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In 1963 Benoit 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 (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward. 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 used by the time of the 2008 financial crisis ruled out these extreme variations and considered they can only happen every 10,000 years. An obvious conclusion from Mandelbrot's work is that greater regulation in financial markets is indispensable. Other contributions of his work for the study of stock market behaviour are the creation of new approaches to evaluate risk and avoid unantecipated financial collapses.[4]
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