快猫短视频

Santa Fe, New Mexico

ON 29 March 1900, in a dusty seminar room at the Sorbonne in Paris, the renowned mathematician Jules Henri Poincar茅 was presiding as a student defended a slightly unusual doctoral dissertation. In Poincar茅`s words, the topic was 鈥渟omewhat remote from those our candidates are in the habit of treating鈥�. This topic was a mathematical treatment of how prices for French government bonds and their options fluctuated on the Paris Bourse. The author of this dissertation, Louis Bachelier, received the insultingly low grade of mention honorable rather than the more usual mention tr猫s honorable, the level needed to be taken seriously as a candidate for an academic post in France. Perhaps the dissertation鈥檚 title, The Theory of Speculation, as well as its frankly commercial character had something to do with the disdain with which the examiners viewed Bachelier鈥檚 efforts. Who can say? What we do know is that this was pioneering work, the academic community鈥檚 initial salvo in the battle to unlock the secrets of speculative markets. And Bachelier鈥檚 ideas were truly visionary, for a century later his conclusions remain at the heart of the dominant paradigm as to how prices fluctuate in such markets.

Bachelier focused his mathematical artillery on changes in prices, rather than on the prices themselves. Prices sometimes go up, of course, and sometimes they go down. But by studying the ups and downs with care, Bachelier discovered that he could say more than that.

The ups and downs

First, he noticed that price changes over one interval have nothing to do with changes over another. If a stock price goes up one week, for example, that doesn鈥檛 mean it is any more likely-or less likely-to go up the next. Changes over different intervals seem to be completely independent of one another, as if the market has absolutely no memory of what it has done in the past.

Second, Bachelier discovered that the statistics of the market鈥檚 ups and downs look the same at all times-this week, next summer or just after Christmas. In other words, there are no special times of year when the market behaves peculiarly.

Bachelier鈥檚 third and final conclusion concerned the specific mathematical form of the probability distribution of price fluctuations. For technical but not terribly important reasons, he looked at changes in the logarithms of the prices. If P is the closing price on the Bourse on one day, and Q is the closing price some days later, then a useful measure of the change in price is log Q- log P. Bachelier found that this quantity follows the famous 鈥渘ormal鈥� distribution, also known as the 鈥渂ell curve鈥�. This last point is crucial.

The outcomes of many irregular phenomena in nature distribute themselves according to this same familiar bell-shaped curve, in which outcomes close to the average occur most frequently, and those far from the average are rare. It describes the normal irregularities occurring in everything from the heights of men or women to the weights of tomatoes or the number of drinks served in a bar on a typical evening. It鈥檚 how things 鈥渘ormally鈥� work. Bachelier claimed that speculative price variation is merely another example of this 鈥渘atural鈥� distribution of irregularities.

As conventional wisdom has it, a price change represents the aggregate response to innumerable lower-level actions on the part of investors. By summing all these unpredictable behaviours, we arrive at the price change. Using this reasoning, Bachelier supposed that a tenet of probability theory called the central limit theorem would apply to these price fluctuations. This theorem states (roughly) that, regardless of the probability law governing a set of independent individual units, the behaviour of an aggregate of such units always follows the bell-shaped curve of the normal distribution. Thus emerged Bachelier鈥檚 conclusion that price changes are normally distributed.

No change

This has immediate implications for investors. First, in the normal distribution, negative price swings of a given amount are just as likely to occur as positive swings of the same size. Moreover, the larger these swings, whether positive or negative, the less likely they become. In fact, for the normal distribution, very large fluctuations are so unlikely as to be virtually impossible. Bachelier also stated that the distribution according to which price changes fluctuate has zero mean, so that the most likely change of price is no change at all.

A time series of price changes satisfying Bachelier鈥檚 three postulates is termed a Gaussian random walk, after the great German mathematician Carl Friedrich Gauss who first saw that probabilities so often fitted the bell-shaped curve. In the 1960s finance professors dubbed Bachelier鈥檚 view of the movement of price changes the 鈥渞andom walk hypothesis鈥�. It describes a market with no memory, no special moments and a manner of irregular fluctuation that resembles thousands of other phenomena in the world.

All in all, Bachelier鈥檚 was a beautiful theory, and he deserved better treatment at the hands of his doctoral examination committee. Still, he wasn鈥檛 quite right. Nearly thirty years ago, the mathematician Benoit Mandelbrot disputed his conclusions, and suggested a crucial modification that until recently was almost completely ignored by the academic Brahmins of finance.

