impotence, baldness, and obesity.
Surveys indicate that the ratio of the remuneration of a U.S. firm’s CEO to that of the average employee of the firm is at an all-time high of around 500, whether the CEO has improved the fortunes of the company or not, and whether he or she is under indictment or not. (If we assume 250 workdays per year, arithmetic tells us the CEO needs only half a day to make what the employee takes all year to earn.) Professor Edward Wolff of New York University has estimated that the richest 1 percent of Americans own half of all stocks, bonds, and other assets. And Cornell University’s Robert Frank has described the spread of the winner-take-all model of compensation from the sports and entertainment worlds to many other domains of American life.
Nero-like arrogance often accompanies such exorbitant compensation. High-tech WorldCom faced a host of problems before its 2002 collapse. Did Bernie Ebbers utilize the company’s horde of top-flight technical people (at least the ones who hadn’t quit or been fired) to devise a clever strategy to extricate the company from its troubles? No, he cut out free coffee for employees to save money. As Tyco spiraled downward, its CEO, Dennis Kozlowski, spent millions of company dollars on personal items, including a $6,000 shower curtain, a $15,000 umbrella stand, and a $7 million Manhattan apartment.
(Even successful CEOs are not always gentlemen. Oracle’s Larry Ellison, a fierce foe of Bill Gates, a couple of years ago admitted to spying on Microsoft. Amusingly, Oracle’s sophisticated snoops didn’t employ state-of-the-art electronics, but tried to buy the garbage of a pro-Microsoft group in order to examine its contents for clues about Microsoft’s public relations plans. I’m talking real cookies here, not the type that Internet sites leave on your computer; scribbled memos and addresses on torn envelopes, not emails and Internet routing numbers; germs and bacteria, not computer viruses.)
What should we make of such stories? Communism, happily, has been discredited, but unregulated and minimally regulated free markets (as evidenced by the behavior of some accountants, analysts, CEOs, and, yes, greedy, deluded, and short-sighted investors) have some obvious drawbacks. Some of the reforms proposed by Congress in 2002 promise to be helpful in this regard, but I wish here only to express disquiet at such enormous and growing economic disparities.
The same steep hierarchy and disproportion that characterize our economic condition affect our media as well. The famous get ever more famous, celebrities become ever more celebrated. (Pick your favorite ten examples here.) Magazines and television increasingly run features asking who’s hot and who’s not. Even the search engine Google has a version in which surfers can check the topics and people attracting the most hits the previous week. The up-and-down movements of celebrity seem to constitute a kind of market in which almost all the “traders” are technical traders trying to guess what everyone else thinks, rather than value traders looking for worth.
The pattern holds in the political realm as well. In general, on the front page and in the first section of a newspaper, the number one newsmaker is undoubtedly the president of
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the United States. Other big newsmakers are presidential candidates, members of Congress, and other federal officials.
Twenty years ago, Herbert Gans wrote in Deciding What’s News that 80 percent of the domestic news stories on television network news concerned these four classes of people; most of the remaining 20 percent covered the other 280 million of us. Fewer than 10 percent of all stories were about abstractions, objects, or systems. Things haven’t changed much since then (except on the cable networks where disaster stories, show trials, and terrorist obsessions dominate). Newspapers generally have broader coverage, although studies have found that up to 50 percent of the sources for national stories on the front pages of the New York Times and the Washington Post were officials of the U.S. government. The Internet has still broader scope, although there, too, one notes strong and unmistakable signs of increasing hierarchy and concentration.
And what about foreign coverage? The frequency of reporting on overseas newsmakers demonstrates the same biases. We hear from heads of state, from leaders of opposition parties or forces, and occasionally from others. The masses of ordinary people are seldom a presence at all. The journalistic rule of thumb that one American equals 10 Englishmen equals 1,000 Chileans equals 10,000 Rwandans varies with time and circumstance, but it does contain an undeniable truth. Americans, like everybody else, care much less about some parts of the world than others. Even the terrorist attack in Bali didn’t rate much coverage here, and many regions have no correspondents at all, rendering them effectively invisible.
Such disparities may be a natural consequence of complex societies. This doesn’t mean that they need be as extreme as they are or that they’re always to be welcomed. It may be that the stock market’s recent volatility surges are a leading indicator for even greater social disparities to come.
Whether haunted by a bad investment or not, we’re all connected. No investor is an island (or even a peninsula). Stated mathematically, this means that statistical independence often fails; your actions affect mine. Most accounts of the stock market acknowledge in a general way that we learn from and respond to one another, but a better understanding of the market requires that one’s models reflect the complexity of investors’ interaction. In a sense, the market is the interaction. Stocks R Us. Before discussing some of the consequences of
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this complexity, let me consider three such sources for it: one micro, one macro, and the third mucro (yup, it’s a word).
