Edgar peters fractal analysis of financial markets pdf. NS
Year of issue : 2004
Genre : finance, forex-forex, trading
Publisher:"Internet trading"
Format: DjVu
Quality : Scanned pages
Number of pages: 304
Description : It's time to take a more holistic look at how markets work. In particular, the time has come to acknowledge the great heterogeneity underlying markets. The participation of all investors is not due to the same reason, and investors do not use their strategies on the same investment horizons. Market stability is inevitably linked to the diversity of investors. The mature market is as heterogeneous as it is old. If all participants had the same investment horizon, if they responded in the same way to the same information and invested with the same goal, instability would rule everywhere. In contrast, mature markets have been remarkably stable over time. The day trader can trade anonymously with the pension fund: the former often trades for short-term profits; the latter trades infrequently and for the sake of long-term financial security. Day trader reacts to technical trends; investments pension fund are based on long-term potential economic growth... And yet, everyone acts at the same time, and each diversifies the other. The reductionist approach, with its rational investor, cannot cope with such heterogeneity without complex multi-element models that resemble the ingenious invention of Rube Goldberg. These models, with many limiting assumptions and requirements, inevitably fail. They are so complex that they lack flexibility, and flexibility is a critical factor in any dynamic system.
The first goal of this book is to present the fractal hypothesis of the market - a basic reformulation of how and why markets function. The second goal of the book is to present tools for analyzing markets within a fractal framework. Many existing tools can be used for this purpose. I will introduce new tools that analysts can add to their toolbox, as well as look at existing tools.
This book is not a story, although the main emphasis is still on conceptual aspects. Analytical methods are scrutinized within the conceptual framework. As in the previous book, I believe anyone with a solid knowledge of business statistics will find a lot of useful information here. The main emphasis is not on dynamics, but on empirical statistics, i.e. on time series analysis to determine what we are dealing with.
Practice shows that dynamics economic processes and phenomena are non-linear and often chaotic (unpredictable). This necessitates the search for alternative modeling methods using non-standard mathematical tools. Today, there are many areas in this area of economic and mathematical science. In the analysis of socio-economic processes, mathematical tools such as fuzzy methods, neural networks, genetic algorithms, etc. are increasingly used. However, when analyzing market dynamics, none of these methods can take into account such a property of the market as self-organization. This problem, to a certain extent, can be solved by the theory of fractals.
Since the 1980s, many Western scientists have been actively involved in the introduction of the theory of fractals into economics, while domestic researchers have begun to consider this theory relatively recently. The use of fractal analysis in economics is described in the works of such outstanding researchers as B. Mandelbrot, E. Peters, W. Arnold, P. Berger, I. Pomeau, C. Vidal, G. Schuster, R. Mantegne, H. Stanley, V. Chow, D. Sornett, A.Yu. Loskutov, A.S. Mikhailov, N.V. Chumachenko, A.I. Lysenko and others.
The use of the mathematical apparatus of the theory of fractals opens up new possibilities in modeling market processes. The key point contributing to this is the self-development of the fractal. This property characterizes a fractal as a mathematical object that most corresponds to the systemic nature of social and economic processes occurring under conditions of nonlinear dynamics of many factors of the external and internal environment.
In the real world, pure, ordered fractals, as a rule, do not exist, and one can only talk about fractal phenomena. They should only be viewed as models that are roughly fractals in a statistical sense. However, a well-constructed statistical fractal model allows one to obtain sufficiently accurate and adequate forecasts.
An example of one of the most effective applications of the theory of fractals in modeling market processes is fractal model stock market ... Due to the peculiarities of the functioning of the market valuable papers, it is rather difficult to predict the dynamics of prices on it. There are many recommendations and strategies, but only the use of fractals allows you to build an adequate model of the stock market behavior. The effectiveness of this approach is supported by the fact that many participants stock exchanges spend a lot of money to pay for the services of specialists in this field.
Fractal analysis of markets, in contrast to the theory of efficient markets, postulates the dependence of future prices on their past changes. Thus, the process of pricing in the markets is globally determined, dependent on "initial conditions", that is, past values. Locally, the pricing process is random, that is, in each specific case, the price has two development options. Fractal analysis of markets directly comes from fractal theory and borrows the properties of fractals to get predictions.
The main properties of fractals on the market:Market charts have a fractal dimension. The fractal dimension of a market chart is always 1
Market charts have the property of scale invariance or scaling. Different time intervals are self-similar.
Market charts always form a specific structure with unique properties.