Mandelbrot, who is better known for his work on self-similar geometrical objects called fractals, observed that real stock prices display fluctuations that are just too large to be associated with a normally distributed random variable. The problem is that the normal distribution has a finite value for a parameter called the variance-a number that mathematicians use to say roughly how much irregularity there is in a particular set of happenings. For financial markets, it describes how much price changes spread out around the average, or expected, change. Bachelier鈥檚 Gaussian distribution implied that extremely large negative or positive fluctuations in prices would be so unlikely as to be practically impossible. But Mandelbrot鈥檚 study of real data showed that in financial markets, radical price swings occur much more frequently than the 鈥渢ails鈥� of the normal distribution would predict. In retrospect, perhaps this isn鈥檛 so surprising, for as investors know, the most interesting price swings are the big ones. Without an adequate supply of booms and busts, the market would lose much of its attraction for speculators.

Prior to Mandelbrot鈥檚 observations, a number of financial investigators in the late 1950s had tried various statistical manoeuvres and mathematical contortions in their attempts to account for the extremes of price changes without ditching Bachelier鈥檚 Gaussian assumption. But Mandelbrot pointed out that despite its inability to account for extreme fluctuations, the assumption serves well as a description for most price shifts. So, he argued, let鈥檚 look for a probability law similar to the Gaussian but with an infinite, rather than finite, variance. Put another way, he was looking for a curve similar to the bell-shaped curve, but with tails that spread out and fall to zero much more slowly.

Mathematics miner

As it turned out, Mandelbrot didn鈥檛 have far to look. In fact, he found that there was already an entire family of them, called the L茅vy stable laws. Paul L茅vy, a contemporary of Bachelier鈥檚, was a French mining engineer and mathematician who invented an alternative theory of probability. His family of probability distributions is characterised by a single parameter D, which ranges between 0 and 2. Every number in this range corresponds to a different probability distribution. The Gaussian distribution, for example, is given when D is 2. But there are infinitely many others. Equally important is the fact that the Gaussian is the only member of the family that has a finite variance; for all values of D other than 2, the corresponding probability distribution has an infinite variance. A fluctuating process with infinite variance is much more prone to very large negative or positive swings, because its distribution is 鈥渇atter鈥� and has bigger tails.

The L茅vy stable laws gave Mandelbrot a huge number of candidates to play with in the search for probabilistic rules that would describe market fluctuations, especially the large ones. He applied them to fluctuations in cotton prices, because those were the data he had at hand. And sure enough, he found that one member of the L茅vy stable family-with D equal to 1.7-fitted the data spot on, much better than the normal distribution. Had he then looked at other markets, such as those trading in stocks or foreign currency, he would have found near perfect fits with L茅vy laws having other values of D.

Speculative markets of all types have large fluctuations more frequently than you might expect, and L茅vy statistics are perfectly suited to describe them. For example, in 1995 Gene Stanley and Rosario Mantegna of Boston University studied fluctuations in the Standard and Poor鈥檚 500 index, which reflects changes in the New York Stock Exchange. Looking at the value of the S&P every minute or so for five years-some 1 447 514 values in all-they, too, found the definitive signs of L茅vy statistics.

Strange rhythms

But money markets are not the only systems displaying L茅vy-type behaviour. These weird statistics also crop up unexpectedly in the unpredictable rhythms of familiar things like heartbeats and leaky taps. Even the wandering albatross seems to know about them.

Albatross don鈥檛 look for food on land, but at sea. When foraging in the South Atlantic, for instance, they occasionally land on the water to eat or rest. In a paper published last May in the journal Nature, Stanley and his colleagues reported their findings regarding the statistics of the birds鈥� flying and landing patterns. Sure enough, their foraging follows L茅vy-type statistics. It turns out that the probability that an albatross flies for a time t before again landing is proportional to 1/(t+1)². This is a typical L茅vy distribution, with 鈥渇at,鈥� slowly decreasing tails, showing that long flights, although less likely than short ones, still occur fairly often. The diagram above left shows a mock-up of an albatross flight path consistent with the measured flight-time statistics. Mostly, the birds flit about randomly, but they make frequent long excursions to new areas. The prevalence of these longer flights is what distinguishes a L茅vy-flight process from normal Gaussian random motion.