The micro example involves insider trading, which has always struck me as an odd sort of crime. Few people who aren’t psychopaths daydream about murder or burglary, but many investors, I suspect, fantasize about coming upon inside information and making a bundle from it. The thought of finding myself on a plane next to Bernie Ebbers and Jack Grubman (assuming they flew economy class on commercial airliners) and overhearing their conversation about an impending merger or IPO offering, for example, did cross my mind a few times. Insider trading seems the limit or culmination of what investors and traders do naturally: getting all the information possible and acting on it before others see and understand what they see and understand.
Insider Trading and Subterranean Information Processing
The kind of insider trading I want to consider is relevant to seemingly unexplained price movements. It’s also related to good poker playing, which may explain why the training program of at least one very successful hedge fund has a substantial unit on the game. The strategies associated with poker include learning not only the relevant probabilities but also the bluffing that is a necessary part of the game. Options traders often deal with relatively few other traders, many of whom they recognize, and this gives rise to the opportunity for feints, misdirection, and the exploitation of idiosyncrasies.
The example derives from the notion of common knowledge introduced in chapter 1. Recall that a bit of information is common knowledge among a group of people if they all know it, know that the others know it, know that the others
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know that they know it, and so on. Robert Aumann, who first defined the notion, proved a theorem that can be roughly paraphrased as follows: Two individuals cannot forever agree to disagree. As their beliefs, formed in rational response to different bits of private information, gradually become common knowledge, the beliefs change and eventually coincide.
When private information becomes common knowledge, it induces decisions and actions. As anyone who has overheard teenagers’ gossip with its web of suppositions can attest, this transition to common knowledge sometimes relies on convoluted inferences about others’ beliefs. Sergiu Hart, an economist at Hebrew University and one of a number of people who have built on Aumann’s result, demonstrates this with an example relevant to the stock market. Superficially complicated, it nevertheless requires no particular background besides an ability to decode gossip, hearsay, and rumor and decide what others really think.
Hart asks us to consider a company that must make a decision. In keeping with the WorldCom leitmotif, let’s suppose it to be a small telecommunications company that must decide whether to develop a new handheld device or a cell phone with a novel feature. Assume that the company is equally likely to decide on one or the other of these products, and assume further that whatever decision it makes, the product chosen has a 50 percent chance of being successful, say being bought in huge numbers by another company. Thus there are four equally likely outcomes: Handheld+, Handheld-, Phone+, Phone- (where Handheld+ means the handheld device was chosen for development and it was a success, Handheld- means the handheld was chosen but it turned out to be a failure, and similarly for Phone+ and Phone-).
Let’s say there are two influential investors, Alice and Bob. They both decide that at the current stock price, if the chances of success of this product development are better
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than 50 percent, they should (continue to) buy, and if they’re 50 percent or less they should (continue to) sell.
Furthermore, they are each privy to a different piece of information about the company. Because of her inside contacts, Alice knows which product decision was made, Handheld or Phone, but not whether it was successful or not.
Bob, because of his position with another company, stands to get the “rejects” from a failed phone project, so he knows whether or not the cell phone was chosen for development and failed. That is, Bob knows whether Phone- or not.
Let’s assume that the handheld device was chosen for development. So the true situation is either Handheld+ or Handheld-. Alice therefore knows Handheld, while Bob knows that the decision is not Phone- (else he would have received the rejects).
After the first period (week, day, or hour), Alice sells since Handheld+ and Handheld- are equally likely, and one sells if the probability of success is 50 percent or less. Bob buys since he estimates that the probability for success is 2/3. With Phone- ruled out, the remaining possibilities are Handheld+, Handheld-, and Phone+, and two out of three of them are successes.
After the second period, it is common knowledge that the true situation is not Phone- since otherwise Bob would have sold in the first period. This is not news to Alice, who continues to sell. Bob continues to buy.
After the third period, it is common knowledge that it is not Phone (neither Phone+ nor Phone-) since otherwise Alice would have bought in the second period. Thus it’s either Handheld+ or Handheld-. Both Bob and Alice take the probability of success to be 50 percent, thus both sell, and there is a mini-crash of the stock price. (Selling by both influential investors triggers a general sell-off.)
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Note that at the beginning both Alice and Bob know that the true situation is not Phone-, but this knowledge is mutual, not common. Alice knows that Bob knows it is not Phone-, but Bob does not know that Alice knows this. From his position the true situation might be Phone+, in which case Alice would know Phone but not whether the situation is Phone+ or Phone-.