Market fractals have a "memory" of their "initial conditions".
The first practitioner who applied fractal theory in the analysis of financial and commodity markets was Bill Williams ... Subsequently, his method of fractal market analysis has spread widely in many countries. This was facilitated by such his works as"Trade Chaos" "New dimensions in stock trading",
"Trading Chaos Second Edition"... Over time, many inattentive traders and analysts thought that behind the beautiful name was a competent PR move of the author rather than the actual use of fractals in the market. The main mistake that leads to the distortion of the analysis results is the misinterpretation of the concept of "overcoming a fractal". The ambiguity of fractal analysis ceases if the word "overcoming" is understood not as a puncture by the price of the fractal level, but as a breakout confirmed by closing the price above or below the fractal level.
Description of the market using fractals.
At the moment, fractal market analysis is the most common on the market.
Even if we take the price movement within one minute, we still get a line that connects the opening and closing prices. The generator for the price movement is another common structure that is well known to the trader -
"Impulse-correction-impulse"which looks like below:There can be an infinite number of these generators on the market, and there may not be two tipping points at all. What information can these figures give a trader? If you look at the price movement of an individual instrument, you can see that the structure of the generator is repeated on all time frames of the instrument (exhibits
fractal properties). Let's take for a given that the intra-annual price movement is a simple structure of two impulses and one correction, as in the figure above. If both impulses and the correction are replaced with the corresponding fractals (generators), we get the following structure:Going deeper and deeper, we come to minute and then tick charts, on which the basic fractal will appear again and again. Tellingly, the relationships between the generator lines will remain fixed at any time frame. The angles between the generator lines on the minute and monthly
graphics will match each other, the ratio 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 the opinion of Mandelbrot himself, "caricatured." It serves us to describe the general principle of the structure of price movement. A real market generator can be much more complicated.
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, consider two other observations of fractal geometry that are easier to understand and give the trader
food for thought.
The market has a memory.
Benoit Mandelbrot's extensive research on the cotton market has led him to conclude that periods of high volatility or "turbulence" tend to gather in
"Clusters" ... This means that events, the probability of which, according to generally accepted financial models, is negligible fractions of a percentage, in many cases occur in succession - one after another. This is fundamentally inconsistent with the "random walk" model that is used all over the world for risk management. According to her, all market events are independent of each other. Mandelbrot shows convincingly. it is not so. Market events tend to remain dependent on each other. He calls this effect -"The Joseph Effect", using as a metaphor the well-known biblical parable of the Pharaoh, who had a dream about seven fat and seven skinny cows (seven fruitful and seven lean years).What is
"Price cluster"? A price cluster means"trend". Trend in the economy - the direction of the predominant movement of indicators. It is usually considered within the framework of technical analysis, where the direction of price movement or index values is implied. Charles Doe noted that with an uptrendthe subsequent peak on the chart should be higher than the previous ones; in a downtrend, subsequent downturns on the chart should be lower than the previous ones (see Dow Theory). Highlight trends upward (bullish), downward (bearish) and lateral (flat ). A trend line is often drawn on the chart, which, in an uptrend, connects two or more price bottoms (the line is under the chart, visually supporting and pushing it up), and on a downtrend, it connects two or more price peaks (the line is located above the chart, visually limiting it and pressing down). Trend lines are the lines of support (for an uptrend) and resistance (for a downtrend). An uptrend (uptrend, bullish trend) is a situation when each new local minimum and local maximum is higher than the previous one.
An example of a growing trend.
![](https://i1.wp.com/m-rush.ru/images/fractalsineconomic/downtrend2.gif)
An example of a downtrend.
Noah effect
And finally, Mandelbrot's third observation is the so-called effect of "Noah" ... We know from the Old Testament that the Flood began unexpectedly, and its destructive power turned out to be very great. The Noah effect is a metaphor for market reversals - stock market crashes and booms. They never happen smoothly, almost always the market soars up or collapses with a force that none of the investors expected.
This always causes panic among the stock market, which is shocked by such price movements. So, in 1987, the Dow Jones fell by 22.6% in one day. After the crash, computer programs were blamed for everything, but Benoit Mandelbrot has a completely different opinion - it's not about software at all, it's about the very nature of the market. It is the intrinsic nature of the market that drives this dynamic. This hypothesis is also new and does not agree 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”, hoping that the market will sooner or later return to the level of opening a trade.
Mandelbrot's summary is that the market is a very risky place, far more risky than is commonly believed. For traders, risk is not a source of danger, but a potential source of profit. If you correctly use knowledge about price movements and find yourself on the "right" side of risk, it will be a blessing, and
not a curse.
