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Forex fractals - what you need to know

It is unlikely that you will find at least one newcomer to the Forex market who would not know what a fractal is. And many people have heard of such a concept outside the market. Fractals have been known for almost a century, are well studied and have numerous applications in life. The basis of this phenomenon is a very simple idea: an infinite number of figures in beauty and variety can be obtained from relatively simple structures with just two operations - copying and scaling.

What is a fractal?

The concept of "fractal" does not have a strict definition. Therefore, this word is not a mathematical term. This is usually called a geometric figure that satisfies one or more of the following properties:

- has a complex structure at any magnification;

- is (approximately) self-similar;

- has a fractional Hausdorff (fractal) dimension, which is greater than the topological;

- can be built by recursive procedures.

History of occurrence

At the turn of the 19th and 20th centuries, the study of fractals was more episodic than systematic. Previously, mathematicians mainly studied objects that could be studied using general methods and theories.

In 1872, the German mathematician Karl Weierstrass constructed an example of a continuous function that is nowhere differentiable. However, its construction was entirely abstract and difficult to perceive. Therefore, in 1904, the Swede Helge von Koch invented a continuous curve that has no tangent anywhere, and it is quite simple to draw. It turned out that it has the properties of a fractal. One variation of this curve is called the Koch snowflake.

The ideas of self-similarity were picked up by the Frenchman Paul Pierre Levy, the future mentor of Benoit Mandelbrot. In 1938, his article "Flat and spatial curves and surfaces consisting of parts similar to the whole" was published, in which another fractal is described - the Levy C-curve. All these fractals listed above can be conditionally assigned to one class of constructive (geometric) fractals.

Another class is dynamic or algebraic fractals, which include the Mandelbrot set. The first studies in this direction date back to the beginning of the 20th century and are associated with the names of the French mathematicians Gaston Julia and Pierre Fatou. In 1918, Julia published almost two hundred pages of work devoted to iterations of complex rational functions, in which Julia sets are described - a whole family of fractals closely related to the Mandelbrot set. This work was awarded the prize of the French Academy, but it did not contain a single illustration, so it was impossible to appreciate the beauty of open objects. Despite the fact that this work glorified Julia among mathematicians of that time, they quickly forgot about it.

Once again, attention was turned to the work of Julia and Fatou only half a century later, with the advent of computers: it was they who made the wealth and beauty of the fractal world visible. After all, Fatou could never look at the images that we now know as images of the Mandelbrot set, because the necessary number of calculations cannot be carried out manually. The first to use a computer for this was Benoit Mandelbrot.

In 1982, Mandelbrot's book Fractal Geometry of Nature was published, in which the author collected and systematized practically all the information on fractals available at that time and outlined it in an easy and accessible manner. Mandelbrot made the main emphasis in his presentation not on heavy formulas and mathematical constructions, but on the geometric intuition of readers.

Thanks to computer-generated illustrations and historical tales, with which the author skillfully diluted the scientific component of the monograph, the book became a bestseller, and fractals became known to the general public. Their success among non-mathematicians is largely due to the fact that with the help of very simple constructions and formulas that a high school student can understand, images that are amazing in complexity and beauty are obtained.

When personal computers became powerful enough, even a whole trend in art appeared - fractal painting, and almost any computer owner could do it. Now on the Internet you can easily find many sites dedicated to this topic.

After this brief digression into history, let us now familiarize ourselves with the classification of fractal types today.

Geometric fractals

It is with them, as you already understood, that the history of fractals began. This type of fractal is obtained by simple geometric constructions. First, the base is depicted. Then some parts of the base are replaced with a fragment. At each next stage, parts of the already constructed figure, similar to the replaced parts of the base, are again replaced by a fragment taken on a suitable scale. Each time the scale decreases. When changes become visually imperceptible, they believe that the constructed figure well approximates the fractal and gives an idea of ​​its shape. To obtain the fractal itself, you need an infinite number of stages. Changing the base and fragment - you can get many different geometric fractals.

Geometric fractals are good in that, on the one hand, they are the subject of sufficient serious scientific study, and on the other hand, they can be seen. Even a person who is far from mathematics will find in them something for himself. Such a combination is rare in modern mathematics, where all objects are defined using obscure words and symbols.