The example can be varied in a number of ways: there needn’t be merely three periods before a crash, but an arbitrary number; there may be a bubble (sellers suddenly switching to become buyers) instead of a crash; there may be an arbitrarily large number of investors or investor groups; there may be an issue other than buying or selling under deliberation, perhaps a decision whether to employ one stock-picking approach rather than another.
In all these cases the stock’s price can move in response to no external news. Nevertheless, the subterranean information processing leading to common knowledge among the investors eventually leads to precipitous and unexpected movement in the stock’s price. Analysts will express surprise at the crash (or bubble) because “nothing happened.”
The example is also relevant to what I suspect is a relatively common kind of insider trading, in which “partial insiders” are privy to bits of insider information but not to the whole story.
Trading Strategies, Whim, and Ant Behavior
A more macro-level interaction among investors occurs between technical traders and value traders. Also contributing over time to booms and busts, this interaction comes through clearly in computer models of the following commonsense dynamic.
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Let’s suppose that value traders perceive individual stocks or the market as a whole to be strongly undervalued. They start buying and, by doing so, raise prices. As prices increase, a trend develops and technical traders, as is their wont, follow it, increasing prices even further. Soon enough, the market is seen as overvalued by value traders, who begin to sell and thereby slow and then reverse the trend. The trendfollowing technical traders eventually follow suit, and the cycle begins over again. There are, of course, other sources of variation (one being the number of people who are technical traders and value traders at any given time), and the oscillations are irregular.
The bottom line of much of this modeling is that contrarian value traders have a stabilizing effect on the market, whereas technical traders increase volatility. So does computergenerated program trading, which tends to produce buying or selling in lockstep. There are other sorts of interaction among different classes of investors leading to cycles of varying duration, all of which have differential impacts on the others on which they are superimposed.
In addition to these more or less rational interactions among investors I must also note influences inspired by nothing more than whim, where behavior turns on a mucro. I recall many times, for example, reluctantly beginning work on a project when a niggling detail about some utterly irrelevant matter came to mind. It may have concerned the etymology of a word, or the colleague whose paper bag ripped open at a departmental meeting revealing an embarrassing magazine inside, or why caller ID misidentified a friend’s telephone number. These in turn brought to mind the next in a train of associations and musings, which ultimately led me to an entirely different project. My impulsively deciding, while browsing in Borders, to make my first margin call on WCOM is another instance.
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When this capriousness extends to influential analysts, the effect is more pronounced. In November 2002 the New York Times reported on such a case involving Jack Grubman, telecommunications analyst and anxious father. In an email to a friend Grubman allegedly stated that his boss, Sanford Weill, the chairman of Citigroup, helped get Grubman’s children into an exclusive nursery school after he raised his rating of AT&T in 1999. Gretchen Morgenson, the article’s author, further reported that Weill had his own personal reasons for wanting this upgrade. Whether these particular charges are true or not is immaterial. It’s very hard to believe, however, that this sort of influence is rare.
Such episodes strongly suggest to me that there will never be a precise science of finance or economics. Buying and selling must surely partake of a similar iffiness, at least sometimes. Butterfly Economics, by the British economic theorist Paul Ormerod, faults these disciplines for not sufficiently taking into account the commonsense fact that people, whether knowledgeable or not, influence each other.
People do not, as chapter 2 demonstrated, have a set of fixed preferences on which they coolly and rationally base their economic decisions. The assumption that investors are sensitive only to price and a few ratios simplifies the mathematical models, but it is not always true to our experience of fads, fashions, and people’s everyday monkey-see, monkey-do behavior.
Ormerod tells of an experiment involving not monkeys but ants that provides a useful metaphor. Two identical piles of food are set up at equal distances from a large nest of ants. Each pile is automatically replenished and the ants have no reason to prefer one to the other. Entomologists tell us that once an ant has found food, it usually returns to the same source. Upon returning to the nest, however, it physically stimulates other ants, who might be frequenting the other pile, to follow it to the first pile.
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So where do the ants go? It might be speculated that either they would split into two roughly even groups or perhaps a large majority would arbitrarily settle on one or the other pile. Their actual behavior is counterintuitive. The number of ants going to each pile fluctuates wildly and doesn’t ever settle down. A graph of these fluctuations looks suspiciously like a graph of the stock market.
And in a way, the ants are like stock traders (or people deciding whether or not to make a margin call). Upon leaving the nest, each ant must make a decision: Go to the pile visited last time, be influenced by another to switch piles, or switch piles of its own volition. This slight openness to the influence of other ants is enough to insure the complicated and volatile fluctuations in the number of ants visiting the two sites.
