Complex systems still largely remain a mystery. Benoit Mandelbrot gave us new language to describe the deep subtleties of these systems — some of which are reflected in this article. We can no longer entertain the illusion that we are in control. We can, however, become more adept in dealing with complexity. ISN works with clients to enhance their ability to cope with disruptive change — seeing possibilities before others do, and being agile when taking advantage of opportunities.
“Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. 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.”
Benoit Mandelbrot began his book The Fractal Geometry of Nature (1982) with these words. He is best known for his contributions to fractal geometry, the mathematics of roughness, and he coined the term ‘fractal.’ Through this work he opened the door to a more nuanced understanding of nature.
Fractals are everywhere – in the shapes of river networks, mountain ranges, coastlines, clouds, and trees; in snowflakes and crystals. Mandelbrot discovered an underlying order in patterns that on the surface looked messy and chaotic. In a similar way, he revealed a deeper order in the dynamic behavior of complex systems by studying financial markets.
Patterns in Market Behavior
Mandelbrot was a polymath who contributed to many fields. Through his analysis of markets he upended traditional models and changed our understanding of how they work. Until then, economists and financial analysts made simplifying assumptions that did not reflect actual market behavior. The ‘random walk’ models commonly used (shown on the left below) included none of the extremes that appeared in real markets (shown on the right).
Analysts used Bell curves to portray the distribution of market values, but that constrained the magnitude of possible events. The curves of power laws were more true to life – a fat-tailed distribution where extreme events were much more likely.
By simplifying reality, financiers and economists were grossly underestimating risk, and there was ample evidence to prove it. Mandelbrot cited a number of examples of extreme market behavior in The Misbehavior of Markets: A Fractal View of Risk, Ruin, and Reward (2004). On August 4, 1998, the Dow Jones Industrial Average fell 3.5 percent. Three weeks later it fell by another 4.4 percent; and on August 31, 1998, by yet another 6.8 percent. The odds of the August 31 collapse were one in 20 million. The odds of three declines like this in the same month were one in 500 billion.
“The seemingly improbable happens all the time in financial markets,” Mandelbrot wrote. “A year earlier, the Dow had fallen 7.7 percent in one day. (Probability: one in 50 billion.) In July 2002, the index recorded three steep falls within seven trading days. (Probability: one in four trillion.) And on October 19, 1987, the worst day of trading in at least a century, the index fell 29.2 percent. The probability of that happening, based on the standard reckoning of financial theorists, was less than one in 10^50 – odds so small they have no meaning.”
Mandelbrot analyzed more than a hundred years of cotton price data and discovered a fractal pattern that was similar at every scale – whether over years of daily trades or a century of monthly trades. The data also followed a power law. More interestingly, the order in which the values occurred was significant. Mandelbrot found that prices were being driven simultaneously by two forces, which he called Noah Effects and Joseph Effects.
Noah Effects are sudden disruptions. He defined a parameter called α to describe a system’s volatility – a measure of how values vary. Low-α systems exhibit wild swings.
Joseph Effects produce a sequence of values where current prices are correlated with historical prices – often far into the distant past. Mandelbrot called this ‘long memory.’ Noah Effects, it turned out, had a persistent impact.
“The signals fade in time,” he said. “But it can take months, years, or decades for a signal to become so weak and remote as to be unremarkable. Such is long-term dependence in an economy. Every event, no matter how remote or long ago, echoes across all other events.”
Mandelbrot credited British hydrologist Harold Hurst with providing an essential clue. Hurst had looked at historical flood patterns of the Nile to determine the ideal height for dams across the river. There were long stretches of high floods, other stretches of low floods, but there was no obvious periodic pattern. Existing methods had underestimated extreme events, and the required height of a dam. They did not adequately describe the river’s behavior.
What emerged through Hurst’s analysis of flood data was a factor Mandelbrot later called ‘H’ (named the ‘Hurst Exponent’ after Hurst) that described system memory. Every system has its own characteristic value. A value of 0.5 signifies no correlation between past and present (the assumption made in previous market models). A value between 0.5 and 1.0 indicates that high values are likely to be followed by another high value, and that values would also tend to be high in the future. A value from 0.0 to 0.5 indicates that a system will alternate between high and low values.
“The changes can be persistent,” Mandelbrot said, “meaning that they reinforce each other; a trend once started tends to keep going. Or they can be anti-persistent, meaning they contradict each other; a trend once begun is likely to reverse itself.” The long memory effect is present in many systems. It has also been detected in climate data.
The Palette of Change
Describing the combined impact of Noah Effects and Joseph Effects, Mandelbrot wrote, “They are two aspects of one reality. One, the other, and usually both can be read in many financial charts. They mix together like two primary colors. The red of one blends with the blue of the other, to produce an infinite palette of purples and violets. Evidence so far suggests each market… may have a different hue, a different mix of the two forms of wildness.” Each market has its own characteristic heartbeat. As he had done with fractals, Benoit Mandelbrot once again opened our eyes to the hidden order in nature.
This article is an excerpt from a book in progress on complexity, collaboration, and the challenges of transformative change. It was first posted on May 8, 2018, on LinkedIn.