The older I get, the more I find myself reveling in complexity. Buried within the overwhelming idea that the world is incredibly complex is a kernel of hope: although complexity frustrates our efforts to make predictions, we can still model it. There are lessons in models of complexity that remain hidden from those who lack the disposition to jump in the figurative pool. Without fail, these lessons are less comforting than the platitudes that are proclaimed in formal classrooms, especially at the high-school level and below. Stepping out of that intellectual comfort zone has been part of a long arc of personal growth for me.
My latest stop on that journey has been Benoît Mandlebrot’s book on finance and markets, The (Mis)behavior of Markets. I picked it up on the recommendation of professor-turned-home-math-guru Mike Lawler. The book is a little bit of economics, math (no equations, though!), history, and personal narrative all rolled into an account that challenges the orthodoxy of finance as it is taught in business schools. Mandelbrot is famous as the inventor of fractal geometry and after reading this book, I am starting to see fractals everywhere.
We’ll get back to fractals in a minute, but let’s start with the prevailing (fractal-free) wisdom. Orthodox financial theory assumes that daily changes in the price of a stock are drawn from a Gaussian or normal distribution. Another way to say this is that the “motion” of a stock price is Brownian. Two key ideas emerge from this assumption. The first is that variations in prices are independent of one another, like subsequent flips of a coin. (A patently absurd notion in practice, but “nice in theory.”) The second is that most changes in price are expected to be small. The vast majority are relatively tiny, within two standard deviations of zero. Very few are more than three standard deviations from the mean. Under the standard theory, large price swings should be extraordinarily rare.
The problem, of course, is that they’re not. Distributions of changes in price for a wide variety of stocks have “fat tails,” meaning events on the extremes are much more likely than the standard theory would have the denizens of Wall Street believe. Those with money in the stock market know this just as well as investment bankers. At one point, Mandelbrot shows a figure containing four price charts: two real and two fabricated. If you’ve looked at a lot of stock charts, the fakes are obvious: their ups and downs are too smooth. Plots of the changes with each step illuminate that the fake charts are built from purely Gaussian noise, while the real charts incorporate what the Gaussian perspective would call “shocks” at an alarming frequency. The underlying model that best reflects the economic reality is not a Gaussian distribution. Instead it is a power-law distribution, for which extreme outcomes are significantly more likely—so much more likely, in fact, that the traditional quantities of mean and variance are not defined!
Furthermore, Mandelbrot goes on to show that price charts look the same at different time scales, reflecting an old Wall Street adage that “all price charts look alike.” That is, whether the time scale is minutes, hours, or days, the zigs and zags of price variations are drawn from probability distributions of the same essential nature. This is where fractals enter the picture.
A fractal is a geometric construct that has the same (or similar) structure at multiple scales. The simplest fractals, such as the Koch snowflake, are constructed by starting with a “trunk” shape and building “branch” structures on the sides of the trunk. The new sides that result are then subjected to the same construction to give a new set of branches at smaller scale. The process is repeated again, and again, and again, infinitely. The resulting structure holds all kinds of insights about complexity in nature (“trunk” and “branches” hold a clue…). For our purposes, the most important is the similarity on multiple scales, a property called self-similarity. In this respect, charts of stock price over time are fractal!
A third insight attacks the assumption that price changes are independent and identically distributed. While a mathematical nicety, this idea is just plain silly in practice. Of course price changes are correlated over time. Bankers see a stock’s price rising and buy more in an effort to cash in by selling high later, pushing the price even higher. Investors sell as they see a stock’s price fall, worried that a further drop in price will cut into their winnings—causing the price to fall even more. Longer-term correlations also play a role: trends in entire industries or the economy as a whole cause large numbers of stocks to move up or down together over long time periods. Such long-term dependence leads to ups and downs at long time scales that the standard model of finance has few satisfactory answers for.
If we combine long-term dependence with the fractal model and a “fat-tailed” power-law distribution of price changes, the result is a multifractal picture of an evolving market with a fugue-like structure. Multiple structures now appear overlaid on one another, each with its own characteristic repeating scale factor. This complex structure can still be reduced to a small number of parameters, just as the musical themes can be distilled from a fugue. We can also work in the other direction, generating the behavior of a market over time starting from a small number of parameters. The result won’t look the same over time (uncertainty and randomness are involved in the “unraveling” process), but all possible price charts will be statistically equivalent.
Mandelbrot argues that the multifractal perspective is a more fundamental approach to modeling markets than the “Franken-Brownian” approach that has emerged from tweaks and adjustments to the standard model. Historically, investment bankers have hit the “real world” with the highly idealized standard model of how markets work that they picked up in business school. To function, they are forced to tweak, adjust, fudge, and essentially torture the standard model until it reflects reality. Why not start from a perspective that better reflects reality? Embrace the complexity, says Mandelbrot.
Speaking of reality, fractals show up in a number of places in nature at a variety of scales: the distribution of galaxies in the universe, growth patterns of plants, even the formation of aggregates in chemistry. It’s not direct evidence for multifractal behavior in markets, but it does hint at some fascinating commonalities between natural phenomena and the global economic system.