Many geometric fractals can be drawn literally on a piece of paper in a cage. It is important to understand that all images obtained are only finite approximations of infinite, inherently fractals. But you can always draw such an approximation that the eye will not be able to distinguish between very small details and our imagination will be able to create a true fractal picture.

For example, having a sufficiently large sheet of graph paper and a supply of free time, you can manually draw such an exact approximation to the Sierpinski carpet that from a distance of several meters the naked eye will perceive it as a real fractal. The computer will save time and paper while still increasing the accuracy of drawing.

Koch snowflake

This is one of the first fractals studied by scientists. The snowflake is obtained from three copies of the Koch curve, which first appeared in an article by Swedish mathematician Helge von Koch in 1904. This curve was invented as an example of a continuous line, to which it is impossible to draw a tangent at any point. Lines with this property were known before, but the Koch curve is remarkable for the simplicity of its design.

The Koch curve is continuous, but nowhere differentiable. Roughly speaking, it was precisely for this that it was invented - as an example of such mathematical "freaks".

The Koch curve has an infinite length. Let the length of the initial segment be 1. At each step of construction, we replace each of the components of the line of segments with a polyline that is 4/3 times longer. This means that the length of the entire polyline at each step is multiplied by 4/3: the length of the line with number n is (4/3) n-1. Therefore, there is nothing left of the limit line, except how to be infinitely long.

Koch snowflake limits the final area. And this despite the fact that its perimeter is endless. This property may seem paradoxical, but it is obvious - the snowflake is completely placed in a circle, therefore its area is deliberately limited. You can calculate the area, and you don’t even need special knowledge for this - the formulas of the area of ​​the triangle and the sum of the geometric progression are held at school.

Koch snowflake "vice versa"

The Koch snowflake "vice versa" is obtained by constructing the Koch curves inside the original equilateral triangle.

Lines of cesaro

Instead of equilateral triangles, isosceles triangles with an angle at the base of 60 ° to 90 ° are used. In the figure below, the angle is 88 °.

Square option

Here squares are being completed.

Koch Pyramid


Construction begins with a unit square. First step: paint a square with 1/2 side in the center in white. Then you need to mentally divide the square into 4 identical ones and in the center of each of them fill in the square with 1/4 side. Further, each of these 4 squares is again divided into 4 parts, a total of 16 squares will be obtained, and with each of them you need to do the same. And so on.

The fractal dimension is shaded white and is equal to log24 = 2. It is everywhere dense in the original square. This means that no matter what point of the square we take, there are shaded points in any of its arbitrarily small neighborhoods. That is, in the end, almost everything turned white - the area of ​​the remainder is 0, and the fractal occupies an area of ​​1. But the length of the border of the filled part is infinite.

H fractal

It all starts with a figure in the form of the letter H, in which the vertical and horizontal segments are equal. Then, to each of the 4 ends of the figure, a copy of it is reduced, halved. A copy of the letter H is reduced to each end (there are already 16 of them), reduced already by 4 times. And so on. In the limit, you get a fractal that visually almost fills a certain square. H-fractal is everywhere dense in it. That is, in any neighborhood of any point of the square there are fractal points. Very similar to what is happening with the T-square. This is not accidental, because if you look closely, it is clear that each letter H is contained in its own small square, which was completed at the same step.

We can say (and prove) that the H-fractal fills its square (English space-filling curve). Therefore, its fractal dimension is 2. The total length of all segments is infinite.

The principle of constructing an H-fractal is used in the production of electronic microcircuits: if it is necessary that in a complex circuit a large number of elements receive the same signal at the same time, then they can be located at the ends of segments of a suitable iteration of the H-fractal and connected accordingly.

Mandelbrot Tree

The Mandelbrot tree is obtained if you draw thick letters H, consisting of rectangles, and not of segments:

Pythagoras tree

It is called so because each triple of pairwise touching squares bounds a right triangle and we get a picture that is often illustrated by the Pythagorean theorem - "Pythagorean pants are equal in all directions."

