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Benoît Mandelbrot – poet in a foreign language (2): Five rules

2012/11/07

The central theme of Benoît Mandelbrot’s work was to find a regular pattern in irregularity and roughness as he described in the videos listed here and in his various books and articles. His mathematics, the fractal geometry, found applications in many different fields. He wrote (p.5 f.):

“It has helped model the weather, study river flows, analyze brainwaves and seismic tremors, and understand the distribution of galaxies. It was … embraced as an essential mathematical tool … by “chaos” theory, the study of order in the seeming-chaos of a whirlpool or a hurricane. It is routinely used today in the realm of man-made structures, to measure Internet traffic, compress computer files, and make movies. It was the mathematical engine behind the computer animation in the movie, Star Trek II: the Wrath of Khan.”

Mandelbrot had a strong interest in economics. He started his empirical research studying daily prices of cotton and other commodities for which very long time series from 1900 onward were available (only long series allowed to explore the patterns he was looking for, and in pre-computer times those were rare), and some of his ideas later became orthodoxy in financial markets worldwide. He raised the awareness of the phenomen of “fat tails” everybody is talking about these days. On the other hand, economists were among his greatest critics as many of his results stood in sharp contrast to their world view.

In the first chapter of his book with Richard L. Hudson on The (mis)Behavior of Markets Benoît Mandelbrot summarised some of his insights into how markets work in five “rules” of market behaviour:

  1. Markets are risky.
  2. Market turbulence tends to cluster.
  3. Markets have a personality.
  4. Markets mislead.
  5. Market time is relative.

Let us have a short look at what is meant by these rules in order to understand both sides of the debate.

Markets are risky

Without further explanation every economist would subscribe to the first rule. Markets ARE risky. But, traditionally, economists regard volatile price movements only as deviations from a “norm”. They see them as stochastic “shocks”, “disturbances” or “news” which enter the picture as mere additions to a deterministic structure which can be explained by fundamental economic developments. Under the assumption that price variations are normally distributed, risks are measurable and foreseeable and thus can be tamed. As Mandelbrot explains:

Most changes, 68 percent, are small moves up or down, within one  ‘standard deviation’ – a simple mathematical yardstick for measuring the scatter of data – of the mean; 95 percent should be within two standard deviations; 98 percent should be within three. Finally … extremely few of the changes are very large. If you line all these price movements up on graph paper, the histograms form a bell shape: The numerous small changes cluster in the center of the bell, the rare big changes at the edges.” (Mandelbrot and Hudson, p. 10)

Take currencies, for example: According to the traditional view, there are fundamental exchange rate movements which are determined by foreign trade and other cross-border economic activities, and there is the daily “noise” produced by the foreign exchange markets. Both overlap, but can be studied separately.

However, in the real world the statistical properties of exchange rate movements, as well as many other financial time series, show “anomalies” which are not compatible with this view. Observed price jumps are too large to fit into the picture of “normality”. When on Monday, 23 September 1985, the day after the Plaza Agreement, the US dollar fell against the D-mark by 5.75 per cent, under the normal distribution a change of this magitude would be expected to occur only once in about 70,000 years (Kaehler 1991). Since then, the world has seen many similar and even more dramatic variations, not only in foreign exchange markets. To cite Mandelbrot once again:

“From 1916 to 2003, the daily index movements of the Dow Jones Industrial Average do not spread out on graph paper like a simple bell curve. The far edges flare too high: too many big changes. Theory suggests that over that time, there should be fifty-eight days when the Dow moved more than 3.4 percent; in fact, there were 1,001. Theory predicts six days of index swings beyond 4.5 percent; in fact, there were 366. And index swings of more than 7 percent should come once every 300,000 years; in fact, the twentieth century saw forty-eight such days.” (Mandelbrot and Hudson, p. 13)

Apparently, large price jumps are the norm. They are part of the underlying “fundamental” dynamics and not an unwanted add-on, aberration or “disturbance”.  (For those who want to try: A detailed recipe how to calculate the cited numbers can be found in Mandelbrot and Hudson, p. 279, note 4.)

Maybe this is one of the most misunderstood aspects of Mandelbrot’s work among economists these days. The conclusion some of them draw, in short, seems to be: “Ok, there are “fat tails”. Disturbances are much larger and more frequent than formerly assumed. We must take account of that.” But, it is not only disturbances that need reconsideration. There is no underlying “fundamental” structure which remains intact and unaffected by these other dynamics. Both short-term price jumps and long-term movements are two sides of the same phenomenon.

Market turbulence tends to cluster …

… or as Mandelbrot writes: “Trouble runs in streaks”. Many financial price series have a “memory”. Big or small price changes are followed by more of the same kind. A trading day which starts volatile will often remain so.  A calm day may be followed by others of few price variations. Market unrest in Tokyo may continue in London and Frankfurt in the morning before spreading to New York in the afternoon and eventually coming back to Japan for no visible reason.