An astonishingly simple formal model of such influence is provided by Stephen Wolfram in his book A New Kind of Science. Imagine a colossally high brick wall wherein each brick rests on parts of two bricks below it and, except for the top row, has parts of two bricks above it. Imagine further that the top row has some red bricks and some green ones. The coloring of the bricks in the top row determines the coloring of the bricks in the second row as follows. Pick a brick in the second row and check the colors of the two bricks above it in the first row. If exactly one of these bricks is green, then the brick in the second row is colored green. If both or neither are green, then the brick is colored red. Do this for every brick in the second row.
The coloring of the bricks in the second row determines the coloring of the bricks in the third row in the same way, and in general, the coloring of the bricks in any row determines the coloring of the bricks in the row below it in the same way. That’s it.
Now if we interpret a row of bricks as a collection of investors at any given instant, green ones for buyers and red ones
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for sellers, then the change from moment to moment of investor sentiment is reflected in the changing color composition of the succeeding rows of bricks. If we let P be the difference between the number of green bricks and the number of red bricks, then P is a rough analogue of a stock’s price. Graphing it, we see that it oscillates up and down in a way that looks random.
The model can be made more realistic, but it is significant that even this bare-bones version, like the ant behavior, evinces a kind of internally generated random noise. This suggests that part of the oscillation of stock prices is also internally generated and is not a response to anything besides investors’ reactions to each other. The theme of Wolfram’s book, borne out here, is that complex behavior can result from very simple rules of interaction.
Chaos and Unpredictability
What is the relative importance of private information, investor trading strategies, and pure whim in predicting the market? What is the relative importance of conventional economic news (interest rates, budget deficits, accounting scandals, and trade balances), popular culture fads (in sports, movies, fashions), and germane political and military events (terrorism, elections, war) too disparate even to categorize? If we were to carefully define the problem, predicting the market with any precision is probably what mathematicians call a universal problem, meaning that a complete solution to it would lead immediately to solutions for a large class of other problems. It is, in other words, as hard a problem in social prediction as there is.
Certainly, too little notice is taken of the complicated connections among these variables, even the more clearly defined
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economic ones. Interest rates, for example, have an impact on unemployment rates, which in turn influence revenues; budget deficits affect trade deficits, which sway interest rates and exchange rates; corporate fraud influences consumer confidence, which may depress the stock market and alter other indices; natural business cycles of various periods are superimposed on one another; an increase in some quantity or index positively (or negatively) feeds back on another, reinforcing or weakening it and being reinforced or weakened in turn.
Few of these associations are accurately described by a straight-line graph and so they bring to a mathematician’s mind the subject of nonlinear dynamics, more popularly known as chaos theory. The subject doesn’t deal with anarchist treatises or surrealist manifestoes but with the behavior of so-called nonlinear systems. For our purposes these may be thought of as any collection of parts whose interactions and connections are described by nonlinear rules or equations. That is to say, the equations’ variables may be multiplied together, raised to powers, and so on. As a consequence the system’s parts are not necessarily linked in a proportional manner as they are, for example, in a bathroom scale or a thermometer; doubling the magnitude of one part will not double that of another—nor will outputs be proportional to inputs. Not surprisingly, trying to predict the precise long-term behavior of such systems is often futile.
Let me, in place of a technical definition of such nonlinear systems, describe instead a particular physical instance of one. Picture before you a billiards table. Imagine that approximately twenty-five round obstacles are securely fastened to its surface in some haphazard arrangement. You hire the best pool player you can find and ask him to place the ball at a particular spot on the table and take a shot toward one of the round obstacles. After he’s done so, his challenge is to make exactly the same shot from the same spot with another ball.
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Even if his angle on this second shot is off by the merest fraction of a degree, the trajectories of these two balls will very soon diverge considerably. An infinitesimal difference in the angle of impact will be magnified by successive hits of the obstacles. Soon one of the balls will hit an obstacle that the other misses entirely, at which point all similarity between the two trajectories ends.
The sensitivity of the billiard balls’ paths to minuscule variations in their initial angles is characteristic of nonlinear systems. The divergence of the billiard balls is not unlike the disproportionate effect of seemingly inconsequential events, the missed planes, serendipitous meetings, and odd mistakes and links that shape and reshape our lives.
This sensitive dependence of nonlinear systems on even tiny differences in initial conditions is, I repeat, relevant to various aspects of the stock market in general, in particular its sometimes wildly disproportionate responses to seemingly small stimuli such as companies’ falling a penny short of earnings estimates. Sometimes, of course, the differences are more substantial. Witness the notoriously large discrepancies between government economic figures on the size of budget surpluses and corporate accounting statements of earnings and the “real” numbers.
