18 March 2020 –Equities markets are not a zero-sum game (Fama, 1970). They are specifically designed to provide investors with a means of participating in companies’ business performance either directly through regular cash dividends, or indirectly through a secular increase in the market prices of the companies’ stock. The efficient market hypothesis (EMH), which postulates that stock prices reflect all available information, specifically addresses the stock-price-appreciation channel. EMH has three forms (Klock, & Bacon, 2014):

- Weak-form EMH refers specifically to predictions based on past-price information;
- Semi-strong form EMH includes use of all publicly available information;
- Strong-form EMH includes all information, including private, company-confidential information.

This essay examines equities-market efficiency from the point of view of a model based on chaos theory (Gleick, 2008). The model envisions market-price movements as chaotic fluctuations around an equilibrium value determined by strong-form market efficiency (Chauhan, Chaturvedula, & Iyer, 2014). The next section shows how equities markets work as dynamical systems, and presents evidence that they are also chaotic. The third section describes how dynamical systems work in general. The fourth section shows how dynamical systems become chaotic. The conclusion ties up the argument’s various threads.

## Stock-Market Dynamism

Once a stock is sold to the public, it can be traded between various investors at a strike price that is agreed upon *ad hoc *between buyers and sellers in a secondary market (Hayek, 1945). When one investor decides to sell stock in a given company, it increases the supply of that stock, exerting downward pressure on the strike price. Conversely, when another investor decides to buy that stock, it increases the demand, driving the strike price up. Interestingly, consummating the transaction decreases both supply and demand, and thus has no effect on the strike price. It is the intention to buy or sell the stock that affects the price. The market price is the strike price of the last transaction completed.

Successful firms grow in value over time, which is reflected in secular growth of the market price of their stocks. So, there exists an arbitrage strategy that has a high probability of a significant return: buy and hold. That is, buy equity in a well-run company, and hold it for a significant period of time, then sell. That, of course, is not what is meant by market efficiency (Chauhan, et al, 2014). Efficient market theory specifically concerns itself with returns in excess of such market returns (Fama, 1970).

Of course, if all investors were assured the market price would rise, no owners would be willing to sell, no transactions could occur, and the market would collapse. Similarly, if all investors were assured that the stock’s market price would fall, owners would be anxious to sell, but nobody would be willing to buy. Again, no transactions could occur, and the market would, again, collapse. Markets therefore actually work because of the dynamic tension created by uncertainty as to whether any given stock’s market price will rise or fall in the near future, making equities markets dynamical systems that move constantly (Hayek, 1945).

Fama (1970) concluded that on time scales longer than a day, the EMH appears to work. He found, however, evidence that on shorter time scales it was possible to use past-price information to obtain returns in excess of market returns, violating even weak-form efficiency. He concluded, however, that returns available on such short time scales were insufficient to cover transaction costs, upholding weak-form EMH. Technological improvements since 1970 have, however, drastically reduced costs for high volumes of very-short-timescale transactions, making high-frequency trading profitable (Baron, Brogaard, Hagströmer, & Kirilenko, 2019). Such short-time predictability and long-time unpredictability is a case of *sensitive dependence on initial conditions*, which Edward Lorentz discovered in 1961 to be one of the hallmarks of chaos (Gleick, 2008). Since 1970, considerable work has been published applying the science of chaotic systems to markets, especially the forex market (Bhattacharya, Bhattacharya, & Roychoudhury, 2017), which operates nearly identically to equities markets.

## Dynamic Attraction

Chaos is a property of dynamical systems. Dynamical-systems theory generally concerns itself with the behavior of some quantitative variable representing the motion of a system in a *phase space.* In the case of a one-dimensional variable, such as the market price of a stock, the phase space is two dimensional, with the variable’s instantaneous value plotted along one axis, and its rate of change plotted along the other (Strogatz, 2015). At any given time, the variable’s value and rate of change determine the location in phase space of a *phase point* representing the system’s instantaneous state of motion. Over time, the phase point traces out a path, or *trajectory*, through phase space.

As a simple example illustrating dynamical-system features, take an unbalanced bicycle wheel rotating in a vertical plane (Strogatz, 2015). This system has only one moving part, the wheel. The stable equilibrium position for that system is to have the unbalanced weight hanging down directly below the axle. If the wheel is set rotating, the wheel’s speed increases as the weight approaches its equilibrium position, and decreases as it moves away. If the energy of motion is not too large, the wheel’s speed decreases until it stops, then starts rotating back toward the fixed equilibrium point, then slows again, stops, then rotates back. In the absence of friction, this oscillating motion continues *ad infinitum.* In phase space, the phase point’s trajectory is an elliptical orbit centered on an *attractor* located at the unbalanced weight’s equilibrium position and zero velocity. The ellipse’s size (semi-major axis) depends on the amount of energy in the motion. The more energy, the larger the orbit.