It is clearly seen that the whole tree is limited. If the largest square is single, then the tree will fit into a 6 × 4 rectangle. Therefore, its area does not exceed 24. But, on the other hand, twice as many triples of squares are added each time as the previous one, and their linear dimensions are √2 times less. Therefore, at each step, the same area is added, which is equal to the area of ​​the initial configuration, that is 2. It would seem that then the area of ​​the tree should be infinite! But, in fact, there is no contradiction here, because quite quickly the squares begin to overlap and the area does not grow so fast. It is still finite, but apparently the exact meaning is still unknown, and this is an open problem.

If you change the angles at the base of the triangle, you will get a slightly different shape of the tree. And at an angle of 60 °, all three squares will be equal, and the tree will turn into a periodic pattern on the plane:

You can even replace squares with rectangles. Then the tree will be more like real trees. And with some artistic processing, pretty realistic images are obtained.

Peano curve

For the first time such an object appeared in an article by the Italian mathematician Giuseppe Peano in 1890. Peano tried to find at least a somewhat vivid explanation for the fact that the segment and the square are equally powerful (if we consider them as sets of points), that is, they have the "same" number of points. This theorem was previously proved by George Cantor in the framework of the set theory he invented. However, such conflicting intuition results caused great skepticism in relation to the new theory. Peano's example — building a continuous mapping from a line segment to a square — was a good confirmation of Cantor’s correctness.

Curiously, Peano's article did not have a single illustration. Sometimes the expression "Peano curve" is not attributed to a specific example, but to any curve that fills part of a plane or space.

Hilbert curve

This curve (Hilbert curve) was described by David Hilbert in 1891. We can only see finite approximations to the mathematical object that is meant - it will turn out in the limit only after an infinite number of operations.

Fractal "Greek Cross"

Another interesting example is the Greek Cross fractal.

Gosper curve

The Gosper curve, or the Gosper snowflake, is another variation of the curved lines.

Levy Curve

Although the object was studied by the Italian Ernesto Cesaro in 1906, his self-similarity and fractal properties were studied in the 1930s by the Frenchman Paul Pierre Levy. The fractal dimension of the boundary of this fractal is approximately equal to 1.9340. But this is a rather complicated mathematical result, and the exact value is unknown.

For its resemblance to the letter "C", written in an ornate font, it is also called the Levy C-curve. If you look closely, you can see that the Levy curve is similar to the shape of the crown of the tree of Pythagoras.

Hilbert cube

And there are also three-dimensional analogues of such lines. For example, a three-dimensional Hilbert curve, or a Hilbert cube.

An elegant metal version of the three-dimensional Hilbert curve (third iteration), created by Carlo Secin, professor of computer science at the University of California, Berkeley.

Sierpinski Triangle

This fractal was described in 1915 by the Polish mathematician Vaclav Sierpinski. To get it, you need to take an equilateral triangle with the inside, draw the middle lines in it and throw out the central of the four formed small triangles. Further, these same steps must be repeated with each of the remaining three triangles, etc. The figure shows the first three steps, and in a flash demonstration, you can practice and get steps up to the tenth.

Ejecting the central triangles is not the only way to get the Sierpinski triangle as a result. You can move "in the opposite direction": take the initially "empty" triangle, then complete the triangle formed by the middle lines in it, then do the same in each of the three corner triangles, etc. At first, the figures will be very different, but with the growth of the iteration number, they will more and more resemble each other, and in the limit coincide.

The next way to get the Sierpinski triangle is even more similar to the usual scheme of constructing geometric fractals by replacing parts of the next iteration with a scaled fragment. Here, at each step, the segments that make up the broken line are replaced by a broken line of three links (it is obtained in the first iteration itself). To postpone this broken line it is necessary alternately to the right, then to the left. It can be seen that the eighth iteration is very close to the fractal, and the further you go, the closer the line will get to it.

Carpet (square, napkin) Sierpinski

The respected mathematician did not stop at the triangles and in 1916 he described a square version. He managed to prove that any curve that can be drawn on a plane without self-intersections is homeomorphic to some subset of this holey square. Like a triangle, a square can be obtained from different designs. The classic method is shown on the right: dividing the square into 9 parts and throwing out the central part. Then the same is repeated for the remaining 8 squares, etc.

Like a triangle, a square has zero area.The fractal dimension of the Sierpinski carpet is log38; it is calculated similarly to the dimension of a triangle.