Aspects of investor behavior too can no doubt be better modeled by a nonlinear system than a linear one. This is so despite the fact that linear systems and models are much more robust, with small differences in initial conditions leading only to small differences in final outcomes. They’re also easier to predict mathematically, and this is why they’re so often employed whether their application is appropriate or not. The chestnut about the economist looking for his lost car keys under the street lamp comes to mind. “You probably lost them near the car,” his companion remonstrates, to which the economist responds, “I know, but the light is better over here.”
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The “butterfly effect” is the term often used for the sensitive dependence of nonlinear systems, a characteristic that has been noted in phenomena ranging from fluid flow and heart fibrillations to epilepsy and price fluctuations. The name comes from the idea that a butterfly flapping its wings someplace in South America might be sufficient to change future weather systems, helping to bring about, say, a tornado in Oklahoma that would otherwise not have occurred. It also explains why long-range precise prediction of nonlinear systems isn’t generally possible. This non-predictability is the result not of randomness but of complexity too great to fathom.
Yet another reason to suspect that parts of the market may be better modeled by nonlinear systems is that such systems’ “trajectories” often follow a fractal course. The trajectories of these systems, of which the stock price movements may be considered a proxy, turn out to be aperiodic and unpredictable and, when examined closely, evince even more intricacy. Still closer inspection of the system’s trajectories reveals yet smaller vortices and complications of the same general kind.
In general, fractals are curves, surfaces, or higher dimensional objects that contain more, but similar, complexity the closer one looks. A shoreline, to cite a classic example, has a characteristic jagged shape at whatever scale we draw it; that is, whether we use satellite photos to sketch the whole coast, map it on a fine scale by walking along some small section of it, or examine a few inches of it through a magnifying glass. The surface of the mountain looks roughly the same whether seen from a height of 200 feet by a giant or close up by an insect. The branching of a tree appears the same to us as it does to birds, or even to worms or fungi in the idealized limiting case of infinite branching.
As the mathematician Benoit Mandelbrot, the discoverer of fractals, has famously written, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” These and many other shapes in nature are near fractals, having
characteristic zigzags, push-pulls, bump-dents at almost every size scale, greater magnification yielding similar but ever more complicated convolutions.
And the bottom line, or, in this case, the bottom fractal, for stocks? By starting with the basic up-down-up and down-up- down patterns of a stock’s possible movements, continually replacing each of these patterns’ three segments with smaller versions of one of the basic patterns chosen at random, and then altering the spikiness of the patterns to reflect changes in the stock’s volatility, Mandelbrot has constructed what he calls multifractal “forgeries.” The forgeries are patterns of price movement whose general look is indistinguishable from that of real stock price movements. In contrast, more conventional assumptions about price movements, say those of a strict random-walk theorist, lead to patterns that are noticeably different from real price movements.
These multifractal patterns are so far merely descriptive, not predictive of specific price changes. In their modesty, as well as in their mathematical sophistication, they differ from the Elliott waves mentioned in chapter 3.
Even this does not prove that chaos (in the mathematical sense) reigns in (part of) the market, but it is clearly a bit more than suggestive. The occasional surges of extreme volatility that have always been a part of the market are not as nicely accounted for by traditional approaches to finance, approaches Mandelbrot compares to “theories of sea waves that forbid their swells to exceed six feet.”
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Extreme Price Movements,
Power Laws, and the Web
Humans are a social species, which means we’re all connected to each other, some in more ways than others. This is especially so in financial matters. Every investor responds not only
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to relatively objective economic considerations, but also in varying degrees to the pronouncements of national and world leaders (not least of those Mr. Greenspan), consumer confidence, analysts’ ratings (bah), general and business media reports and their associated spin, investment newsletters, the behavior of funds and large institutions, the sentiments of friends, colleagues, and of course the much-derided brother- in-law.
The linkage of changes in stock prices to the varieties of investor responses and interactions suggests to me that communication networks, degrees of connectivity, and so-called small world phenomena (“Oh, you must know my uncle Waldo’s third wife’s botox specialist”) can shine a light on the workings of Wall Street.
First the conventional story. Movements in a stock or index over small units of time are usually slightly positive or slightly negative, less frequently very positive or very negative. A large fraction of the time, the price will rise or fall between 0 percent and 1 percent; a smaller fraction of the time, it will rise or fall between 1 percent and 2 percent; a very small fraction of the time will the movement be more than, say, 10 percent up or down. In general, the movements are well described by a normal bell-shaped curve. The most likely change for a small unit of time is probably a minuscule jot above zero, reflecting the market’s long-term (and recently invisible) upward bias, but the fact remains that extremely large price movements, whether positive or negative, are rare.
