Mandelbrot Fractals

We all live in the reality of our ideas. The market is a mirror reflecting our own ideas. Every person associated with the market never deals with reality, but always only with his or her own perceptions. Sometimes the ideas become massive and we talk about trends, support lines or graphical figures. Analysts often remark that "rising profits led to buying stocks" and "oil went down on investor selling.

But in reality none of us knows what is going on. We all use only a model, a simplified version, a copy of the market. Our success depends solely on how similar our copy is to the original.

The beginning of the 21st century seriously shook all modern financial theories. The bankruptcy of major investment corporations, the financial crisis - all this has called into question the models used by analysts and traders of the entire world. Investment advisors either throw up their hands or cling tenaciously to outdated theories, but reality can no longer be ignored - markets are not described by random walk, Gaussian curve, Markowitz model and Black-Scholz formulae.

The copy turned out to be a crude forgery. That is why interest in alternative theories describing the behavior of the financial market has grown rapidly.
One such theory, which we will discuss today, is fractal geometry. This theory is not new and is actively used in various fields of activity.

Fractal Geometry

The term "fractals" in Russian traders is traditionally associated with the name of Bill Williams. "Williams' Fractals. are known to everyone and they are even included in the list of indicators of the famous MetaTrader 4 platform. But not many people know the name of the real author of this concept - Benoit Mandelbrot, a famous mathematician, the creator of fractal geometry. Probably, this article would have never existed if Benoit Mandelbrot had not seriously started to apply fractals in financial markets.

So, what is a fractal?

Fractal - is a form or "structure" that has the property to self-repeating at different scales. The best way to explain this term is to use an example. Look at Figure 1. What do you see?

Your attention is obviously drawn to the general triangular shape of the figure. If you look closer, you will see that the triangle in turn consists of three more triangles inscribed into it, with three more smaller triangles inscribed into each of them, and so on.

The example given is a popular figure, also known as "Sierpinski's napkin". At any level of the figure, each element is similar to an element at a lower or higher scale. The building material for a fractal or the shape underlying it is called "initiator", the same structure or a self-repeating pattern - "generator.". The initiator for a "Sierpinski wipe" can be a point, and the generator can be a triangle.
But what does this give us for understanding financial markets?

Observation 1. Describing the market with fractals

As it turned out, market behavior can be described by of fractals. The most basic graphical element of the market is a straight line pointing downwards or upwards. It is well understood by every trader - the price is either rising or falling, this process occurs over time.
Thus, we have an initiator that looks like this (see Figs. 2, 3).

Even if we take the price movement within one minute, we still get a line that connects the opening and closing price. The generator for the price movement, on the other hand, is another common structure well known to the trader - "pulse-correction-pulse"which looks like the one below (see Figs. 4, 5).

There can be an infinite number of generators on the market, and there may not be two breaking points at all. Which one information can these figures give a trader?

If you look at the price movement of a single instrument, you can see that the structure of the generator is repeated on all time scales of the instrument. Let's take it as a given that the intraannual price movement is a simple structure of two impulses and one corrections as in Figure 2 or 3. If we replace both pulses and correction with corresponding fractals (oscillators), we get the following structure (see Fig. 6):

Going deeper and deeper, we will get to the minute and then tick charts, on which again and again will appear basic fractal.

What is characteristic, the ratios between the lines of the generator will remain fixed on any time structure. The angles between the generator lines on the minute and monthly chart will correspond to each other, the correlation of their lengths as well. This amazing discovery gives us a completely new perspective on the usual price movement.

Of course, this understanding is simplistic and, in Mandelbrot's own words, "cartoonish. It serves us to describe the general principle of the structure of the price movement. The real market generator can be much more complex.

In modeling market behavior, Mandelbrot uses a more complex "multifractal" model that uses three dimensions and the so-called "fractal cube". We will not dwell on it in detail. Instead, we will consider two other observations of fractal geometry that are easier to understand and give the trader food for thought.

Observation 2. The market has a memory

Extensive research on the cotton market led Benoit Mandelbrot to the following conclusion: periods of high volatility or "turbulence" tend to gather in "clusters.

What is it "price cluster"? I'm sure you guessed it's a "trend". For you and me, as traders, this is certainly good news. As long as there are trends, a trader's work will pay well.

Observation 3: The Noah Effect

Finally, Mandelbrot's third observation is the so-called The Noah Effect. We know from the Old Testament that the Flood began unexpectedly, and its destructive power was very great. The "Noah effect" is a metaphor for market reversals. stock market panics and upsurges. They never happen smoothly, almost always the market soars up or collapses with such force that none of the investors expected.

This always causes panic among the stock market public, which is shocked by such price movements. In 1987, for example, the Dow fell 22.6% in one day. After the crash they blamed computer programs, but Benoit Mandelbrot has a different opinion - it's not about the programs, it's about the the nature of the market. It is the intrinsic nature of the market that causes such dynamics. This hypothesis is also new and inconsistent with the efficient market hypothesis, according to which the market should change smoothly and consistently.
This property of the market should be remembered by traders who work without "stops", relying on the fact that the market sooner or later will return to the level of the opening deal.

Conclusions

The summary Mandelbrot makes is as follows: market - a very risky place, much riskier than is commonly believed. For traders risk - not a source of danger, but a potential source of profit. If you use your knowledge of price movements correctly and get on the "right" side of risk, it will be a blessing, not a curse.

Application Studies of fractal market models The field has only just begun, but it is already quite clear that it has a great future. It only remains to admit that we are living in an interesting time, a time when new financial theories are being born. Perhaps in 10 years' time our terminals will be installed with completely different indicators.

Good luck to you.