MATH OR ART?: This Mandelbrot set with colored environment. Each pixel is associated with a certain sequence of complex numbers. The index for which the absolute value of all following numbers exceed 1000 increases from each colored stripe to the next towards the Mandelbrot set by the amount of 1.
Image: Created by Wolfgang Beyer with the program Ultra Fractal 3.
Editor's Note: This story was originally published in the February 1999 edition of Scientific American. We are posting it in light of recent news involving Lehman Brothers and Merrill Lynch.
Individual investors and professional stock and currency traders know better than ever that prices quoted in any financial market often change with heart-stopping swiftness. Fortunes are made and lost in sudden bursts of activity when the market seems to speed up and the volatility soars. Last September, for instance, the stock for Alcatel, a French telecommunications equipment manufacturer, dropped about 40 percent one day and fell another 6 percent over the next few days. In a reversal, the stock shot up 10 percent on the fourth day.
The classical financial models used for most of this century predict that such precipitous events should never happen. A cornerstone of finance is modern portfolio theory, which tries to maximize returns for a given level of risk. The mathematics underlying portfolio theory handles extreme situations with benign neglect: it regards large market shifts as too unlikely to matter or as impossible to take into account. It is true that portfolio theory may account for what occurs 95 percent of the time in the market. But the picture it presents does not reflect reality, if one agrees that major events are part of the remaining 5 percent. An inescapable analogy is that of a sailor at sea. If the weather is moderate 95 percent of the time, can the mariner afford to ignore the possibility of a typhoon?
The risk-reducing formulas behind portfolio theory rely on a number of demanding and ultimately unfounded premises. First, they suggest that price changes are statistically independent of one another: for example, that today’s price has no influence on the changes between the current price and tomorrow’s. As a result, predictions of future market movements become impossible. The second presumption is that all price changes are distributed in a pattern that conforms to the standard bell curve. The width of the bell shape (as measured by its sigma, or standard deviation) depicts how far price changes diverge from the mean; events at the extremes are considered extremely rare. Typhoons are, in effect, defined out of existence.
Do financial data neatly conform to such assumptions? Of course, they never do. Charts of stock or currency changes over time do reveal a constant background of small up and down price movements—but not as uniform as one would expect if price changes fit the bell curve. These patterns, however, constitute only one aspect of the graph. A substantial number of sudden large changes—spikes on the chart that shoot up and down as with the Alcatel stock—stand out from the background of more moderate perturbations. Moreover, the magnitude of price movements (both large and small) may remain roughly constant for a year, and then suddenly the variability may increase for an extended period. Big price jumps become more common as the turbulence of the market grows—clusters of them appear on the chart.
According to portfolio theory, the probability of these large fluctuations would be a few millionths of a millionth of a millionth of a millionth. (The fluctuations are greater than 10 standard deviations.) But in fact, one observes spikes on a regular basis—as often as every month—and their probability amounts to a few hundredths. Granted, the bell curve is often described as normal—or, more precisely, as the normal distribution. But should financial markets then be described as abnormal? Of course not—they are what they are, and it is portfolio theory that is flawed.
Modern portfolio theory poses a danger to those who believe in it too strongly and is a powerful challenge for the theoretician. Though sometimes acknowledging faults in the present body of thinking, its adherents suggest that no other premises can be handled through mathematical modeling. This contention leads to the question of whether a rigorous quantitative description of at least some features of major financial upheavals can be developed. The bearish answer is that large market swings are anomalies, individual “acts of God” that present no conceivable regularity. Revisionists correct the questionable premises of modern portfolio theory through small fixes that lack any guiding principle and do not improve matters sufficiently. My own work—carried out over many years— takes a very different and decidedly bullish position.
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28 Comments
Add CommentMany of the illustration links are broken.
Reply | Report Abuse | Link to thisThis article is completely useless: the random nature of stock markets has been recognized for some time, and a fractal description provides no useful information.
