How do you measure the rough and jagged coastline of the United Kingdom? Or the sharp, seemingly arbitrary rise and fall of a stock-price? To the layperson, the answer to the first question might seem a straightforward matter of getting on a boat and making a trip. 1 The answer to the second question might be observing the markets for long periods of time and trying to discern patterns within the graphs (much like technical analysts do today). However, mathematicians aren’t known for their love of fieldwork. This is the story of a rogue mathematician’s search for an answer to questions like these, questions which have to do with how we measure ‘roughness’ in the world around us: from the sharp edges of a stock price graph to the uneven surface of a cauliflower. 1 It tells the story of how different kinds of ‘roughness’ can be described by different kinds of statistical distributions, and how we may have been using the wrong distribution to price our bonds and derivatives all along.
Mandelbrot’s Story: From World War II France to ‘60s America
Mandelbrot was born in Warsaw in 1924 and privately tutored by an uncle who despised rote learning. His first exposure to algebra and his self-discovery as a mathematician followed his family’s relocation to France in 1936. After being relegated to the countryside at the onset of war in 1944, he was hidden by French resistance members in a Lyon school. For every question the professor asked, Mandelbrot would describe a geometrical approach to yield a fast, simple solution. He passed in this way through a series of elite French universities as well as Caltech. He returned to Paris for his PhD, then proceeded to the Institute for Advanced Study. It was during these heady years that Mandelbrot developed the fractal, to which we now turn. 1
Fractals: An Introduction to Mandelbrot’s Beautiful Creations
The concepts behind the math Mandelbrot developed during those years and which he applied to fields as varied as economics and thermodynamics are easy to learn, and the results are often visually stunning. Perhaps the most important concept for our purpose in fractal geometry is that of the fractal dimension. Since the time of Euclid, a point in any space has had no dimension. A straight line has had one. A curve has had two and so on. But what about a fractional dimension? The utility of such a concept is that, unlike integer-valued dimensions, it can be used to measure roughness. 1,2
Think about is this way: Measuring a single line requires one ruler, while approximating a curve requires many smaller rulers. A rougher figure, such as the British coastline, requires even more. There is no single answer to the question of how long the coastline is. Unlike the smooth curve, the rough coastline does not provide one best estimate of length. Your answer to the question depends on the scale of the map you are drawing. 1
Mandelbrot’s solution is startlingly simple and very illuminating: for each ‘ruler’ you use, write down the estimate of the length of the coastline you get. Then halve the size of the ruler. Write down the new, increased estimate. And so on. Observing the results of this exercise reveals a startlingly simple truth: the length we are recording increases at a more or less stable rate! This rate is Mandelbrot’s fractal dimension. 1
Bachelier And Those Who Came After
The most important effect of Mandelbrot’s math was to rework how financial theorists and traders viewed price change and volatility. To understand this effect and how radical it was, we must travel back in time, once again, to Paris. The year is 1900. The French mathematician Louis Bachelier, in his PhD thesis ‘Theorie de la Speculation’ asked the million-dollar question: ’How do bond prices move?’ He made an ingenious observation: the volatile pattern with which heat moved from particle to particle in a physical system was like the volatile pattern of bond price movement. In both systems it was possible to derive a probability distribution that broadly described its behavior.1,2,3
With this in mind, Bachelier tried to calculate the odds of a bond price going up or down. He assumed that there were two viewpoints to consider: after the event had occurred and before the event had occurred. After the price change, it was possible to attribute some form of cause and effect to the price change. But before the event had occurred, traders assumed that the market had already taken account of all relevant information and that demand would equal supply. Unless new information about a bond emerged, there was no reason to assume its price would change. The next change was as likely to be up as down. In essence, prices followed a random walk. Each price fluctuation is independent, driven by a mysterious statistical process that drives markets. Bachelier discovered that if one were to plot the price changes of a bond over a period of a month or a year, the many small price changes would cluster in the middle, while the fewer large price changes would be at the edges: a Gaussian distribution, so ubiquitous in modern statistics that it is known as the ‘normal curve.’ This opened up the standard mathematical toolkit for use in finance. 1,3
Later theorists built up a towering structure based on the assumption that bond prices were like a fair game where the long-run payoff is zero. Eugene Fama of the University of Chicago formulated the efficient markets hypothesis: a stock price always reflects all relevant information that the market has to offer. Many advances were made in deriving pricing mechanisms for risk and assets based on the idea that each price change was independent of the last. Modern finance as we know it emerged. 1
Challenging Bachelier: Mandelbrot’s New View of Volatility
There is a very old and very simple game that mathematicians like to play. Two brothers, Harry and Tom, bet on the toss of a coin. Each toss is pure luck. Harry wins one Swiss franc on every heads. Tom wins one Swiss franc on every tails. If they play the game long enough, probability theory says that their payoffs will converge to some expected value. In this case, each brother would expect to win half the time. 1,2