It’s been clear for some time, however (that is, since Mandelbrot made it clear), that extreme movements are not as rare as the normal curve would predict. If you measure commodity price changes, for example, in each of a large number of small time units and make from these measurements a histogram, you will notice that the graph is roughly normal near its middle. The distribution of these price movements, howA
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ever, seems to have “fatter tails” than the normal distribution, suggesting that crashes and bubbles in a stock, an index, or the entire market are less unlikely than many would like to admit. There is, in fact, some evidence that very large movements in stock prices are best described by a so-called power law (whose definition I’ll get to shortly) rather than the tails of the normal curve.
An oblique approach to such evidence is via the notions of connectivity and networks. Everyone’s heard people exclaim about how amazed they were to run into someone they knew so far from home. (What I find amazing is how they can be continually amazed at this sort of thing.) Most have heard too of the alleged six degrees of separation between any two people in this country. (Actually, under reasonable assumptions each of us is connected to everyone else by an average of two links, although we’re not likely to know who the two intermediate parties are.) Another popular variant of the notion concerns the number of movie links between film actors, say between Marlon Brando and Christina Ricci or between Kevin Bacon and anyone else. If A and B appeared together in X, and B and C appeared together in Y, then A is linked to C via these two movies.
Although they may not know of Kevin Bacon and his movies, most mathematicians are familiar with Paul Erdos and his theorems. Erdos, a prolific and peripatetic Hungarian mathematician, wrote hundreds of papers in a variety of mathematical areas during his long life. Many of these had co-authors, who are therefore said to have Erdos number 1. Mathematicians who have written a joint paper with someone with Erdds number 1 are said to have Erdos number 2, and so on.
Ideas about such informal networks lead naturally to the network of all networks, the Internet, and to ways to analyze its structure, shape, and “diameter.” How, for example, are
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the Internet’s nearly 1 billion web pages connected? What constitutes a good search strategy? How many links does the average web page contain? What is the distribution of document sizes? Are there many with, say, more than 1,000 links? And, perhaps most intriguingly, how many clicks on average does it take to get from one of two randomly selected documents to another?
A couple of years ago, Albert-Laszlo Barabasi, a physics professor at Notre Dame, and two associates, Reka Albert and Hawoong Jeong, published results that strongly suggest that the web is growing and that its documents are linking in a rather collective way that accounts for, among other things, the unexpectedly large number of very popular documents. The increasing number of web pages and the “flocking effect” of many pages pointing to the same popular addresses, causing proportionally more pages to do the same thing, is what leads to a power law.
Barabasi, Albert, and Jeong showed that the probability that a document has k links is roughly proportional to 1/k3— or inversely proportional to the third power of k. (I’ve rounded off; the model actually predicts an exponent of 2.9.) This means, for example, that there are approximately one- eighth as many documents with twenty links as there are documents with ten links since 1/203 is one-eighth of 1/103. Thus the number of documents with k links declines quickly as k increases, but nowhere near as quickly as a normal bellshaped distribution would predict. This is why the power law distribution has a fatter tail (more instances of very large values of k) than does the normal distribution.
The power laws (sometimes called scaling laws, sometimes Pareto laws) that characterize the web also seem to characterize many other complex systems that organize themselves into a state of skittish responsiveness. The physicist Per Bak, who has made an extensive study of them, described in his book
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How Nature Works, claims that such l/km laws (for various exponents m) are typical of many biological, geological, musical, and economic processes, and that they tend to arise in a wide variety of complex systems. Traffic jams, to cite a different domain and seemingly unrelated dynamic, also seem to obey a power law, with jams involving k cars occurring with a probability roughly proportional to l/km for an appropriate m.
There is even a power law in linguistics. In English, for example, the word “the” appears most frequently and is said to have rank order 1; the words “of,” “and,” and “to” rank 2, 3, and 4, respectively. “Chrysanthemum” has a much higher rank order. Zipf’s Law relates the frequency of a word to its rank order k and states that a word’s frequency in a written text is proportional to 1/k1; that is, inversely proportional to the first power of k. (Again, I’ve rounded off; the power of k is close to, but not exactly 1.) Thus a relatively unusual word whose rank order is 10,000 will still appear with a frequency proportional to 1/10,000, rather than essentially not at all as would be the case if word frequencies were described by the tail of a normal distribution. The size of cities also follows a power law with k close to 1, the kth largest city having a population proportional to 1/k.