Reply | Report Abuse | Link to thisThe article is far from useless. Randomness does not imply a lack of structure. The modern portfolio analysis software for valuing derivatives assumes a normal distribution of portfolio motion. That is, changes tend to be small, and large changes are increasing infrequent with a rapidly falling percentage. Mandelbrot is arguing in this article that this assumption is wrong. The appropriate statistics should be fractal statistics in which large moves are much more likely to interrupt long stretches of smaller movements. In other words, the pricing models are bogus, and in light of current events, this seems to be true. (No, this article does not tell you how to beat the market, just how to evaluate certain pricing strategies.)
Reply | Report Abuse | Link to thisInternet traffic also has fractal statistics which can lead to serious bursts of traffic and make it hard to build enough capacity to manage peak traffic. Telephone systems were designed using what are called Erlang statistics which are smoother. These were used in the early days of network design, and this led to lots of slow downs and sometimes router crashes. More modern models provide for better performance using better statistics.
Statistics are useful in a predictive capacity: for instance the statistical mechanics of gas molecules allow us to predict the response of gases to changes in temperature and pressure. The statistics of chaotic systems can lead to the identification of attractors, leading to an assessment of the probability that the system will have a particular value (although it will not tell you WHEN that value will occur). Fractals do not help in any sense to predict the likelihood of market behaviour, on the other hand studies of social phenomena that lead to flocking in birds show promise.
Reply | Report Abuse | Link to thisit is shame that sciam started to publish this kind of articles. are edtors lost good sense of science?
Reply | Report Abuse | Link to thisthe power of every theory is in its prediction and our ability to act on this prediction.
i. schagaev
or is is deliberate misleading?
I wonder about fractal models. It seems to me that all they tell us, is that we can take a complex phenomenon and approximately compress it. Does the compressed representation tell us anything much about the original?
Reply | Report Abuse | Link to thisWe have criminals in charge and they ran a pyramid scam on the American people disguised as a real estate "boom". You don't need a complicated mathematical model to predict how a pyramid scam will end, unless you're stupid.
Reply | Report Abuse | Link to thisexactly! thanks for this. i am bit tired from pseudo-scientific papers....
Reply | Report Abuse | Link to thisI suggest you look into Elliott Wave Analysis, and remember to factor in 1) expectation, 2) mass human psychology, and 3) perhaps part of 1 & 2, greed.
Reply | Report Abuse | Link to thisJim C. La Mesa, Calif.
For more and better information abour fractals please have a look at http://www.fractal.org
Reply | Report Abuse | Link to thisJules Ruis.
Failure of regulators to control runaway behavior of a few smart guys playing with the market using too many layers of derivatives to camouflage and present a complex financial structure building up mythical pyramids of wealth - ended in the individual players making tons of money. Why no one questions the regulators? The solution again is to make the tax payer pay for the failure of the regulators whom also the tax payer funded. Well, knowingly a magical feel-good- feeling was allowed to prevail without sound financial diligence checks and the entire financial world on the planet are now questioning the maturity of managing markets in USA, the largest economy built by sheer hard work over decades by people believing in simple values of following ethics in business. Fraudsters are getting rewarded now.
Reply | Report Abuse | Link to thisNone of the illustration links work on this article, and even moving from one page to the next sometimes draws a blank. For a magazine about science, your web design seems embarrassingly lacking in technical quality. And my computer is not at fault - it accesses stuff faultlessly on more technically comples sites.