But other aspects of the game get more complicated. At any given moment in the game, either Harry or Tom might have accumulated far more winnings than the other. Willey Feller, author of a widely used 1950s college textbook on probability, actually graphed each brother’s winnings over 10,000 coin tosses. Most readers did not pay this graph any attention, but Mandelbrot did: he found that the times at which each brother’s purse emptied out were clustered together. It was an irregular pattern: a few long, up and down cycles, with shorter cycles riding on top of them: it seemed that price changes were not independent of one another, which meant that the efficient markets hypothesis (which assumed a ‘random walk’) was based on a fallacious assumption, as were Black-Scholes’ equations. 1
The French maverick mathematician discerned an important pattern: he found that the irregular, rough pattern in the many pennies game implied a different kind of randomness, a kind of randomness wilder than the politely shaped Gaussian distribution. 1
It was the winter of 1961. When he made this discovery, Mandelbrot was at IBM, studying income distribution patterns between the rich and poor. The Harvard economics department invited him to speak about his work. He walked into the office of his host that day to a surprise. On the chalk-board was a figure with a convex shape that opened to the right. He immediately turned to Professor Hendrik Houthhakker and asked why his diagram was already drawn. Houthhakker was perplexed: ‘These are graphs of cotton prices.’ 1,2
The puzzling similarity in pattern between income distribution and cotton prices got Mandelbrot thinking. Was it pure coincidence that the two were spitting images of one another, or was there a deeper truth in the strange connection between the two pictures? And so it was that Mandelbrot was propelled into investigating the mysteries of finance. 1,2
Different Kinds of Randomness as Different Kinds of Volatility
The difference between a Gaussian distribution and the kind of distribution that the shape of the Feller graph implied for Mandelbrot can be demonstrated aptly with the following parable. Imagine an archer shooting between two horizontal lines drawn in white paint on a wall 100 yards away. In the first scenario, the archer is blindfolded. Naturally, his arrows are all over the place. The statistician sitting safely behind the archer graphs each of the archer’s shots on a frequency vs. distance from target graph. The pattern is rough and irregular: a Levy distribution. Now imagine scenario two. The same archer minus the blind-fold. He obviously manages many more shots closer to the target, and comparatively fewer shots far away from the target. This time, the statistician’s graph looks far more familiar: a normal curve, a Gaussian bell.1
Mandelbrot compared each distribution from the above scenario to the Feller graph. He found that, over a long period of time such as that represented by 10,000 coin tosses, the distribution corresponded to the Levy distribution, and not the bell curve! 1
The Role of Fractals: Conclusion
Mandelbrot began, in a series of subsequent papers, to measure the roughness of different Levy distributions (as bond prices, of course) using fractals. And so it came to be known, at least in a select circle of the academic world, that the fractal dimension of a Levy distribution is a far more accurate measure of volatility than a Gaussian distribution. 2
While Mandelbrot’s finding may seem innocuous to some, the implications are profound. What is very, very unlikely for a system that follows a Gaussian distribution is far more so for a Levy distribution. This means that recessions, big price changes, and all the other things which modern financial theory posited were ‘six-sigma’ events, events that shouldn’t have happened, are the results of models based on inaccurate assumptions. These models, Mandelbrot’s body of work suggests, have caused us to misperceive risk in a dangerous way. His work is a potential explanation to unusual market volatility: it suggests that our notions of ‘usual’ might be incorrect. There are academics who have taken up the baton from Mandelbrot, who died last October. These scholars work to build fractal descriptions of markets, models that take into account the Levy distribution. It remains to be seen how long it will take Wall Street to begin using these Levy-based models. 1
References:
- Benoit Mandelbrot and Richard Hudson, ‘The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward,’ Basic Books, August 2004
- Benoit Mandelbrot, ‘The Variation of Certain Speculative Prices,’ The Journal of Business, Vol. 36, No. 4, October 1963, pp. 394-419
- Louis Bachelier (Author), Mark Davis (Translator), and Alison Etheridge (Translator), ‘Louis Bachelier’s Theory of Speculation: The Origins of Modern Finance,’ Princeton University Press, September 2006
- Benoit Mandelbrot, ‘Fractals and Scaling in Finance: Discontinuity, Concentration, Risk,’ Springer, 1997
- Image (CC-BY-SA): Ken Teegardin, “Graph with Stacks of Coins,” Flickr, taken March 4, 2011, accessed April 2, 2012, http://www.flickr.com/photos/teegardin/6093690339/.
- Image (CC-BY-NC-ND): Jack Keene, “Cracked,” Flickr, taken July 5, 2006, accessed April 2, 2012, http://www.flickr.com/photos/whatknot/2804669838/
- Image (used with permission): Jez Liberty, “Why Trend Following Works: Look at the Distribution,” Au.Tra.Sy blog — Automated Trading System, last modified October 21, 2009, accessed April 2, 2012, http://www.automated-trading-system.com/why-trend-following-works-look-at-the-distribution/
Akshat Goel is a second-year student at the University of Chicago majoring in economics and sociology. Follow The Triple Helix Online on Twitter and join us on Facebook.