Markets are non-linear, dynamic systems characterized with an inherent instability often referred to as chaos. Chaos in dynamic systems is essentially periodic and typically arises from a sensitive dependence on initial conditions. This is what makes market systems so difficult to predict. The study of chaos reveals hidden patterns of order fluctuations that often characterize complex systems and/or markets.

Although a perfect prediction of chaotic market system is impossible, there exists rigorous ways to measure the amount of chaos associated with a market. These are Lyapunov exponent, fractal dimension, entropy and other advanced measures. Although tedious in their rigorous mathematical definitions, they often have an easy visual interpretation and can play a crucial practical role in making timely trading decisions.

# Lyapunov exponent

One of the most widespread measures of chaos is Lyapunov exponent. In a chaotic system, neighboring trajectories separate exponentially fast with time. Lyapunov exponent generally characterizes the divergence.

Consider two close points at step n, xn and xn+dxn. At the next time step they will have diverged, namely to xn+1 and xn+1+dxn+1. It is this average rate of divergence (or convergence) that the Lyapunov exponent captures. Another way to think about the Lyapunov exponent is as the rate at which information about the initial conditions is lost.

There are as many Lyapunov exponents as dimensions of the phase space associated with a market. The Lyapunov exponent can be calculated for each dimension. When talking about a single exponent one is normally referring to the largest.

Lyapunov exponent has a simple and practical use in trading:

- If the Lyapunov exponent is positive, then the system is chaotic and unstable. This normally means an upcoming period of high volatility and suggests exit from open positions.
- If the Lyapunov exponent is less than zero then the system attracts to a fixed point or stable periodic orbit. This corresponds to an existing market trend and suggests an entry order to follow the trend.
- If the Lyapunov exponent is near to zero then the system is neutrally stable. No change in open positions is necessary in this case.

These simple rules outline practical trading using market chaos measures.

The Quant Trade Price Beam indicator is an excellent way to get a visual impression on the density of phase trajectories involved in estimation of Lyapunov exponent for a given market.

# Fractal dimension

Another characteristic of chaotic markets is fractal dimension. Fractals are objects which are “self-similar” in the sense that the individual parts are related to the whole. In market price action, as you look at monthly, weekly, daily and intraday bar charts, the structure has a similar appearance. Fractal dimension measures a degree of this self-similarity.

Fractal dimension is closely related with Lyapunov spectrum of the market. Roughly speaking, the sum of Lyapunov exponents is proportional to a conservative upper estimate of fractal dimension.

Rigorous inference involves the Kaplan-Yorke formula, which provides an upper bound dimension of the attractor and approximate information on the number of the active degrees of freedom. In fact, in a typical market the phase-space dimension is infinite, but the number of independent variables that are necessary to uniquely identify the different points of the attractors can be finite and sometimes even small.

While fractal dimension is a technical indicator, which is wide spread now, its standard evaluation in technical analysis is very far from original provisions of the chaos theory. We apply the original mathematical formalism found in modern theory of dynamic systems to promote indicators of market chaos to new level of accuracy and practical applicability in everyday trading practice.

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