From Chaos Theory to Market Finance

Business woman in front of computer. Forex market concept.

 

“Between the deterministic approach for which the same causes produce the same effects, the present situation dictating the future situation and the random walk for which the too great complexity of the present situation makes it impossible to predict the future, there is a transitional state, neither deterministic nor random: Chaos theory“.

The main characteristic of a deterministic system when given chaotic behaviour is that it can produce results that are unpredictable at any point. This occurs when the deterministic system is extremely sensitive to the initial conditions.

Billiards is a perfect example to illustrate this. Let’s imagine for a moment two people trying to break a pool game. The slightest change in the holding of the cue, in the angle between the cue and the white ball when touching the 2 objects or in the power with which the white ball will be struck will have a significant impact on the dispersion of the balls across the table.

This is also known as the “butterfly effect” or “the flapping of a butterfly’s wings in Asia can cause a storm in the United States”.

However, even if such a system remains unpredictable, if its behaviour were to be traced over a relatively long period of time, we would notice that it follows a “kind of model”: Lorenz’s pattern.

No matter where the system starts, we will find that it will mold itself into a behaviour that looks almost identical but actually differs greatly (extreme sensitivity to initial conditions). The system follows an orbit comparable to the orbit that the moon could follow around the Earth.

This orbit exists because the Earth emits gravity against the moon, preventing it from drifting endlessly through space. This attraction that the Moon has for the Earth is the result of an attractor. The attractor can be a point or curve to which the system tries to conform and which defines the behaviour of the system.

theorie du chao

The Moon moves around this attractor in an orbit similar to that of an ellipse. This orbit is “pretty”, smooth, predictable because the attractor in question is continuous everywhere, the Moon has no difficulty in complying with it.

However, in systems that are extremely sensitive to initial conditions, let us call them “chaotic systems“, the attractor is no longer a simple point or a smooth, continuous curve. In the case of chaotic systems, the attractor – often referred to as “strange” or “split” – could be an infinity of unconnected points (Cantor set) or a smooth curve but with mathematical discontinuities.

These strange attractors are no longer defined in full dimension spaces (dimension 2 or dimension 3 for example), they are defined in fractional dimension spaces such as π, 1/3,5.47 or any other space of positive dimension strictly not full.

A system trying to comply with such an attractor will only be able to react in a chaotic way, in a way that is not predictable.

Try to roll a tennis ball on a pebble beach for a moment. The ball will bounce incessantly and repeatedly with unpredictable behavior. It will be impossible to roll it. This surface, defined by all these rollers, is somehow defined in a dimension between dimension 2 and dimension 3.

Physical systems as we know them in our 3-dimensional space (4 if we count time), cannot be predictable if they try to conform to a defined attractor in a fractional dimension space. This dichotomy between the integer and the fractionated inevitably results in a chaotic result.

These strange attractors can be graphically represented, using significant computing power. This graphical representation is called “fractal“. Here are some of them:

  • The Mandelbrot set
  • Sierpinski’s triangle
  • The fractal sheet

A fractal image is therefore only the visual representation of a strange attractor, the latter defining the orbit of a system behaving in a chaotic manner.

The fractal image has the interesting feature of having an identical pattern regardless of the scale at which it is viewed.

This is where I will create the bridge between chaos and market finance. In technical analysis, there is also a structure with this characteristic, namely the Elliott waves. Regardless of the time horizon over which they are viewed (minute, hour, week, month, etc.), Elliott waves will always have this structure in wave 1, 2, 3, 4, 5, a, b, c. Elliott waves are a fractal figure.

If we now take the reasoning a little further and try to draw an analogy between chaos and market finance, we will find that the strange attractor, what attracts the system is nothing more than the P&L, the result of the positions taken by the trader. Different traders are attracted ton earings and each will define themselves differently according to their own initial conditions.

Everyone will have their own way of trading with their own temporality and different tools. A distinction is made between scalp traders, day traders, swing traders, portfolio managers, etc. Each of them will define their own initial conditions and consequently will have their own orbits and find their own balance points. All these orbits will define the Lorenz pattern.

The market, which appears disordered but which in reality is in a chaotic state, then knows a succession of balance points linking each other.

It is this succession of equilibrium phases that gives chaos its adaptive value.

Arnaud LUU

Sources :

  • « Performer votre Trading» par Nicolas Bourhis-Mariotti
  • Ted Wrigley– Philosophe & Scientifique

 

Arnaud LUU


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