This observation is in contradiction to the Efficient Market Hypothesis which dates back to Eugene F. Fama (who was one of Mandelbrot’s doctoral students) and is held in high esteem in traditional economics. In a short version, according this hypothesis in an ideal market prices reflect all relevant information. Price changes, which occur only if new information becomes available, are statistically independent of each other.

Markets have a personality

Maybe this rule is even harder to accept for mainstream economists than the former ones. It states that there is an emerging property of markets beyond the effects of real-world events and news, a dynamic resulting from the way the various participants interact which is “greater than, and different from, the sum of its parts”. As Mandelbrot writes:

“In substantial part, prices are determined by endogenous effects peculiar to the inner workings of the markets themselves, rather than solely by the exogenous action of outside events.” (Mandelbrot and Hudson, p. 21)

Furthermore, this underlying internal market mechanism shows a strong persistence, largely unaffected by developments outside. The same broad patterns were found to reign independent of wars and peace, crashes and crises and a changing institutional environment.

This persistence is a phenomenon researchers identified as one characteristic of so-called dissipative complex systems in general. By definition, a dissipative system is an open system that is exposed to friction losing, to use a physics’ metaphor, some form of “energy”. Only continuous exchange with its environment is keeping it going. Financial markets are dissipative systems. Without constant interaction with their environment incentives to trade would cease to exist and they would rapidly dry up. These systems are in contrast to closed conservative ones without interaction with the outside world, where nothing is lost or added. Traditionally, in analogy to physics, economies are modelled as conservative systems (see Mirowski 1991).

Grégoire Nicolis and Ilya Prigogine gave an example contrasting the two kinds of systems in their reaction to external disturbances. Take the movement of an idealised swinging pendulum in a conservative world without friction. Give it a push and it will swing with a wider angle for all eternity. The same holds in economic models. An oil price increase or a change in government spending will have a lasting effect on the economy as long as nothing happens to offset it.

The Nicolis/Prigogine example for a dissipative system is the adjustment of a chemical to a disturbance, the so-called Belousov-Zhabotinsky reaction. There is a transition period during which a different behaviour before and after the disturbance is observed, but then the system resumes its former path. The disturbance is in a sense “digested”.  Studying the system’s behaviour before and after gives no clues to its history and tells nothing about its underlying structure and functioning. This is bad news for economic modelling and forecasting:

A conservative system would adjust to a new equilibrium after a disturbance. Its movement could be easily reversed in time by an opposite effect of equal strength. An oil price decline of the same magnitude as the former increase would turn the clock back. Thus, observing the reaction of the economy to an event in the past would make it possible to draw conclusions about its future behaviour under similar circumstances, even if the “true” underlying model is not known. Not so in markets with “personality”.

Markets mislead

This rule refers to pseudo-cycles generated by market activity which make actors believe they can predict and exploit future price changes. Mandelbrot emphasised that models of purely random processes can be used to generate charts of trends and cycles, bubbles and crashes like those observed in the real world. But, investigating time series by spectral analyses unveils that these cycles are mere “artefacts” (Mandelbrot 1969: p. 83). The periodicity seen in the data depends on the length of the sample. Different time spans of the same series exhibit cycles of different length. Nothing to bet on here.

Mandelbrot called these patterns “the fool’s gold of financial markets”. But markets believe in what they think they see. This does not only hold for “chartists” and their various, in parts highly sophisticated techniques to discover regularities, but also for “fundamentalists” looking for economic determinants of price changes. Furthermore, it holds for economists in universities and think tanks studying price dynamics and trying to separate the wheat from the chaff, namely the deterministic explanations of financial prices from the stochastic disturbances blurring their picture – in a world of pseudo-cycles and inner workings of markets independent of outside influences a rather pointless exercise.

Market time is relative

What is meant if Mandelbrot states that markets are operating on their own “trading time” which is distinct from the linear “clock time” in which we normally think (Mandelbrot and Hudson, p. 22)? Again, his focus is on patterns in price series.

Trading time, he writes, is faster than clock time when market volatility is high, and slower in periods of stability. He gives an example:

“Mathematically, I can write an equation showing how one time frame relates to the other and use it to generate the same kind of jagged price series (my emphasis) that we observe in real life.”

And:

“Professional traders often speak of a “fast” market or a “slow” one, depending on how they judge the volatility at that moment. … Likewise, a bit of market folk-wisdom holds that all charts look alike (my emphasis): Without the identifying legends, one cannot tell if a price chart covers eighteen minutes, eighteen months, or eighteen years. This will be expressed by saying that markets scale.”

Here we have the idea of self-similarity on all scales which influenced research in so many fields, but left almost no marks in economics. Fifty years after Mandelbrot wrote his article on The Variation of Certain Speculative Prices, most economists still cling to the view of market equilibrium. They calculate “equilibrium exchange rates” for the euro and US dollar, explain the extent to which currencies are “overvalued” or “undervalued” and model how policy can steer the economy out of a crisis towards a stable path. Very few joined the promising interdisciplinary strand of research on complex systems “far from equilibrium” in which, instead of point equilibria or steady states, the invariant characteristic is described by Benoît Mandelbrot’s fractal geometry. But this is another story.

Stay tuned!

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