Understanding Instability: Mandelbrot, Fractals, and Financial Crises

Alejandro Nadal

Lightning in the sky does not follow a straight line. The irregular patterns in a cauliflower or the capricious forms of a tree’s branch are a challenge to the clean geometric figures we learn in school. Neither the straight lines, nor the smooth curves of that geometry exist in nature. But after the wonderful work of Benoit Mandelbrot it is now possible to get closer to a theory of the manifold wrinkles and rough surfaces that are the stuff of our universe. And our economies.

Ten days ago this great mathematician, the creator of fractal geometry and other wonders closely related to chaos theory, passed away.

The word fractal, coined by Mandelbrot, denotes a logical semi-geometric figure that can be divided as many times as desired and every time you zoom in on these smaller fractions you end up looking at a replica of the original figure. The best example of this is the famous Koch snowflake, in which the wrinkles are intimately related to patterns of affinity between the parts and the whole. Another example is the cauliflower: no matter how many times one breaks it up, when the pieces are magnified, the same ruggedness and wrinkles of the whole reappear. The property of self-similarity emerges even in the tiniest crumbles.

In 1963 Mandelbrot analyzed the variations of cotton prices on a time series starting in 1900. There were two important findings. First, price movements had very little to do with a normal distribution in which the bulk of the observations lies close to the mean (68% of the data are within one standard deviation). Instead, the data showed a great frequency of extreme variations. Second, price variations followed patterns that were indifferent to scale: the curve described by price changes for a single day was similar to a month’s curve. Surprisingly, these patterns of self-similarity were present during the entire period 1900-1960, a violent epoch that had seen a Great Depression and two world wars.

Mandelbrot used his fractal theory to explain the presence of extreme events in Wall Street. In 2004 he published his book on the “misbehavior” of financial markets. The basic idea that relates fractals to financial markets is that the probability of experiencing extreme fluctuations (like the ones triggered by herd behavior) is greater than what conventional wisdom wants us to believe. This of course delivers a more accurate vision of risk in the world of finance.

The central objective in financial markets is to maximize income for a given level of risk. Standard models for this are based on the premise that the probability of extreme variations of asset prices is very low. These models rely on the assumption that asset price fluctuations are the result of a well-behaved random or stochastic process. This is why mainstream models (such as the infamous Black-Scholes model) use normal probabilistic distributions to describe price movements. For all practical purposes, extreme variations can be ignored.

Mandelbrot thought this was an awful way to look at financial markets. For him, the distribution of price movements is not normal and has the property of kurtosis, where fat tails abound. This is a more faithful representation of financial markets: the movements of the Dow index for the past hundred years reveals a troubling frequency of violent movements. Still, conventional models rule out these extreme variations and consider they can only happen every 10,000 years!

An obvious conclusion from Mandelbrot’s work is that greater regulation in financial markets is indispensable. Today, three years after the global crisis erupted, reforms for the financial system are clearly insufficient (whether in the guise of the Volcker rule or the Dodd-Frank act).

Mandelbrot confirmed what we know about the instability of markets, especially financial markets. But his analysis didn’t focus on the causes of this instability or the origins of financial crises. Many toy with the idea of applying Mandelbrot’s fractals to economic analysis. But for all those that look at fractals, chaos theory, complex and non-linear systems, it is perhaps useful to recall Koopmans’ admonition (Three Essays on the State of Economic Science): economists must forge the concepts of their discipline before going to look for new mathematical tools.

Mandelbrot was not an economist. Perhaps this explains why he tried to study how markets really work. This is better than the flawed 200-year-old research program of an economic theory obsessed with the aim of proving (fruitlessly) that markets are efficient. In the context of today’s crisis, Mandelbrot opened the window and let some fresh air in.

The beauty of fractal geometry knows no boundaries. Perhaps it was already inscribed in the first verses of Blake’s famous poem Auguries of Innocence:

To see the world in a grain of sand,
And a heaven in a wild flower;
Hold infinity in the palm of your hand,
And eternity in an hour.