Sierpinski Pyramid

One of the three-dimensional analogues of the Sierpinski triangle. It is constructed in the same way, taking into account the three-dimensionality of what is happening: 5 copies of the initial pyramid, compressed twice, make up the first iteration, its 5 copies make up the second iteration and so on. The fractal dimension is log25. The figure has zero volume (at each step, half of the volume is ejected), but the surface area is preserved from iteration to iteration, and the fractal is the same as the initial pyramid.

Menger Sponge

Generalization of the Sierpinski carpet in three-dimensional space. To build a sponge, you need an endless repetition of the procedure: each of the cubes that make up the iteration is divided into 27 three times smaller cubes, from which the central and its 6 neighbors are thrown. That is, each cube generates 20 new ones, three times smaller. Therefore, the fractal dimension is log320. This fractal is a universal curve: any curve in three-dimensional space is homeomorphic to some subset of the sponge. The sponge has zero volume (since at each step it is multiplied by 20/27), but there is an infinitely large area.

There are still a lot of geometric fractals, and the surface area of ​​this page, unfortunately, is not infinite. Therefore, let's move on to the next type of fractals - algebraic.

Dynamic (algebraic) fractals

Fractals of this type arise in the study of nonlinear dynamical systems (hence the name). The behavior of such a system can be described by a complex nonlinear function (polynomial) f (z).

The Julia Sets

Take some starting point z0 on the complex plane. Now we consider an infinite sequence of numbers on the complex plane, each of which is obtained from the previous one: z0, z1 = f (z0), z2 = f (z1), ... zn + 1 = f (zn). Depending on the starting point z0, such a sequence can behave differently: tend to infinity as n → ∞; converge to some end point; take a series of fixed values ​​cyclically; more complex options are possible.

Thus, any point z of the complex plane has its own character of behavior during iterations of the function f (z), and the entire plane is divided into parts. Moreover, the points lying on the boundaries of these parts have the following property: at an arbitrarily small displacement, the nature of their behavior changes dramatically (such points are called bifurcation points). So, it turns out that sets of points having one particular type of behavior, as well as sets of bifurcation points often have fractal properties. These are the Julia sets for the function f (z).

Mandelbrot set

It is built a little differently. Consider the function fc (z) = z2 + c, where c is a complex number. We construct a sequence of this function with z0 = 0, depending on the parameter c, it can diverge to infinity or remain bounded. Moreover, all the values ​​of c for which this sequence is bounded form precisely the Mandelbrot set. It was studied in detail by Mandelbrot himself and other mathematicians, who discovered many interesting properties of this set.

It can be seen that the definitions of the Julia and Mandelbrot sets are similar to each other. In fact, these two sets are closely related. Namely, the Mandelbrot set is all the values ​​of the complex parameter c for which the Julia set fc (z) is connected (the set is called connected if it cannot be divided into two disjoint parts, with some additional conditions).

Halley Fractal

Such fractals are obtained if the Halley formula is used as a rule for constructing a dynamic fractal to search for approximate values ​​of the roots of a function. The formula is rather cumbersome, so anyone who wants to can see it on Wikipedia. The idea of ​​the method is almost the same as that used to draw dynamic fractals: we take some initial value (as usual, we are talking about complex values ​​of variables and functions) and apply the formula to it many times, getting a sequence of numbers. Almost always, it converges to one of the zeros of the function (that is, the value of the variable at which the function takes the value 0). The Halley method, despite the cumbersomeness of the formula, works more efficiently than the Newton method: the sequence converges to zero faster.

Newton fractal

Another type of dynamic fractals is Newton's fractals (the so-called pools). The formulas for their construction are based on the method of solving nonlinear equations, which was invented by the great mathematician back in the 17th century. Using the general formula of the Newton method zn + 1 = zn - f (zn) / f '(zn), n = 0, 1, 2, ... to solve the equation f (z) = 0 to the polynomial zk - a, we obtain a sequence of points: zn + 1 = ((k - 1) znk - a) / kznk-1, n = 0, 1, 2, .... Choosing various complex numbers z0 as initial approximations, we obtain sequences that converge to the roots of this polynomial. Since it has exactly k roots, the whole plane is divided into k parts - areas of attraction of the roots. The boundaries of these parts have a fractal structure (note in parentheses that if we substitute k = 2 in the last formula and take z0 = a as the initial approximation, we get a formula that is actually used to calculate the square root of a in computers). Our fractal is obtained from the polynomial f (z) = z3 - 1.