At the end of the article, we also mention the use of fractals in time series modeling... In particular, such a characteristic of the time series as the fractal dimension makes it possible to determine the moment at which the system becomes unstable and is ready to move to a new state.
An example of a time series.
Thus, the theory of fractals provides a qualitatively new approach to modeling the economy. However, its novelty and inconsistency with classical methods complicate its widespread use. One of the main constraining factors is the randomness of the fractal model, which is due to the exclusive interdependence of its input and output parameters. Even the smallest change in the input parameter or the smallest mistake when setting it can lead to completely unpredictable behavior of the model. At the same time, due to the insufficiently developed mathematical apparatus of the theory itself, it is absolutely impossible to check (evaluate) the results obtained in fractal modeling. At the same time, this is indeed the most promising modern direction of mathematics from the point of view of applied research in economics.
Sources: fortrader.ru, Wikipedia and other materials from the Internet ..
In the previous article, we briefly reviewed the basic principles that Trade Chaos relies on. In fact, Williams perfected the Elliott Wave Theory, supplementing it with specific criteria for identifying the moment of completion and beginning of waves.
But for the picture to be complete today we will continue to consider Bill's trading techniques that significantly increase profits from speculation, and let's start with fractals.
In one of our earlier publications, we have already touched upon the topic of identifying and constructing fractals (including using indicators). Therefore, we will not repeat the theory again, we only note that a fractal is a formation, the central extremum of which is located above (below) the corresponding extrema of four neighboring bars.
The entire logic of fractal analysis in the Trading Chaos is based on the search for breakouts of extremes, but unlike later trading strategies developed by other traders, the original Williams model consists of strictly three elements:
- Fractal start - the first extremum preceding the signal;
- Signal fractal - is formed in the opposite direction to the start fractal;
- A fractal stop is the largest top on a downtrend (or trough on an uptrend) of the last two fractals.
Thus, fractal analysis completely eliminates uncertainty in decision-making and at the same time allows you to weed out a lot of false signals (but only if the prevailing trend is reliably known).
Speaking of trends, in Williams' Chaos theory, this issue is resolved by itself, since fractals become an integral part of wave analysis, and a wave (or wave structure) is a trend. At the same time, to maximize profits and further create a pyramid, it is permissible to switch to a lower timeframe after the start of the wave.
Pyramiding is an increase in a position in the direction of a trend after a floating profit on the first deal allows you to move a stop order for a set of orders to breakeven, while the volume of each new deal is either constant or divided by a certain coefficient.
For example, suppose that the third wave has begun on the market, for which all wave leaders are hunting, in this case, the trader's action algorithm will be as follows:
Besides, without fractal analysis any attempt to search for wave structures is doomed to failure - this is a fact proven by several generations of traders, although Bill warned about it. In his "five bullets," outlined in Chapter 9 of Trading Chaos, Williams listed the main signs that a trend is ending:
- Divergence appeared on the MACD between the third and fifth waves;
- The current price is located in the target zone, i.e. the fifth wave according to the approximate counting should have already begun (but not the fact that it will be formed completely), as a rule, beginners use Fibonacci levels to build zones, but much more often the situation is assessed visually;
- A fractal has formed at the next top during a bullish trend and at the bottom during a bearish trend;
- Among the three maximum (minimum) bars, a “squat” appeared (see the previous publication);
- The bars of the MACD histogram crossed the signal line in the direction opposite to the last trend.
In conclusion, we note that, despite the universality and good practical results, there is something to complain about in Williams' theory. For example, Bill argues that the market does not obey traditional physical laws, but at the same time behaves similarly to the tides of the sea, which, in fact, are associated with the gravitational influence of the Moon and the Sun on the Earth - is this not a law?
Therefore, one should not look for a hidden meaning in the Trading Chaos, Williams was simply able to describe the behavior of the market crowd for the first time using technical analysis tools, i.e., roughly speaking, mathematics, which deserves respect in any case.
Currently, there are many methods of analyzing financial markets, on the basis of which traders create their trading strategies.
Among the various analysis and forecasting tools, fractal analysis stands a little out of the way. This is a separate, versatile and interesting theory for discussion and study. The first impression speaks of the simplicity of the subject matter, but dig deeper and you will see many hidden nuances.
Understanding fractals is the key to seeing hidden information about the market. But it is she who is one of the key factors in the market success of a speculator and the key to a large stable profit.