Reply | Report Abuse | Link to thisThis article does not explain price fluctuations, it describes them
Reply | Report Abuse | Link to thisMost of our scientific knowledge fail to explain the why part-but only try to describe and fail rigorous test of cause and effect relation ships. Surprisingly logic has the fallacy described as post hoc ergo propter hoc which roughly means after this, therefore because of this...But all our knowledge if you introspect deeply, falls under this category. Trends can be foreseen but not spikes and timing. But what can be monitored is the ethical governance which is within human control, and which when we over look on a massive scale, can result in the current stock market melt down. Instead of describing the symptoms accurately with mathematical models, if we look for underlying driving forces and how the initial innocent belief that market forces will work in unbiased manner FAIRLY, is not proven sound and we do need HONEST regulation, not human based but may be digitally enforced ruthlessly, all will be better off. Interestingly if you mandate a group of us to lay down a process, we all do it very well but when one or two of us are asked to implement the same, discretion and human weaknesses allow the process to be compromised initially as one time exception and then the precedence slowly opens the flood gates of vitiating the process totally! Entropy is real in human ethical behavior while once we accept the process to be implemented by un-emotional silicon chips, this danger is eliminated. Yes some will crib that creativity will be stunted. But we observe that creativity is always viewed with suspicion in all other human endeavors except financial world, unless proven beyond doubts by the society, to protect itself. Society hence must now learn to restrict the over zealous and daily creativity obsessed boys walking out of our business schools through electronically enforced ethical practices. Human regulators fail for obvious reasons. No amount of covering tracks explaining away by mathematical models based on different assumptions can replace simple observance of basic value based systems and enforcement in the markets for balanced and sustainable growth. Else this bleeding periodically appears the only way to correct our thinking and after sometime we go back to our old ways till next bleeding session.
Reply | Report Abuse | Link to thismight as well just write off the market's volatility to brownian motion. or perhaps just that bad news travels WAY faster than good news and the slightest hint of bad news sends investors heading for the SELL in a huge hurry.
Reply | Report Abuse | Link to thisperhaps greenspan's presence in the fed was better for the markets. when financial sector businesses like lehman are dropping like flies, something is clearly wrong with regulators' enforcement and governmental fiscal policy. please explain to me how we fit that into mathematical fractal equations of any kind.
Reply | Report Abuse | Link to thiswell said!!
Reply | Report Abuse | Link to thiswell said
Reply | Report Abuse | Link to thisTom, Yountville, Calif
Well, fractals explain many things in nature. It's time staticians start using imaginary numbers in their prediction models.
Reply | Report Abuse | Link to thisWhat are the odds of a company succeeding vs. a company going bankrupt anyhow?
Reply | Report Abuse | Link to thisWhy do people always set up ridiculous "straw men" about economics and then, surprise surprise, tear them down ? Why do they make preposterous claims about what economists believe, insulting the intelligence of many serious scholars ? If Mandelbrot were onto something he wouldn't be publishing in Scientific American, sorry.
Reply | Report Abuse | Link to thisWith the end of the George W Bush era, hubris is going out of style. Get with it Mandelbrot.
Oh yeah, the financial crisis is evidence market participants took excessive social risks for personal profit, at no personal risk. Not necessarily evidence the models are fundamentally flawed.
Reply | Report Abuse | Link to thisWhen Mandelbrot says something like "statisticians don't like invariants," I wonder if he took a random sample of statisticians to find that out.
Mandelbrot kind of reminds me of the "fuzzy math" guy who has a big chip on his shoulder that the academic community doesn't think he has a very deep result and should garner more respect.
Enjoyable, nicely written. In fact it iselequentely written.
Reply | Report Abuse | Link to thisI believe that the links to Illustrations 3 and 4 are, indeed, the same as to Illustration 2, just further down the page.
Reply | Report Abuse | Link to thisI agree with Kaleberg. I used to be involved with statistical analysis of bond portfolios. We would run 200 random-walk scenarios based on the log-normal distribution. After reading this article back in 1999, we discussed trying fractals, but we did not have the software support to accomplish it.
Reply | Report Abuse | Link to thisIgor, you said prediction??
Reply | Report Abuse | Link to thisthis is one of the thing not possible for human being.
trying to predict financial markets is sure the wrong way to go
prediction ??
Reply | Report Abuse | Link to thisI am amazed at how much the economy continues to struggle. I think this could be why <a href="http://burtonwrightresidential.org">new life services</a> are becoming so successful. People need a new start and need it now.
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