The use of fractals in industry and everyday life

Scientists are very passionate personalities. Don’t feed them bread, let’s fantasize on abstract topics. But we are practical people, and, having read everything that is written above, many probably already have a reasonable question: "so what?". So, what did this knowledge bring to the world?

Firstly, fractals are used in computer systems, and very densely. The most useful use of fractals in computer science is fractal data compression. This type of compression is based on the fact that the real world is well described by fractal geometry. At the same time, the pictures are compressed much better than is done by conventional methods (such as jpeg or gif). Another advantage of fractal compression is that when the image is enlarged, there is no pixelation effect (increasing the size of dots to sizes that distort the image). With fractal compression, after enlarging, the picture often looks even better than before.

Secondly, it is the mechanics of liquids and, as a consequence, the oil industry. The fact is that the study of turbulence in flows adjusts very well to fractals. Turbulent flows are chaotic and therefore difficult to accurately model. And here the transition to their fractal representation helps, which greatly facilitates the work of engineers and physicists, allowing them to better understand the dynamics of complex flows. Using fractals, you can also simulate the tongues of flame. Porous materials are well represented in fractal form due to the fact that they have a very complex geometry. It is used in petroleum science.

Thirdly, coming home from the factory in the evening, lying on your favorite combat sofa, you turn on the TV, which is also related to fractals. The fact is that antennas having fractal shapes are used to transmit data over distances, which greatly reduces their size and weight.

The use of fractal geometry in the design of antenna devices was first applied by an American engineer Nathan Cohen, who then lived in the center of Boston, where the installation of external antennas on buildings was prohibited. Cohen cut out a Koch curve shape from aluminum foil and then pasted it onto a piece of paper and then attached it to the receiver. It turned out that such an antenna works no worse than usual. Although the physical principles of such an antenna have not yet been studied, this did not stop Cohen from establishing his own company and arranging their serial production. Currently, the American company Fractal Antenna System has developed a new type of antenna. Now you can refuse to use protruding external antennas in mobile phones - the so-called fractal antenna is located directly on the main board inside the device.

In addition, fractals are used to describe the curvature of surfaces. A rough surface is characterized by a combination of two different fractals. They are also used in the development of biosensor interactions, studies of heartbeat, modeling of chaotic processes, in particular when describing animal population models and so on.

Fractal market structure

All this ode to fractals would be in vain if it were not for the fractal nature of financial markets. Yes, finally we got to the discussion of the very issue for which I wrote this article.

So, at present, there are many ways to analyze financial markets, on the basis of which traders create their trading strategies. Among the various analysis and forecasting tools, fractal analysis is on the sidelines. This is a separate versatile and interesting theory for discussion and study. The first impression speaks of the simplicity of the subject, but dig deeper, and you will see many hidden nuances.

Understanding fractals is the key to seeing hidden market information. But it is she who is one of the key factors of the market success of the speculator and the key to a large stable profit.

On October 14, 2010, Benoit Mandelbrot passed away - a man who in many ways changed our understanding of the objects around us and enriched our language with the word "fractal".

As you already know, it is thanks to Mandelbrot that we know that fractals surround us everywhere. Some of them are constantly changing, like moving clouds or flames, while others, like coastlines, trees or our vascular systems, preserve the structure acquired in the process of evolution. Moreover, the real range of scales where fractals are observed extends from the distances between molecules in polymers to the distance between clusters of galaxies in the Universe. The richest collection of such objects is collected in Mandelbrot's famous book "Fractal geometry of nature."

The most important class of natural fractals are chaotic time series, or time-ordered observations of the characteristics of various natural, social and technological processes. Among them there are both traditional (geophysical, economic, medical), and those that have become known relatively recently (daily fluctuations in the level of crime or traffic accidents in the region, changes in the number of hits of certain sites on the Internet, etc.). These series are usually generated by complex nonlinear systems that have a very different nature. However, for all, the behavior pattern is repeated at different scales. Their most popular representatives are financial time series (primarily stock prices and exchange rates).