Pyramids
It can be seen that these pyramids are composed of several similar pyramids on a smaller scale. These, in turn, consist of even smaller objects of the same shape. This image clearly demonstrates the essence of fractals.
The whole consists of many parts, which in their structure resemble this whole. All fractals are united by such a pattern, which is also found in nature.
The fractal theory was created because not everything can be described by exact sciences. If we take geometry, it immediately becomes clear that geometric shapes, or rather the dimensions of their shapes, are characterized by one or two parameters.
The size of the circle directly depends on the size of the diameter, the size of the square is set on one of its sides, the size of the triangle is based on the lengths of all its sides. But in fractal theory, everything is different - the type of fractal does not depend on the scale at all.
B. Mandelbrot
This pattern was discovered by Benoit Mandelbrot. This is a French mathematician who studied linguistics, aeronautics, game theory, economics and other areas of science. In his research into economics, Mandelbrot found that externally arbitrary price fluctuations can follow a hidden mathematical order in time... Moreover, this order cannot be described by standard curves.
In the future, he applied his developments to the financial markets. Studying the statistics of cotton prices for a very long period of time (more than 100 years), Mandelbrot was able to identify the trend of their changes. With this, he made an important discovery for economists and became very famous in the economic environment.
Market
What is the financial market? What does he look like?
The market is presented to us in the form of graphs of price fluctuations over time. The image shows a typical market chart.
But what can fractals do in the market?
It can be seen that the price is constantly fluctuating, forming a structure of a repeating nature. It can be seen in all markets, regardless of the time scale.
What happens if you compare 4 different price charts?
Picture A the EUR / USD chart of 1 hour scale is presented;
Image B- This is a 4 hour chart of gold prices;
Image C- USD / JPY on a scale of 5 minutes;
Image D shows price changes for IBM daily stocks.
Without signatures and explanations, hardly anyone will be able to distinguish the 4-hour chart of gold (B) from the 5-minute chart of USD / JPY (C).
These 4 graphs are not exactly alike, but they have some common patterns. For a given period of time, the price moves in one direction, then reverses its direction and partially restores the previous movement, and then turns around again.
It doesn't matter what timeframe is used for charts - they all look about the same (constant fluctuations), just like fractals.
Fluctuations form market waves. What is a Wave? This is an impulse and a correction to it (movement-reversal-movement in the opposite direction, partially restoring the previous one). Such movements form waves.
Waves
The image shows these movements, which form waves. Several of these waves form a large wave of the same shape (impulse correction). Several small waves form one medium-sized wave.
Waves
Medium sized waves form one large 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 a higher time order. As in the case of waves, small movements form one average, etc. This is how short-term trends are distinguished, medium-term and long-term.
The essence of fractal theory is simple and straightforward. How to use it for your own purposes, to make a profit?
The well-known trader described the use of fractals in financial markets in his book "Trading Chaos" Bill Williams.
He wrote that identifying the fractal structure of the market allows one to find a way to understand price behavior. It's a way to see order and predictability.
Williams introduced the concept of a market fractal, or as it is also called the B. Williams fractal. He focused on such a property as self-similarity. What is a Williams fractal?
B. Williams' fractals
This is a 5 bar combination. The central bar, on both sides of which there are 2 bars with rising and falling extremes. There are 2 types of fractals - buy and sell.
On the left, the image shows a sell fractal - a central bar that forms a low. On the left side of it there are 2 bars with decreasing lows, on the right side - with rising lows.
A buy fractal is shown on the right side of the image. The central bar forming the high. The bars to the left and right of it have rising and falling highs.
Technical analysis examines 4 existing fractal formations:
- true buy fractal;
- false fractal for buying;
- true fractal for sale;
- false fractal for sale.
You should know that a fractal is a formation that determines the current state of the market and shows the prevailing mood of its participants... But this mood can be determined only after the final formation of the fractal. It is truth and falsity that shows this mood.
What does the formation of a fractal in the market say?
True fractal to buy
The image shows that the formation of a fractal indicates the presence of resistance, which the bulls are not yet able to overcome. If buyers overcome this resistance, then the fractal will turn out to be true. And this will indicate the prevailing bullish sentiment of market participants.
False buy fractal
But this image shows a false buy fractal. Those. fractal with no potential. As in the previous example, the price, moving up, met resistance that the bulls could not overcome. A buy fractal has formed. However, in this example, it can be seen that the strength of buyers dried up, and the bulls could not overcome the resistance. As a result, the fractal turned out to be false, the market moved in the opposite direction.
