A Quick and Dirty Primer on Chaos Theory

“If the flap of a butterfly’s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado.” -Edward Lorenz

Interesting to consider, no? Pop culture is very familiar with the adage about a butterfly’s wings flapping in one part of the world ultimately causing a tornado in a completely different part. We often interpret this on a kind of karmic scale, considering the far-reaching consequences of seemingly insignificant actions. However, mathematicians, economists, and sometimes meteorologists, interpret this far differently. Mathematicians and economists think about the butterfly effect in terms of chaos, and naturally set out to quantify it.

Chaos Theory was first discovered in 1961 by meteorologist Edward Lorenz as he was attempting to predict the weather by using a program of twelve recursive equations. This means that each equation used the information gleaned from the previous equation. While trying to recreate a previous weather pattern, Lorenz started mid-cycle and input his data. To his surprise, the resulting pattern was very different from the original one (see below). Inquiry into this phenomenon led him to the realization that while his original data had started at .506127, he had tried to recreate the pattern with the starting point of .506. This led him to the foundational principle of Chaos Theory – these systems are highly sensitive to initial conditions.

There are several layers to the definition of a chaotic system. The system must be dynamical and dynamic – that is, each state of the system is dependent on its previous state and the components of the system are fluid. The system must also be nonlinear. That means the system’s inputs are not necessarily proportional to its outputs: small changes could affect the system in a big way and big changes could fail to affect the system at all. Think of a firm whose production and workers are at equilibrium. If we add one worker we could see production rise, but we could also add five workers and see production stay the same. There are a host of variables that would affect each possible outcome. Chaotic systems can be deterministic or nondeterministic. Deterministic systems are systems with no random processes and non-deterministic systems do include random processes. Given these parameters, chaos theory seeks to answer several central questions. How sensitive to its initial conditions is the system? What happens when we impose an iterative process? Can we find the underlying order in a system that looks completely random on the surface?

It’s easier to first think about this on a small scale. Fractals are built on self-similarity and theoretically iterate to infinity. Apply the same function over and over and the fractal will form a picture. Zoom in on any one portion of the fractal and it will look exactly like the larger picture. This can be done deterministically with software, where the end product is only affected by the iterated function. If the function is changed, the picture will change. Some of the more famous fractals are the Mandelbrot fractal and the Koch curve:


image from fractal-explorer.com                                            

 image from fractal.institute

Nature is also saturated in fractals. Trees, snowflakes, ferns, and cabbage are fractals. Sunflowers, hurricanes, and pinecones are special cases called Fibonacci fractals. It makes sense then that a fractal found in nature is necessarily non-deterministic as it is subject to any number of random processes that will affect its outcome in any number of ways. Look at any deciduous tree in the winter and the elements have clearly played a significant role in what it grew to be. The salient question remains, what about Chaos Theory? What happens when we change one thing in the function? Can we predict the outcome? Well, yes…and no. The underlying functions of our deterministic fractals can be manipulated in the software to create many different patterns. Since these fractals are essentially grown in a vacuum, we can predict exactly what they will look like for any given number of iterations. It gets more complicated when our fractals are subject to the elements, but we can add in many different variables and come up with a probability that the result will fall in a given range. That is, if we want to predict how high our tree will grow, we can use information such as average rainfall, average height of that species, sun exposure, probability of extreme weather events, probability of disease, etc. to come up with a probability that the tree will reach a certain height.

Things get infinitely more complicated when the system gets more complicated. That is, what if the chaotic system in question is the economy? Currently, it is common thought that economic fluctuations are due to some exogenous factor. Barnett, Serletis and Serletis (2015) point out that most economic theory assumes that equilibria exist and when there is an absence of exogenous shocks to the system, the market will tend to a steady state. However, thinking of the economy as a chaotic system means that shocks aren’t exogenous but endogenous. The implication of this is the ability to assign a probability to what the economy will look like at a given point in time, if one were able to nail down the initial conditions to perfection and assign probabilities of stochastic variables within an arbitrarily small degree of confidence. The impossibility of that is not the only issue. Economists aren’t yet sure the economy even falls under the definition of a chaotic system. It’s incredibly difficult to definitively say whether a given economic shock is exogenous and random or endogenous and nonlinear. Further, testing an infinite system with finite data presents a problem, and Litimi, BenSaida, Belkacem, and Oussama (2019) find there to be too much noise to be of practical use.