The self-similar structure of such series has been known for a very long time. In one of his articles, Mandelbrot wrote that his interest in stock market quotes began with the statement of one of the stock exchanges: “... The price movements of most financial instruments are outwardly similar at different scales of time and price. The observer cannot tell by the appearance of the chart, the data refer weekly, daily, or hourly changes. "

Mandelbrot, who occupies a very special place in financial science, had the glory of a "subverter of the foundations", causing among economists a clearly ambiguous attitude towards himself. Since the advent of modern financial theory based on the concept of general equilibrium, he was one of its main critics and tried to find an acceptable alternative to it until the end of his life. However, it was Mandelbrot who developed the system of concepts, which, with appropriate modifications, as it turned out, allows not only to build an effective forecast, but also to offer, apparently, the only empirical substantiation of the classical theory of finance at the moment.

The main characteristic of fractal structures is the fractal dimension D, introduced by Felix Hausdorff in 1919. For time series, the Hurst index H is often used, which is associated with a fractal dimension by the ratio D = 2 - H and is an indicator of persistence (the ability to maintain a certain tendency) of a time series.

Usually there are three fundamentally different regimes that may exist in the market: at H = 0.5, price behavior is described by a random walk model; when H> 0.5, prices are in a trend (directional movement up or down); at H <0.5, prices are in a flat state, or frequent fluctuations in a fairly narrow price range. However, reliable calculation of H (as well as D) requires too much data, which excludes the possibility of using these characteristics as indicators that determine the local dynamics of the time series.

As you know, the basic model of financial time series is the random walk model, first obtained by Luis Bachelier to describe the observation of stock prices on the Paris Stock Exchange. As a result of the rethinking of this model, which is sometimes observed in the behavior of prices, the concept of an effective market arose in which the price fully reflects all available information.

For the existence of such a market, it is enough to assume that it has a large number of fully informed rational agents that instantly respond to incoming information and adjust prices, bringing them into equilibrium. All the main results of the classical theory of finance (portfolio theory, CAPM model, Black-Scholes model and others) were obtained in the framework of just such an approach. Currently, the concept of an effective market continues to play a dominant role in both financial theory and financial business.

Nevertheless, by the beginning of the 60s of the last century, empirical studies showed that strong changes in market prices occur much more often than the basic model of an effective market predicted (the random walk model). Mandelbrot was one of the first who subjected the concept of an effective market to comprehensive criticism.

Indeed, if it is correct to calculate the value of the H indicator for any stock, then it will most likely be different from H = 0.5, which corresponds to the random walk model. Mandelbrot found all possible generalizations of this model, which may be related to real price behavior. As it turned out, these are, on the one hand, the processes that he called Levy’s flight, and on the other, processes that he called the generalized Brownian motion.

To describe price behavior, the concept of a fractal market is usually used, which is usually considered as an alternative to an effective market. The concept assumes that the market has a wide range of agents with different investment horizons and, therefore, different preferences. These horizons vary from a few minutes for intraday traders to several years for large banks and investment funds.

A stable position in such a market is a regime in which "the average profitability does not depend on scale, except for multiplying by the corresponding scale factor". In fact, we are talking about a whole class of modes, each of which is determined by its value of the indicator H. Moreover, the value of H = 0.5 turns out to be one of many possible and, therefore, equal with any other value. These and other close considerations gave rise to serious doubts about the existence of a real equilibrium in the stock market.

Look at the price charts below:

It can be seen that the price makes constant fluctuations, thus forming a structure of a repeating nature.It is visible in all markets, regardless of the time scale.

The image shows charts: BRN M30, BTCUSD H1, DAX30 D1, EURSGD M5, USDCHF H1, XAUUSD M15. Without signatures and explanations, hardly anyone can distinguish them from each other.

These graphs are not exactly alike, but they share some common patterns. At a given period of time, the price moves in one direction, then changes its direction to the opposite and partially restores the previous movement, then reverses again. It does not matter which timeframe is used for the charts - they all look about the same (constant fluctuations), just like the fractals.

Fluctuations form market waves. What is a wave? This is an impulse and correction to it (movement-reversal-movement in the opposite direction, partially restoring the previous one). Such movements form waves.