Similarly, with fractals for sale. Their truth indicates the prevailing bearish sentiment of market participants. Falsity to the contrary.
As you can see, fractals show the behavior of market traders and their mood.
Fractal theory of the Forex market is the author's attempt to look at the foundations of financial markets through the prism of concepts such as synergetics, chaos, Mandelbrot set, Hurst exponent and Brownian motion. The book will help the reader uncover these concepts as applied to the Forex market and change their perception of things like quotes and prices. The theory of fractals can be successfully applied in combination with both technical and fundamental analysis. And the information that the reader will receive by studying the material will give him a solid foundation for interpreting the charts of exchange rates and securities.
Alexey Almazov. Fractal theory of the Forex market. - St. Petersburg: Admiral Markets, 2009 .-- 296 p.
Download the abstract (summary) in the format or
Chapter 1. Synergetics
Nowadays, when it becomes clear that financial markets are nonlinear systems, synergetics has made it possible to extend nonlinear concepts to the economic analysis of markets and to more clearly explain their nature, incl. ways of their further evolution.
A self-organizing system cannot be closed. The foreign exchange market is an open system. Millions of sources of information every day influence the course of quotes. Another condition for self-organization is the initial deviation from equilibrium. Such a deviation can be a consequence of a directed influence from the outside, but it can also arise in the system itself in a random way, stochastically. The third condition: all processes in the system (processes amenable to statistical analysis) occur cooperatively, self-consistently. In the market, this condition is fulfilled in accordance with the timescales. Graphs at various scales are consistent with each other.
Why does a system that develops according to well-defined laws behave chaotically? Chaos is generated by the intrinsic dynamics of a nonlinear system - its property of exponentially quickly spreading arbitrarily close trajectories. As a result, the shape of the trajectories depends very strongly on the initial conditions.
Since in a real physical experiment it is possible to set the initial conditions only with a finite accuracy, it is impossible to predict the behavior of chaotic systems for a long time. Henri Poincaré, in his Science and Method (1908), says: “In unstable systems,“ an absolutely insignificant cause, eluding us by its smallness, causes a significant effect that we cannot foresee. (…) Prediction becomes impossible. "
Mandelbrot, Prigogine and others found that on the border between conflicts of opposing forces there is not the birth of chaotic, disordered structures, as previously thought, but a spontaneous emergence of self-organization of a higher order occurs (see, for example,).
Wave synergetics theory. The essence of the theory is that the price chart has a certain structure of behavior. The role of the structure is played by the model of Brownian motion.
Chapter 2. Linear and nonlinear paradigms in the Forex market
If a system or organism wants to survive, it must evolve and be far from equilibrium. Therefore, a healthy economy and market do not strive for equilibrium, they strive for growth and development.
The linear paradigm states that markets are efficient. This theory argues that since current prices reflect all public information, no one market participant can have an advantage over another, thereby extracting excess profits. However, Robert Shiller has shown that some price changes are due to changes in fundamental information and uncertainty about future cash flow trends. Only 27% of the volatility of earnings in the US stock markets is explained in terms of fundamental information (see).
It is believed that traders are rational and obey the made estimates of exchange rates, while not taking into account the psychological characteristics of the participants. The linear paradigm postulates that exchange rates describe trajectories of a random walk (Brownian motion), and their distribution is normal and bell-shaped. The width of the bell (its sigma, or standard deviation) reflects how far price changes deviate from the mean; edge events are considered extremely rare.
However, the financial data does not match these assumptions. The magnitude of price movements can remain roughly constant for a year, and then suddenly the volatility can rise for a long time. Large price jumps have become common. The Gaussian normal distribution model does not reflect the real picture of what is happening in the financial markets. The value of currencies is not postulated by efficient market theory.
The old methods must be replaced by new ones that do not imply independence or normality. New methods should include fractals and nonlinear dynamics. The non-linear paradigm must admit the concept of long-term memory into market theory: an event can affect markets for a long time. The impossibility of development in linear systems occurs due to the fact that deterministic statistical systems have a small number of degree freedom, which significantly limits their adaptive capabilities, they are forced to yield in the process of development to more adaptive competitors.
A very interesting theory was proposed by Peters in his book, according to her, markets remain stable when many investors participate in them and have different investment horizons (investment terms).