If, on the other hand, the wheel’s motion has too much energy, it carries the unbalanced weight over the top (Strogatz, 2015). The wheel then continues rotating in one direction, and the oscillation stops. In phase space, the phase point appears outside some elliptical boundary defined by how much energy it takes to drive the unbalanced weight over the top. That elliptical boundary defines the attractor’s *basin of attraction.*

## Chaotic Attractors

To illustrate how a dynamic system can become chaotic requires a slightly more complicated example. The pitch-control system in an aircraft is particularly apropos equities markets. This system is a feedback control system with two moving parts: the pilot and aircraft (Efremov, Rodchenko, & Boris, 1996). In that system, the oscillation arises from a difference in the speed at which the aircraft reacts to control inputs, and the speed at which the pilot reacts in an effort to correct unintended aircraft movements. The pilot’s response typically lags the aircraft’s movement by a more-or-less fixed time. In such a case, there is always an oscillation frequency at which that time lag equals one oscillation period (i.e., time to complete one cycle). The aircraft’s nose then bobs up and down at that frequency, giving the aircraft a porpoising motion. Should the pilot try to control the porpoising, the oscillation only grows larger because the response still lags the motion by the same amount. This is called *pilot induced oscillation (PIO)*, and it is a major nuisance for all feedback control systems.

PIO relates to stock-market behavior because there is always a lag between market-price movement and any given investor’s reaction to set a price based on it (Baron, Brogaard, Hagströmer, & Kirilenko, 2019). The time lag between intention and consummation of a trade will necessarily represent the period of some PIO-like oscillation. The fact that at any given time there are multiple investors (up to many thousands) driving market-price fluctuations at their own individual oscillation frequencies, determined by their individual reaction-time lags, makes the overall market a chaotic system with many closely spaced oscillation frequencies superposed on each other (Gleick, 2008).

This creates the possibility that a sophisticated arbitrageur may analyze the frequency spectrum of market fluctuations to find an oscillation pattern large enough (because it represents a large enough group of investors) and persistent enough to provide an opportunity for above-market returns using a contrarian strategy (Klock, & Bacon, 2014). Of course, applying the contrarian strategy damps the oscillation. If enough investors apply it, the oscillation disappears, restoring weak-form efficiency.

## Conclusion

Basic market theory based on Hayek’s (1945) description assumes there is an equilibrium market price for any given product, which in the equity-market case is a company’s stock (Fama, 1970). Fundamental (i.e., strong-form efficient) considerations determine this equilibrium market price (Chauhan, et al, 2014). The equilibrium price identifies with the attractor of a chaotic system (Gleick, 2008; Strogatz, 2015). Multiple sources showing market fluctuations’ sensitive dependence on initial conditions serve to bolster this identification (Fama, 1970; Baron, Brogaard, et al, 2019; Bhattacharya, et al, 2017). PIO-like oscillations among a large group of investors provide a source for such market fluctuations (Efremov, et al, 1996).

## References

Baron, M., Brogaard, J., Hagströmer, B., & Kirilenko, A. (2019). Risk and return in high-frequency trading. *Journal of Financial & Quantitative Analysis*, *54*(3), 993–1024.

Bhattacharya, S. N., Bhattacharya, M., & Roychoudhury, B. (2017). Behavior of the foreign exchange rates of BRICs: Is it chaotic? *Journal of Prediction Markets*, *11*(2), 1–18.

Chauhan, Y., Chaturvedula, C., & Iyer, V. (2014). Insider trading, market efficiency, and regulation. A literature review. *Review of Finance & Banking*, *6*(1), 7–14.

Efremov, A. V., Rodchenko, V. V., & Boris, S. (1996). *Investigation of Pilot Induced Oscillation Tendency and Prediction Criteria Development (No. SPC-94-4028)*. Moscow Inst of Aviation Technology (USSR).

Fama, E. (1970). Efficient capital markets: A review of theory and empirical work. *The Journal of Finance*, *25*(2), 383-417.

Farazmand, A. (2003). Chaos and transformation theories: A theoretical analysis with implications for organization theory and public management. *Public Organization Review*, *3*(4), 339-372.

Gleick, J. (2008). *Chaos: Making a new science*. New York, NY; Penguin Group.

Hayek, F. A. (1945). The use of knowledge in society. *American Economic Review*, *35*(4), 519–530.

Klock, S. A., & Bacon, F. W. (2014). The January effect: A test of market efficiency. *Journal of Business & Behavioral Sciences*, *26*(3), 32–42.

Strogatz, S. H. (2018). *Nonlinear dynamics and chaos.* Boca Raton, FL: CRC Press.