The image shows these movements that form the waves. Several of these waves form a large wave of a similar shape (impulse correction). Several small waves form one medium-sized wave.

Medium-sized waves form one big wave. This is the essence of fractal theory in financial markets.

A series of such waves form directional movements in the market - trends. Such trends, in turn, form directional movements of an older temporal order. As in the case of waves, small movements form one mean, etc. This distinguishes between short-term trends, medium-term and long-term. This is a classic understanding of the fractal nature of the market.

Fractals Bill Williams

As I said, market fractals are one of the indicators in the Bill Williams trading system. It is believed that it was he who first introduced this name to trading, but, as you know, this is not so. When trading on fractals, in combination with its Alligator indicator, the author found local highs or lows of the market. He also wrote that determining the fractal structure of the market allows you to find a way to understand the behavior of prices.

In general, the theory of Williams fractals at one time caused heated debate, primarily because the author, as many believe, inserted a lot of scientific terminology (fractal, attractor, and so on) into his theory and did it not quite correctly.

In general, Williams fractals appear on the market quite often and on almost all timeframes and are, in fact, simple local extremes on a segment of 5 bars and practically do not correspond to the mathematical theory of fractals. Thomas Demark second-order TD points are exactly the same formation on the graph. However, despite all these coincidences - this theory is very popular to this day.

Williams technical analysis examines 4 existing fractal formations:

  • true fractal to buy;
  • false fractal to buy;
  • true fractal to sell;
  • false fractal for sale.

We will talk about true and false fractals and how to distinguish between them below.

Fractals indicator in the MetaTrader trading terminal

Bill Williams indicators do not require installation and are included in the standard set of indicators available to the trader out of the box. In order to attach the fractal indicator in the MetaTrader 4 terminal to the chart, you must: in the main menu (or in the "Navigator" window) select the menu item "Insert" - "Indicators" - "Bill Williams" - "Fractals":

The standard indicator for MT4 has no settings other than color. Its use with a fixed period of "5" negates all the possibilities and advantages of this tool. But for the MetaTrader platform, there are many custom indicators that will help solve this problem.

The problem of falsehood and truth of fractals

During trading using fractals, there is one important nuance - the appearance on the chart of a large number of signals, some of which are false. To filter them, Bill Williams developed another indicator called "Alligator", which can also be found in the standard set of indicators in MT4.

The problem of false fractals is the main source of errors, similar to estimates of the truth of the breakdown of support / resistance. Regardless of the specific methodology, the general principle for determining reliability is as follows - any deviations from the classic look should be in doubt. As in the entire technical analysis, a decrease in the timeframe leads to an increase in false signals and clutter the chart. Examples of unstable fractals are shown in the figure below.

When practicing large patterns, it is better to open positions at the moments of correction of the last price impulse, which are on the left side of the formation. Inside the pattern, standard Fibonacci corrections work reliably at 38% (0.382), 50% (0.500) and 62% (0.618). If you “stretch” levels through neighboring indicator signals, you can open through limit orders near key levels.

In the same way, you can protect the transaction from unpredictable reverse breakdown by gradually moving the stoploss to control the opposite maximum or minimum of the last and penultimate candles. When the structure is just forming, the stop should be at least 5-10 points above or below the last signal that the Fractal indicator gave. Then, with minor rollbacks, we remain in the market, and if there is a complete change of trend, the transaction closes with minimal loss.

There is another way to determine that we have false fractals - when they are pierced by a bar with a long shadow and a small body (pin bar). The longer its “nose”, the stronger the reversal signal, which means that the market failed to change the level of the last pattern the first time. If the breakdown has taken place, and the next candle is closed above High (for sale) or below Low (for purchase) of the nose, then with a high probability you can skip the signal and wait for the next one. A similar situation can happen in 3-5 bars, but we pay attention only to the bar that has broken the Fractals indicator.

The practical use of fractals

Bill Williams advised using fractals in strategies that are based on the breakdown of important price levels. The price movement above or below at least one point from the level of the previous fractal, according to the author of this indicator, already speaks of breaking through this level by the price.