Harold Edwin Hirst (1880-1978) - English physicist, famous for researching the Nile floods. Hirst introduced a new statistical technique based on the expression R (t, d) / S (t, d). This method was called R / S analysis. Hirst's discovery lies in the fact that R / S diagrams related to empirical chronicles, in the general case, consist of curves closely entwining some straight line, but the slope H this line varies from case to case. Different curves behave very differently, they are located near a certain straight line, the angle of inclination of which, H, often exceeds 0.5 (ie, does not correspond to the normal distribution; Fig. 1).
Rice. 1. Estimation of the Hurst exponent
A wavy line depicts a time series (a set of observed parameters of the studied system in time) of prices. The straight line corresponds to the indicator H(Hirst). When H = 0.5, the graph will correspond to the normal distribution and be random. At 0.5< Н < 1, процесс является персистентным. Если мы наблюдаем восходящую тенденцию, то в будущем она продолжит свой рост. Когда Н возрастает от 0,5 до 1, устойчивость становится все заметнее. С практической точки зрения это выражается в том, что возникающие разнородные «циклы» различаются все яснее. В частности, большую важность становятся медленные циклы. Если 0 < Н < 0,5, то процесс является антиперсистентным. Когда восходящая тенденция сменяется нисходящей или наоборот.
Chapter 3. Introduction to Fractals
In 1975, Benoit Mandelbrot first introduced the concept of a fractal - from the Latin word fractus, a broken stone, split and irregular. It turns out that almost all natural formations have a fractal structure. What does it mean? If you look at a fractal object as a whole, then at a part of it on a larger scale, then at a part of this part, etc., it is easy to see that they look the same.
A fractal is a geometric shape that can be divided into parts, each of which is a smaller version of the whole.
Properties of fractals.Irregularity. If a fractal is described by a function, then the property of irregularity in mathematical terms will mean that such a function is not differentiable, that is, not smooth at any point. Self-similarity. A fractal is a recursive model, each part of which repeats in its development the development of the entire model as a whole and is reproduced on various scales without visible changes. Self-similarity means that the object does not have a characteristic scale: if it had such a scale, you would immediately distinguish the enlarged copy of the fragment from the original image. Self-similar objects have infinitely many scales for all tastes.
In finance, the movements of a stock or currency are superficially similar, regardless of the time frame and price. The observer cannot tell from the appearance of the chart whether the data is for weekly, daily, or hourly changes.
The third property of fractals is that fractal objects have a dimension that is different from Euclidean. It bears the name of the Hausdorff-Besicovitch dimension. This dimension increases with increasing tortuosity, while the topological dimension stubbornly ignores all changes that occur to the line. If this is a curve with topological dimension equal to 1 (straight line), then the curve can be complicated by an infinite number of bendings and branches to such an extent that its fractal dimension approaches two, i.e. will fill almost the entire plane. (fig. 2).
Rice. 2. (a) A strongly curved line is capable of filling a plane with itself; (b) transition from dimension 1 to dimension 1.5
In multifractals, the value H.
If the fractal dimension is less than 1.4, the system is influenced by one or more forces that move the system in one direction. If the dimension is about 1.5, then the forces acting on the system are multidirectional, but more or less compensate each other. If the fractal dimension is much more than 1.6, the system becomes unstable and is ready to move to a new state. Hence, we can conclude that the more complex the structure we observe, the more and more the probability of powerful movement increases (Fig. 3).
Rice. 3. Modeled lines with different dimensions
When we apply classical models (for example, trend, regression, etc.), we say that the future of the object is unambiguously deterministic, i.e. completely depends on the initial conditions and lends itself to a clear forecast. You can do one of these models yourself in Excel. And fractals are used when the object has several development options and the state of the system is determined by the position in which it is at the moment. That is, we are trying to simulate chaotic development. The interbank foreign exchange market is just such a system.
Chapter 4. The theory of Elliott waves as the founder of the theory of fractals
Technical analysis of markets is a method of predicting further price trend behavior based on knowledge of the history of price development. Technical analysis for forecasting uses the mathematical properties of trends, and not the economic indicators of various countries to which this or that currency pair belongs. Technical analysis is based on 3 postulates:
- The market takes everything into account.
- Price movement is subject to trends. Bullish trend - upward price direction. Bearish trend - the downward direction of the price. Flat - lateral (horizontal) market movement.
- History repeats itself.
In Elliot's theory, numbers are used to denote the five wave trend, and letters are used for the opposite three wave trend. If a wave is directed towards the main trend and consists of five wave movements, then it is called impulsive. If the direction of the wave is opposite to the main trend, and it consists of three wave movements, then it is called corrective (Fig. 4).