Perhaps that’s overcomplicating things – it’s too general to get a handle on. The efficient market hypothesis is widely cited in economic theory. The most salient assumption of the EMH is that asset prices always reflect all the available information in the market. Thus, most of the players in the market don’t beat it but are instead crowded around the median (this isn’t to say that it isn’t possible to beat the market under the EMH, just that most don’t. Since investors make up “the market”, if most investors were to beat the market, that would mean the market beat the market.). Another way of thinking about this is the marginal gain from using information equals the marginal cost of acquiring it. However, there are clear problems with this. The EMH rests on the assumption that asset prices are rationally set by full information, which is never the case in the real world. Kristoufek (2012) also astutely notes that different groups can interpret the same piece of information in very different ways. That is, a buy signal at one price to one group can be a sell signal to another. This brings up the question of liquidity, something for which EMH does not account. If one side of the transaction dominated the market and there were no investors on the other side, the asset would become illiquid and prices could collapse. Enter instead the fractal market hypothesis, which is based on liquidity and attempts to address the weaknesses of the EMH.

It is a well-established fact that investors are in the market for a myriad of reasons. Each investor has his or her own risk profile, cash needs, industry preferences, and time horizons, among other things. The FMH zones in on the assumption that investors have investment time horizons that span anywhere from mere minutes to many years. The general idea then is when there is a buyer for every seller and vice versa, which occurs most of the time in a market where the players have many different timelines, that liquidity works to smooth pricing and stabilize the market. What does this have to do with Chaos Theory? The FMH works much like a fractal, building itself with the same self-similar structure and feedback loops. A buy to one group is a sell to another and a sell to one group is a buy to another, and on and on it goes. If there is ever a time in which a group at one spot in the time horizon vacates or investors begin to cluster around a specific time, the fractal will destabilize and even break down because the underlying function is no longer continuous, or smooth. On the surface, this exists in a vacuum with no exogenous shocks to the system (whether random or not). FMH does account for these shocks, the discussion of which is outside the scope of this paper.

Where are we now? Economists participate in a lively back-and-forth about the future practicality of Chaos Theory. There isn’t even a consensus when it comes to definitively stating if the financial markets even fall under the definition of a chaotic structure. If a consensus is ever reached, there are realistically far too many variables to ever accurately predict even one data point long term within a reasonable confidence interval. Simply put, Chaos Theory isn’t there yet, and it may never be…or it may just join the ranks of the thousands of other ideas whose practical purpose did not come until much later. At least we have cool pictures to look at while we’re waiting.

Image from Wikipedia

Sources:

Adrangi, B., Allender, M., & Raffiee, K. 2010, ‘Exchange Rates and Inflation Rates: Exploring Nonlinear Relationships’, Review of Economics and Finance. Available at https://pdfs.semanticscholar.org/edb5/9aa144327363644b360409a78f5551d63b….

Barnett, W., Serletis, A., & Serletis, D. 2015, ‘Nonlinear and Complex Dynamics in Economics’, Macroeconomic Dynamics, vol. 19, no. 8, pp. 1749-1779.

Cottrell, P. 2016, ‘Chaos Theory and Modern Trading’. Available at SSRN: https://ssrn.com/abstract=2761874.

Kristoufek, L. 2012, ‘Fractal Markets Hypothesis and the Global Financial Crisis: Scaling, Investment Horizons and Liquidity’, Advances in Complex Systems, vol. 15, no. 6, art. 1250065.

Litimi, H., BenSaida, A., Belkacem, L., & Oussama A. 2019, ‘Chaotic Behavior in Financial Market Volatility’, Journal of Risk, vol. 21, no. 3, pp. 27-53.

Peters, E.E. (1994). Fractal Market Analysis: Applying Chaos Theory to Investment and Economics. New York, NY: John Wiley & Sons, Inc.

Tziperman, E., ‘Chaos Theory: A Brief Introduction’. Available at https://courses.seas.harvard.edu/climate/eli/Courses/EPS281r/Sources/Cha…