# On Chaos in Markets

“Reading this volume will not make you rich. But it will make you wiser—and may thereby save you from getting poorer” - Mandelbrot

Recently I’ve seen too many countless articles trying to apply the pillars of modern finance (Capital Asset Pricing Model, Modern Portfolio Theory, and Black Scholes and Merton Framework) in such a deregulated market such as cryptocurrency. Although it is exciting to see that there may be some sort of correlation between classical physics and the market, too many people assume them to be right because it may be easier to believe than doubt. These people try to apply traditional models, as disputed in reputation as they are, to a market that seems as though governed by no precedent. This got me thinking about a book I read a while back written by Benoit Mandelbrot called Misbehavior of Markets. In this he discusses how fundamental analysis, technical analysis, and the random walk theory are quite inadequate in characterizing market fluctuations. Instead, he provides a less popular point of view in analyzing market risk using fractals, commonly used in explanation of Chaos Theory.

A fractal is described as a pattern whose parts parallel the whole. A fern leaf for example looks like a fern frond, a few feet of coastline may look like a few hundred miles of coastline, and similarly a stock market chart over one week looks like a chart over a few years if proportions are kept the same. So fractals are prevalent in many different parts of the natural world. The fractal dimension is defined as the ratio of a logarithm 5 to a logarithm 3 for example, which would give a value less than 2 but greater than 1. This makes sense since the curve is not straight, but it only slightly jagged so it fills more space than a one dimensional figure, but less space than the entire two dimensional plane. In application to Chaos Theory, the slightest change in initial conditions of a fractal can generate immensely different patterns a few iterations later. Since we lack sufficient knowledge of the details in the initial conditions of a complex system, it is near impossible to predict the ultimate outcome of the system. If the initial error is even 0.001 percent, a couple iterations later would already yield amplified results. This argument is similarly used for the Butterfly Effect, as it is impossible to quantify the number of “butterflies” in the entire world. Mandelbrot, who was a distinguished mathematician, believed this to be a better way to think about markets.

We have established the non-linearity of fractals, but what happens when humans are thrown into the mix? If you see a person walking at point A, and half an hour later you see the person walking at point B, most would assume that the person has covered the distance between the two points. However, what lays behind the curtain is that the person could have taken an Uber halfway, and jumped to point B. This capability for jumps is a discontinuity that uncovers the principal difference between classical physics and economics. Therefore, when humans are involved, in most cases the scenario is non-linear.

Let us establish also that fundamentally, humans crave understanding of the natural world. We create understanding even when there is none, and patterns when they do not exist. Take religion for example. There is an increasingly popular theory of the God of the Gaps, which states that religion is an attempt by humans to fill the void of understanding. This wishful thinking has crafted a system that has build upon itself for hundreds, if not thousands of years. However, recent studies has shown that religion is becoming less and less a part of civilization. As people develop into more advanced beings, that gap will slowly be filled and thus religion will cease to have such a dominance in our civilization. As for the stock market, people too create pseudo-understanding of how the gears turn. In how many cases have people attempted to justify a market trend with technical analysis and fancy math for both the stock market and crypto market alike? Even studies carried out testing so called “experts” of the market against the S&P Index revealed that almost none of them beat the market.

With the lack of precedent to study with the cryptocurrency market, it’s confusing to see self-proclaimed experts trying to advise people to diversify according to the MPT or trying to value options with the Black Scholes formula. It is exhilarating crossing such a thing as Brownian motion in physics, used for modeling the random movement of particles, with market variations, but each one of these “efficient” market hypothesis can be derived from the Gaussian curve outlook of the world. This however, is also faulty at modeling deviations. It’s noted by Mandelbrot that big price changes occurred too often compared to the Gaussian model. These large diversions from the average were greater than five standard deviations, and occurred 2000 times more frequently than under Gaussian rules. This would drastically understate the real outlook of market risk.

Another model constantly used to simulate markets is the random walk model, which attempts to depict the randomness and unpredictability of markets. This model is structured behind three foundational claims. The first is the martingale condition, which hinges the best guess for tomorrow’s price on today’s price. The second is that future prices are independent of previous prices. The third is that a compilation of price changes in order from least to greatest would produce a mild bell-curve distribution. However, there has been plentiful evidence by a majority of economists that short-term dependence is real. If the market rises more in a certain time period (say a few months), the market will most likely fall even further in the next correction. This demonstrates the “momentum” effect and a power law. Although different stocks and cryptocurrencies obey power laws to different powers, some may diverge randomly according to gaussian distribution but many would end up not. This directly refutes the idea of independence. On the other hand, the bell curve has never been adequate as a model. The daily index movements of the Dow Jones spread out with far edges that dilate too high, and although according to theory, the chance of the index swinging more than 7 percent is once in 300,000 years, there were 48 such occurrences in the 20th century alone.

Albeit, the alternative framework that Mandelbrot suggests using Power Laws/Cauchy Distribution has a few flaws in itself when determining asset return variance. In general, asset return variance has higher kurtosis than modeled by log-normality, but adopting a Cauchy distribution would still be an overstatement relative to reality.

Random Walk proposes that returns are demonstrated by a normal process:

R ~ N(w, d)

Cauchy proposes that returns are demonstrated by the quotient of two normal processes:

R ~ N(w1, d1) / N(w2, d2)

Scaled random walk model can be demonstrated by a normal process multiplied with an exponential distribution, F(I), that reflects the liquidity intensity:

R ~ N(w, d) * F(I)

The Cauchy and Scaled Random Walk Model both contain fat tails, but Cauchy describes a world where mean, variance, correlation or any statistical data can’t be measured on asset price returns. From an ontological perspective, this makes no sense, since in a Cauchy scenario the days that contain major price changes would be the “normal” days. But in reality this is not the case.

Admittedly, it is beyond my scope to understand the happenings of markets to structure a solution. This topic, however, needs to remain in discussions as models are carelessly brought from one environment, to another much less familiar one. Mandelbrot rightly addressed the key issue that most people missed in the random walk model, but more people need to realize the dangers of these traditional methods.

Furthermore, the fractal view of the market grants a different perspective of its cyclical nature. Although cyclical, it shouldn’t distract us from the fact that just like the roughness and chaotic nature of clouds, mountains, cauliflowers and ferns, the market might not be as predictable as many traders make it out to be. However, in such a chaotic system, fractal mathematics can tell us that certain events will happen, although the when and to what extent is impossible to predict.

To Mandelbrot, “If you have a large number of price changes, a small number of days are important. The other days, you might as well go to the beach and do nothing”.