Rice. 4. Elliott Wave Cycle
Based on the definition of a fractal, Eliot was the first to notice that waves of a smaller order are similar to waves of a higher order and that the system is self-similar. But when most of us are faced with the reality of the data, and not with the simple scheme that is described in detail in the wave theory, many are disappointed that they do not find this cycle in its original form.
Elliot proposed a self-similar model of price behavior, which in its essence is a fractal, but it does not reflect all the properties inherent in this concept and what actually happens in the financial markets.
In the foreign exchange market, time is multifractal, and in the role of price we observe Brownian movement, generalized or fractional!
Elliot just laid the foundation and proposed a simplified form of price action.
Chapter 5. Benoit Mandelbrot model
Benoit Mandelbrot proposed a fractal model, which has already become classical and is often used to demonstrate both a typical example of the fractal itself, and to demonstrate the beauty of fractals, which also attracts researchers, artists, simply interested people. This model, which received the name "Mandelbrot set", laid the foundation for the development of fractal geometry (Fig. 5).
Rice. 5. Mandelbrot set
The Mandelbrot model has characteristic properties. Self-similarity is perhaps one of the most important properties of this model.
The next property that our model has is its dimension (detail). In relation to the market, you can see that the weekly price scale has the most detailed data, which makes its structure clearer than the minute charts (Fig. 6).
Rice. 6. Dimension (detail) of the model: (a) weekly chart, (b) minute chart
A characteristic property of the Mandelbrot set is its irregularity. The Mandelbrot model randomly chooses the direction of the further development path, which looks like a separation of trajectories. This point is usually called the bifurcation point. The most amazing property of the Mandelbrot set is infinite variance.
When analyzing Forex charts, the golden ratio and can be applied.
Chapter 6. Generator - the Holy Grail in the Forex market
By a model, we mean a regularly built price structure formed in a complete cycle. Forex charts can be generated using diagonal self-affinity. Transformations are said to be affine when they use translation and reduction operations.
I build all models based on the Weierstrass-Mandelbrot function:
Parameter b determines how much of the curve is visible when the argument t changes in the specified interval. Parameter D takes the values 1< D < 2 и является показателем размерности фрактальной кривой. Например, при D = 1,5 и b =1,5 мы имеем модель, названную мной, как «модель 1.5» (рис. 7).
Rice. 7. Model 1.5
The notion of dimension can be easily correlated with the volatility of exchange prices. That is, using the parameter D, when we match our model to a similar one on the market, we can adjust it so that price volatility and the dimension of the model become almost identical.
Models can be obtained using a program that can generate them by specifying parameters D and b... Models 1.43, 1.5, 1.6, 1.7, 1.9 are closest to the market realities.
Chapter 7. Initial conditions and main stages of development of models
For example, we will use model 1.9 (Fig. 8). Waves are delimited by vertical lines. Model 1.9 includes the most complete and standardized list of items. All other models are derived from this structure.
Rice. 8. Structure of the model 1.9
This Origin has the following features:
- This structure begins after a downward movement.
- As a rule, the last wave in the origin wave is quite pronounced (shown by an arrow in the figure).
- The retracement from the origin wave should not cross its base.
- The key cancellation level of this structure will be the breakdown of the 23.6 Fibonacci level.
The main features of the trident wave:
- Unlike origin, it does not start from the next low of the downtrend.
- The beta point must never cross the bottom of the origin wave.
- If the slope between the alpha and beta points is steep, then the trend will be quite powerful and impulsive. If it is flat, then the trend will go not at an angle, but in a horizontal direction.
Impulse wave features:
- It is the most noticeable of all waves, which is expressed in the duration and speed of its movement.
- It almost always reaches the 161.8 level from the origin wave.
- All indicators show the maximum value at the stage of the end of the impulse wave.
- This wave can consist of two cycles.
The salient features of the revival wave:
- Unlike the trident wave, in this structure, the alpha and beta points are not particularly important, since the beta level can be significantly lower than alpha, which does not mean the cancellation of the upward movement.
- In this structure, it is very important to ensure that the revival highs do not become the maximum levels of the entire model as a whole.
In order to learn how to correctly define the cycle, we must be able to vary the time scales. The ability to work with different scales is the beginning of the path to professional trading! If we do not see the development of the model at a certain price scale, then we have a part of a larger cycle. This is why we will not see the whole pattern after every downward (upward) move. Cycles that develop on a characteristic price scale for them have the highest dimension than a part of a larger cycle located on the same scale.
