Alexander Maydanyuk, student

National Technical University of Ukraine “Kyiv Polytechnic Institute”,

Department of Applied Mathematics

The Fractal Market Hypothesis in Chaos Theory

Hypothesis of fractal market as alternative to the hypothesis of efficient market.

Fractals are everywhere in our world and play a substantial role, including, and in the structure of financial markets, which are locally casual, but globally determined. The methods of fractal analysis of markets of shares will be considered, bonds and currencies, methods of distinction of independent process, nonlinear stochastic process and nonlinear determined process and influence of these distinctions is investigational on user investment strategies and design capabilities. Such strategies and design capabilities are closely related to the type of assets and investment horizon of user.

The Fractal Market Hypothesis emphasizes the impact of liquidity and investment horizons on the behavior of investors. To make the hypothesis as general as possible, it will place no statistical requirements on the process. The purpose of the Fractal Market Hypothesis is to give a model of investor behavior and market price movements that our observations.

Markets exist to provide a stable, liquid environment for trading. Investors wish to get a good price, but that would not necessary be a “fair” price in the economic sense. For instance, short covering rarely occurs at a fair price. Markets remain stable when many investors participate and have many different investment horizons.

Economic capital markets, like stocks and bonds, have a short-term fractal statistical structure superimposed over a long-term economic cycle, which may be deterministic. Currencies, being a trading market only, have only the fractal statistical structure.

Information itself would not have a uniform impact on prices; instead, information would be assimilated differently be the different investment horizons. A technical rally would only slowly become apparent or important to investors with long-term horizons. Likewise, economic factors would change expectations. As long-term investors change their valuation and begin trading, a technical trend appears and influences short-term investors. In the short term, price changes can be expected to be noisier because general agreement on fair price, and hence the acceptable band around fair price, is a larger component of total return. At longer investment horizons, there is more time to digest the information, and hence more consensus as to the proper price. As a result, the longer the investment horizon, the smoother the time series.

The Fractal Market Hypothesis proposes the following:

1. The market is stable when it consists of investors covering a large number of investment horizons. This ensures that is ample liquidity for traders.

2. The information set is more related to market sentiment and technical factors in the short term than in the longer term. As investment horizons increase, longer-term fundamental information dominates. Thus, price changes may reflect information important only to that investment horizon.

3. If an event occurs that makes the validity of fundamental information questionable, long-term investors either stop participating in the market or begin trading based on the short-term information set. When the overall investment horizon of the market shrinks to a uniform level, the market becomes unstable. There are no long-term investors to stabilize the market by offering liquidity to short-term investors.

4. Prices reflect a combination of short-term technical trading and long-term fundamental valuation. Thus, short-term price changes are likely to be more volatile, or “noisier,” than long-term trades. The underlying trend in the market is reflective of changes in expected earnings, based on the changing economic environment. Short-term trends are more likely the result of crowd behavior. There is no reason to believe that the length of the short-term trends is related to the long-term economic trend.

5. If a security has no tie to the economic cycle, then there will be long-term trend. Trading, liquidity, and short-term information will dominate.

The Fractal Market Hypothesis (FMH) says that information is valued according to the investment horizon of the investor. Because the different investment horizons value information differently, the diffusion of information will also be uneven. At any one time, prices may not reflect all available information, but only the information important to that investment horizon.

The FMH owes mush to the Coherent Market Hypothesis (CMH) of Vaga and the K-Z model of Larrain. Like the CMH, the FMH is based on the premise that the market assumes different states and can shift between stable and unstable regimes. Like the K-Z model, the FMH finds that the chaotic regime occurs when investors lose faith in long-term fundamental information. In many ways, the FMH combines these two models through the use of investment horizons; it specifies when the regime changes and why markets become unstable when fundamental information loses its value. The key is that the FMH says the market is stable when it has no characteristic time scale or investment horizon. Instability occurs when the market loses its fractal structure and assumes a fairly uniform investment horizon.

We have outlined a new view on the structure of markets. Unfortunately, most standard market analysis assumes that the market process is, essentially, stochastic. For testing the Efficient Market Hypot