One of the most intriguing consequences of the Barabasi- Albert-Jeong model is that because of the power law distribution of links to and from documents on web sites (the nodes of the network), the diameter of the web is only nineteen clicks. By this they mean that you can travel from one arbitrarily selected web page to any other in approximately nineteen clicks, far fewer than had been conjectured. On the other hand, comparing nineteen with the much smaller number of links between arbitrarily selected people, we may wonder why the diameter is as big as it is. The answer is that the average web page contains only seven links, whereas the average person knows hundreds of people.
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Even though the web is expected to grow by a power of 10 over the next few years, its diameter will likely grow by only a couple of clicks, from nineteen to twenty-one. The growth and preferential linking assumptions above indicate that the web’s diameter D is governed by a logarithmic law; D is a bit more than 2 log(N), where N is the number of documents, presently about 1 billion.
If the Barabasi model is valid (and more work needs to be done), the web is not as unmanageable and untraversable as it often seems. Its documents are much more closely interconnected than they would be if the probability that a document has k links were described by a normal distribution.
What is the relevance of power laws, networks, and diameters to extreme price movements? Investors, companies, mutual funds, brokerages, analysts, and media outlets are connected via a large, vaguely defined network, whose nodes exert influence on the nodes to which they’re connected. This network is likely to be more tightly connected and to contain more very popular (and hence very influential) nodes than people realize. Most of the time this makes no difference and price movements, resulting from the sum of a myriad of investors’ independent zigs and zags, are governed by the normal distribution.
But when the volume of trades is very high, the trades are strongly influenced by relatively few popular nodes—mutual funds, for example, or analysts or media outlets—becoming aligned in their sentiments, and this alignment can create extreme price movements. (WCOM often led the Nasdaq in volume during its slide.) That there exist a few very popular, very connected nodes is, I reiterate, a consequence of the fact that a power law and not the normal distribution governs their frequency. A contagious alignment of this handful of very popular, very connected, very influential nodes will occur
more frequently than people expect, as will, therefore, extreme price movements.
Other examples suggest that the exponent m in market power laws, l/km, may be something other than 3, but the point stands. The trading network is sometimes more herdlike and volatile in its behavior than standard pictures of it acknowledge. The crash of 1929, the decline of 1987, and the recent dot-com meltdown should perhaps not be seen as inexplicable aberrations (or as “just deserts”) but as natural consequences of network dynamics.
Clearly much work remains to be done to understand why power laws are so pervasive. What is needed, I think, is something like the central limit theorem in statistics, which explains why the normal curve arises in so many different contexts. Power laws provide an explanation, albeit not an airtight one, for the frequency of bubbles and crashes and the so-called volatility clustering that seem to characterize real markets. They also reinforce the impression that the market is a different sort of beast than that usually studied by social scientists or, perhaps, that social scientists have been studying these beasts in the wrong way.
I should note that my interest in networks and connectivity is not unrelated to my initial interest in WorldCom, which owned not only MCI, but, as I’ve mentioned twice already, UUNet, “the backbone of the Internet.” Obsessions fade slowly.
Economic Disparities and Media Disproportions
WorldCom may have been based in Mississippi, but Bernie Ebbers, who affected an unpretentious, down-home style,
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wielded political and economic influence foreign to the average Mississippian and the average WorldCom employee. For this he may serve as a synecdoche for the following.
More than a mathematical pun suggests that power laws may have relevance to economic, media, and political power as well as to the stock market. Along various social dimensions, the dynamics underlying power laws might allow for the development of more centers of concentration than we might otherwise expect. This might lead to larger, more powerful economic, media, and political elites and consequent great disparities. Whether or not this is the case, and whether or not great disparities are necessary for complex societies to function, such disparities certainly reign in modern America. Relatively few people, for example, own a hugely disproportionate share of the wealth, and relatively few people attract a hugely disproportionate share of media attention.
The United Nations issued a report a couple of years ago saying that the net worth of the three richest families in the world—the Gates family, the sultan of Brunei, and the Walton family—was greater than the combined gross domestic product of the forty-three poorest nations on Earth. The U.N. statement is misleading in an apples-and-oranges sort of way, but despite the periodic additions, subtractions, and reshufflings of the Forbes 400 and the fortunes of underdeveloped countries, some appropriately modified conclusion no doubt still holds.
(On the other hand, the distribution of wealth in some of the poorest nations—where almost everybody is poverty- stricken—is no doubt more uniform than it is here, indicating that relative equality is no solution to the problem of poverty. I suspect that significant, but not outrageous, disparities of wealth are probably more conducive to wealth creation than is relative uniformity, provided the society meets some minimal conditions: It’s based on law, offers some educational and
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other opportunities, and allows for a modicum of private property.)