It may seem bizarre that wandering albatross should mimic the behaviour of flighty investors in speculative markets. But at the University of Texas, Harry Swinney and his co-workers have stumbled upon these strange statistics somewhere else as well-in the way fluids flow. In their experiments, they pumped a fluid from below into a rotating circular channel. Imagine a pig鈥檚 watering trough, bent around to connect to itself end to end, and then set spinning. As a result of this motion, the fluid in the trough formed six swirling vortices spaced equally around the trough. Swinney and his colleagues placed tiny dust particles into the fluid and followed their motion as they were carried along. They found that the particles tended to spend time bouncing around in the vicinity of one vortex, but then occasionally they would suddenly jump over to another, and begin bouncing around there. The pattern, like that of the albatross鈥� flight, suggests a L茅vy random motion in two dimensions. Swinney鈥檚 group also noticed that each time you let a dust particle go, it traces a different path. Measuring the scatter, or variance, of the dust particle鈥檚 position after longer and longer times, they found that it increased in exactly the way expected for L茅vy-like motion.

All these results raise an obvious question. Why is L茅vy motion so common in natural and human phenomena? And some old mathematics suggests an answer. L茅vy distributions turn up when we ask the following simple question. Suppose each step of a random walk has length x, and that each value of x occurs with probability p(x). The question is, when does the probability of the distribution of distances travelled after N steps obey the same distribution as the length of a single step, p(x)? This question is really one of fractals, since we are asking when the whole looks like its parts.

It turns out that the sum of N Gaussian-distributed variables (the N steps, where the length of each one follows a Gaussian distribution) is again a Gaussian distribution. So the Gaussian distribution is one possible answer for p(x). What L茅vy discovered early this century is it is not the only answer. The L茅vy stable laws all follow the same rule. So in this respect, the two kinds of distributions are extraordinarily similar.

Powerful laws

But they are also very different. In a Gaussian random walk, the steps have a typical, 鈥渁verage鈥� size, and fluctuate about this average with a finite variance. But in a L茅vy random walk, or in a two-dimensional L茅vy flight, there is no characteristic size for the lengths of the movements, and the variance is infinite. For L茅vy motion, the length of each step is distributed according to what mathematicians call a 鈥減ower law鈥�: p(s) = 1/KsD, where s is the length and K and D are numbers that remain fixed (see 鈥淲hen size doesn鈥檛 matter鈥�). If steps are distributed in this way, then there is no 鈥渢ypical鈥� step size. Essentially, the steps come in all sizes, and it is impossible to predict even roughly the size of next step.

L茅vy鈥檚 theorem implies that if a huge set of small events all contribute to make one larger one, then if the small events have finite variance, the larger one will follow the normal, Gaussian distribution. But if the statistical distributions of the small, contributing events have infinite variance, then so will the resulting distribution for the larger, cumulative event. It will follow a L茅vy stable law. It鈥檚 straightforward mathematics. Lots of random events together lead to either the 鈥渘ormal鈥� distribution, or to a L茅vy distribution. There is no other choice.

So there is a good reason why L茅vy鈥檚 irregularities are popping up all over-in markets, in fluid flows and in the flight patterns of birds. Even meandering ants seem to take their cues from L茅vy. These 鈥渨eird鈥� statistics are just as normal as the 鈥渘ormal鈥� ones-more so even. So before you invest your money, take a while to watch critters rambling through the garden. You may learn a valuable secret or two.

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When size doesn鈥檛 matter

A mathematical curve can have essentially any shape. But out of all the possibilities, curves of a special kind are known as 鈥減ower laws鈥�. In these curves, the height is inversely proportional to the distance raised to some power. One such curve, for example, has the formula height = 1/(distance)². The formula doesn鈥檛 have to involve a 2. If p is any nonzero number, then height = 1/(distance)^p is a power-law curve too. The choice p = 2 is just one possibility.

For mathematicians and physicists, what is fascinating about power law curves is that they have no 鈥渋ntrinsic scale鈥�. To see what this means, suppose that you didn鈥檛 know the shape of a curve, and began making measurements to find out. If you were a very small being, you would use a very short ruler to measure heights and distances over a small portion of the curve.

Suppose you found that it followed a power law with p = 2. Now, if you were a big being measuring the same curve, you鈥檇 use a much longer ruler, and as a result measure over a much larger portion of the curve. But you would still come to the same conclusion-it鈥檚 a power-law curve with p = 2. This isn鈥檛 true of other mathematical curves, such as the 鈥渂ell-shaped鈥� Gaussian curve that is used in the normal distribution of statistics. Using bigger or smaller rulers to find the shape of that curve would yield completely different equations, because it looks different when viewed on different scales.

It is this 鈥渟cale invariance鈥� that makes power laws interesting. For if they turn up in the study of a particular phenomenon-like the foraging behaviour of albatross, for instance-it immediately implies that self-similarity is at work. For the albatross, it means that the birds鈥� eating and flying patterns over short times look much like they do over long times.