Breaking through the level of the previous fractal is called a breakthrough of buyers in case the price rises above the previous fractal, directed upwards. In the opposite case, when the price falls below the previous fractal, directed downwards, they speak of a breakthrough of sellers. Bill Williams advised to regard the breakthrough of buyers or sellers as a signal to open a position.

Usually, traders place pending Stop orders several points above or below the fractal to open a position in case of breaking through this level. In such cases, the stop loss is usually set at the level of the penultimate opposite fractal.

In a classic interpretation, Bill Williams advises filtering trading signals generated by fractals using the Alligator indicator. So, to open a buy position, it is necessary that a fractal is located above the red line (the so-called "Alligator's teeth"). The author of the strategy advised entering the market immediately after breaking the fractal up or using a pending BuyStop order. Entrance to the market for sale occurs in case of breaking a fractal below the red line.

You can read more about this strategy in the article about the Bill Williams system Profitunity. And we will analyze the main practical ways to use fractals in isolation from this vehicle.

Fractal Breakout Trading

This method is classic, proposed by Bill Williams. As the name implies, the trade is breakdown in nature and is designed to continue the current trend. Entrance to the transaction is carried out by a pending stop order for the breakdown of the fractal closest to the price. An example you can see in the picture above.

According to the author himself, this trading methodology will give a lot of false inputs, so Bill suggests filtering signals using the Alligator indicator. In principle, the Alligator indicator can be replaced with ordinary moving averages and can also be used as a filter. But I repeat that it does not make sense to consider fractals and the Alligator separately from other Williams tools, so we will not dwell on this and move on.

Fractals as support / resistance levels

If you have encountered support / resistance levels at least once, then you know how difficult it is to build them, especially if you are a beginner. And all this complexity arises from the subjectivity of this tool. When we build levels, we cannot say with certainty whether we built them correctly or not. Bill Williams with his fractals gives us a great tool for finding and building meaningful levels of support and resistance.

Let's put an indicator on some chart and analyze it in terms of levels.

This is a USDCHF D1 chart with a classic fractal. Yes, the schedule is simply replete with these arrows. If a horizontal line is drawn through each extremum highlighted by the indicator, then the chart itself will not be visible behind these lines.

Let's increase the number of periods and look at the result:

As you can see, the chart has become better and really significant extremes remain, through which levels quite suitable for trading can be drawn. Pay attention to how the price "respects" and fulfills these levels. I am sure that in the future, when the price approaches them, we will again see a reaction to them.

Fractals and trend lines

Another pretty good method for applying the fractals indicator is to define reference points for plotting trend lines:

I threw the indicator on the chart, increasing the number of bars in the settings. Then he drew several trend lines through some fractals. Indeed, the lines turned out to be quite interesting, and the price interacts with them. Naturally, the trader should have basic knowledge in the field of technical analysis and building trend lines. But I am sure that this indicator will be a good help in practice for a beginner currency speculator.

Determining a trend using an indicator

Using fractals, we can also determine the dominant trend in the market. It is very easy to do. If we recall the definition of a trend, which states that an uptrend is a sequence of growing local highs and lows, and a downtrend is a sequence of decreasing extremes. Let’s throw our indicator on the chart and see that in an uptrend, buy fractals will update (break through) more often than sell fractals.

Definition of flat movement

If the price could not overcome the previous fractal, this can serve as a signal for the start of a flat movement. To confirm the signal, it is necessary to wait for the formation of the opposite fractal.

If he also could not break through the previous fractal, then we should expect a flat in the range between the upper and lower fractals, which will end after breaking through at the price of one of these levels.


The Fractal indicator and its modifications build on the chart many potential entry points for every taste, most of them seem quite reliable. In fact, this analysis technique is not so simple and unambiguous. Beginners are not recommended to use it as the only factor for making a decision.

Fractals cannot be used to predict prices. Even Williams considered them, at least, only the third confirming factor. Please note that the standard Fractal indicator, which is part of the basic set of trading platforms, has no parameters, so choose modifications where the number of settlement bars changes. So you can more accurately tune in to a specific asset.

Use will have a positive result only in combination with other indicators at time intervals of an hour or more. Strategies that include the Fractals indicator must definitely analyze several timeframes. However, do not discard this indicator.

Watch the video: What are Fractals? (December 2019).

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