The dynamic whereby the rich get richer is nowhere more apparent than in the pharmaceutical industry, in which companies understandably spend far, far more money researching lifestyle drugs for the affluent than life-saving drugs for the hundreds of millions of the world’s poor people. Instead of trying to come up with treatments for malaria, diarrhea, tuberculosis, and acute lower-respiratory diseases, resources go into treatments for wrinkles, impotence, baldness, and obesity.
Surveys indicate that the ratio of the remuneration of a U.S. firm’s CEO to that of the average employee of the firm is at an all-time high of around 500, whether the CEO has improved the fortunes of the company or not, and whether he or she is under indictment or not. (If we assume 250 workdays per year, arithmetic tells us the CEO needs only half a day to make what the employee takes all year to earn.) Professor Edward Wolff of New York University has estimated that the richest 1 percent of Americans own half of all stocks, bonds, and other assets. And Cornell University’s Robert Frank has described the spread of the winner-take-all model of compensation from the sports and entertainment worlds to many other domains of American life.
Nero-like arrogance often accompanies such exorbitant compensation. High-tech WorldCom faced a host of problems before its 2002 collapse. Did Bernie Ebbers utilize the company’s horde of top-flight technical people (at least the ones who hadn’t quit or been fired) to devise a clever strategy to extricate the company from its troubles? No, he cut out free coffee for employees to save money. As Tyco spiraled downward, its CEO, Dennis Kozlowski, spent millions of company dollars on personal items, including a $6,000 shower curtain, a $15,000 umbrella stand, and a $7 million Manhattan apartment.
(Even successful CEOs are not always gentlemen. Oracle’s Larry Ellison, a fierce foe of Bill Gates, a couple of years ago admitted to spying on Microsoft. Amusingly, Oracle’s sophisticated snoops didn’t employ state-of-the-art electronics, but tried to buy the garbage of a pro-Microsoft group in order to examine its contents for clues about Microsoft’s public relations plans. I’m talking real cookies here, not the type that Internet sites leave on your computer; scribbled memos and addresses on torn envelopes, not emails and Internet routing numbers; germs and bacteria, not computer viruses.)
What should we make of such stories? Communism, happily, has been discredited, but unregulated and minimally regulated free markets (as evidenced by the behavior of some accountants, analysts, CEOs, and, yes, greedy, deluded, and short-sighted investors) have some obvious drawbacks. Some of the reforms proposed by Congress in 2002 promise to be helpful in this regard, but I wish here only to express disquiet at such enormous and growing economic disparities.
The same steep hierarchy and disproportion that characterize our economic condition affect our media as well. The famous get ever more famous, celebrities become ever more celebrated. (Pick your favorite ten examples here.) Magazines and television increasingly run features asking who’s hot and who’s not. Even the search engine Google has a version in which surfers can check the topics and people attracting the most hits the previous week. The up-and-down movements of celebrity seem to constitute a kind of market in which almost all the “traders” are technical traders trying to guess what everyone else thinks, rather than value traders looking for worth.
The pattern holds in the political realm as well. In general, on the front page and in the first section of a newspaper, the number one newsmaker is undoubtedly the president of
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the United States. Other big newsmakers are presidential candidates, members of Congress, and other federal officials.
Twenty years ago, Herbert Gans wrote in Deciding What’s News that 80 percent of the domestic news stories on television network news concerned these four classes of people; most of the remaining 20 percent covered the other 280 million of us. Fewer than 10 percent of all stories were about abstractions, objects, or systems. Things haven’t changed much since then (except on the cable networks where disaster stories, show trials, and terrorist obsessions dominate). Newspapers generally have broader coverage, although studies have found that up to 50 percent of the sources for national stories on the front pages of the New York Times and the Washington Post were officials of the U.S. government. The Internet has still broader scope, although there, too, one notes strong and unmistakable signs of increasing hierarchy and concentration.
And what about foreign coverage? The frequency of reporting on overseas newsmakers demonstrates the same biases. We hear from heads of state, from leaders of opposition parties or forces, and occasionally from others. The masses of ordinary people are seldom a presence at all. The journalistic rule of thumb that one American equals 10 Englishmen equals 1,000 Chileans equals 10,000 Rwandans varies with time and circumstance, but it does contain an undeniable truth. Americans, like everybody else, care much less about some parts of the world than others. Even the terrorist attack in Bali didn’t rate much coverage here, and many regions have no correspondents at all, rendering them effectively invisible.
Such disparities may be a natural consequence of complex societies. This doesn’t mean that they need be as extreme as they are or that they’re always to be welcomed. It may be that the stock market’s recent volatility surges are a leading indicator for even greater social disparities to come.
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