Introduction to chaos - Fractals geometry

What is it?

  • A pioneering approach to estimate with high precision market reality
  • It analysis details to find the hidden order behind the seemingly random movements
  • It studies the logic behind natural phenomena: Unpredictability, Complexity…
  • Its inventor, Benoit Mandelbrot, is recognized to be one of the most brilliant mathematicians of the last century

Main Idea

The main idea behind chaos theory is that even simple deterministic systems can sometimes produce completely unpredictable results. This happens when a deterministic system has sensitivity to initial conditions: basically, that if we take two starting states which are very similar to each other, over time the states will diverge and re-converge in unpredictable ways.

In other words, the main percept behind this theory is the underlying notion of small currencies significantly affecting the outcomes of seemingly unrelated events. This is one of the reason why Chaos theory is often referring to a “non-linear dynamics systems”.

3 Factors

  • Chaos Dynamical Systems: Disorder, Unpredictability, Turbulence…
  • Fractals Natural Phenomena: Repeating pattern at every scale
  • Geometry Dimensions: Shape, Size, Volume, Lengths, Area…


Relative to Financial Markets, we believe that price is the very last thing to change for a stock or any other securities listed on the markets. Price changes can be determined by stringent mathematical differential equations based on predefined assumptions as following:

  • Financial Markets are irrational: if investors are looking for maximizing their gain, turbulent movements such market collapse are stimulated and exacerbated by emotional and irrationals trading decisions.
  • Market Prices do not follow a linear and normal distribution: The Modern Portfolio Theory or Mean Variance Theory as described by the famous Economist Harry Markowitz in 1952 base its theory on the assumption that markets movements are following a Gaussian curve which implied that the probability to have extreme markets fluctuations are almost non-existent.
  • The Capital Market is imperfect: with a Gaussian approach, it is impossible to consider markets imperfections, which implies that historical prices do not impact future prices. The Chaos-Fractals approaches demonstrate the opposite. Indeed, Financial Markets have a memory and current prices have a direct influence on future prices. An important number of volatility on many and different time frame increase the risk of high volatility on the same future time frame.

What is trying to address?

Modern Finance and Portfolio Theory have seen a lot of innovation decades, but they still suffer from the same underlying assumptions, that markets are random walk, i.e., are normally distributed and tail events are rare.

Value-at-Risk assumes that risk could be control by looking only at the modest market moves occurring on 99% of days, without worrying about the much-larger jumps that could occur during the remaining 1% of trading days.

Chaos-Fractal Geometry predicts that major crashes/standard deviations happen often, roughly once per decade (1987, 1998, 2008…)

Euclidian Versus Chaos Theory

Euclidian Geometry “Notion of Smooth”

  • Represents and measure reality through smoothness.
  • Roughness is the imperfection of an ideal shape.

Euclidian Geometry

Application to Finance: There is a normal situation (smooth) and markets aberrations (rare of rough events)

Fractal Geometry “Notion of Roughness”

  • Represents and measures reality through roughness.
  • Roughness is not the imperfection of an ideal shape, but the heart of it.

Fractal Geometry

Application to Finance: What are Everything is about turbulence (roughness)

What are Fractals?

In one sentence, Fractals measure roughness, revealing order in the apparent disorder.

Definition: The whole can be divided in smaller parts, each repeating the whole like an echo.

The first four stages construction of the Von Koch snowflake

The Fractal:

  • Appropriates tool track causes
  • Reveals structure in the apparent turmoil
  • Follows the evolution equations
  • Tracks complex configurations formation

Application of Chaos Theory to the Financial markets

Applied to the stock market, an archetypal chaotic system, fractal geometry is very useful: It shows that the price of a stock is not independent from its price the day before. A tiny price variation can lead to an important modification of the stock dynamic. Therefore it’s possible to produce equations of the disruptions and chaos of a stock price.

What can we say about Market Efficiency?

Benoît Mandelbrot conceptualizes three kind of market randomness:

  • Benign Random: (Efficient Market)
    Large-scaled regularity; for instance Brownian motion belongs to benign random
  • Slow Random: (useless)
    The usual statistical rules remain applicable but evolution is so slow that knowing if the system will finally become regular doesn’t help solve concrete problems.
  • Savage Random: (inefficient Market)
    “Noah Effect”: remarkable break in comparison to what is considered standard. It depends on the event scope: harsh variations, thick “tail distributions” and sudden discontinuity (October 1987). The tendency of persistent time series (0.50 <H<1.00) to have abrupt, and discontinuous changes.

“Joseph Effect”: strong long-term dependence between two or more values of a series. The successive variables (remaining in the norm) have a price variance interdependence during long periods. This directional persistence makes forming the average impossible: problem of interdependence, or “Joseph” Effect (Example with the stock market).

The term “Joseph Effect” alludes to an Old Testament story about Joseph, where Egypt would experience seven years of feast followed by seven years of famine.

Geoffrey Ducournau Executive Director of Dimtech