Chapter 8. Definition of cycles in the foreign exchange market
Is there a cycle in the market? To this day, there is no specific answer to this question. This is how Mandelbrot describes the presence of cycles in financial markets: “… all periodicities are 'artifacts', not a characteristic of a process, but rather a cumulative result depending on the process itself, the sample length and the judgment of an economist or hydrologist. The first of the factors mentioned is external to the observer, the second (depending on the specific case) can be assumed in advance or chosen arbitrarily, and the third is subjective in all cases, that is, it is a product of human perception and a subject of disagreement. (However, these disagreements often concern only details, which may be of interest from the point of view of the theory of perception.) "
Mandelbrot suggested using the Hurst exponent as the definition of the dimension of self-affine processes:
(2) H = logP / logT
The fractal dimension in this case is defined as:
(3) D = Dm - H
and characterizes how the object fills the space. The higher D, the more noise on the graphs. minute and hour charts. The shadows that we see at different time scales determine the amount of noise in the time series! The longer the shadows of the candlesticks, the more noisy the pair is, which is reflected in the deviation of the price at the moment of receipt of new information from the true value (structure). Hence, we can conclude that the scale in the foreign exchange market is a kind of filter that filters out all unnecessary information and determines the more important (Fig. 9).
Rice. 9. Scales as filters
A distinctive feature of the cycles that are present in the foreign exchange market is their non-periodicity. This means that the cycle does not have a specific standard length. Peters in his book "Chaos and Order in the Capital Market" gives the following definition: "The average cycle length is the duration after which the memory of the initial conditions is lost." Peters attempts to determine cycle length using R / S analysis. He found the average S & P500 cycle length to be 4 years. However, the author makes an allowance for the fact that this is a kind of statistical cycle and that it is of absolutely no interest for practical trading.
The fractal theory is applicable to the market, and those irregular curves that we daily observe on our monitors are nothing more than a fractal time series. This gives rise to the concept of multifractal exchange time. Einstein found that the mean square of the distance a randomly wandering particle moves away from its origin is proportional to time. The mean square of the distance for a fractal medium turns out to be proportional to some fractional power of time, the exponent of which is related to the fractal dimension of the medium α :
(4) α = 1 / H
(5) dP ~ (dt) N
If H = ½, the model is characteristic of an efficient market. Where it is postulated that the price distribution process is Gaussian. Here dP- price change corresponding to the time interval dt... The most common value, the Hurst exponent for foreign exchange markets, fluctuates around 0.58 - 0.6, which corresponds to α = 1,7 ("Model 1.7"). Because the H constantly changing, it will multifractal... The prefix multi means that we have not one, but several values of H at different time intervals.
Chapter 9. How to combine fractal theory with other types of analysis
Set support and resistance levels based on the highs and lows of prices. The distance between the level and the shadows should be at least 1-2 points. How do you know if this is really the maximum price or just another surge? This is done by identifying the highs and lows that price has made in the past. Fractal market theory assumes that past values of the price correlate with its future values.
When trading in the foreign exchange market, a trader does not even think about the relationship between individual currencies and is limited to just one pair. Many people believe that the foreign exchange market is a system divided into many separate elements (currency pairs) that are essentially unrelated to each other! You are missing out on huge opportunities by not using the structure of the behavior of different currency pairs. I am not urging you to open deals in 2, 3, and even more so in 10 pairs at the same time. You can always open deals only on one currency pair, but do it so that the signals to enter the market do not contradict another currency pair associated with it.
For example, the Euro / Dollar and Pound / Dollar currency pairs are unidirectional, and we observe similar price movement patterns. At the same time, they have different volatility, and hence different key levels. One of them reaches levels much faster than a similar currency. We just have to set the key levels and wait for their breakdown. As soon as they are broken, we can confidently assume that for the currency where the price has not even approached them, a breakdown of the key level is possible.
The essence of the market is the price movement, or rather its structure. A trader who is influenced by various sources of information, and also tries to apply indicators, runs the risk of getting away from a correct forecast. I am not at all opposed to indicators and fundamental analysis, I only call for the fact that, using these tools, we do not need to forget about the subject itself. The use of fractal theory helps to determine the direction of the price, however, taking into account the fluctuations in the volatility of each pair separately, the indicators will help to most accurately orient us in the current situation. Yes, they are more convenient in the sense that they can be used to find the most key points for entering or exiting the market. But the indicators show the general direction of the price very poorly, which is their significant disadvantage.
Chapter 11. Psychology of trade
Working on real accounts, a person no longer thinks, he works. Traders imperceptibly cease to reasonably think about their losses and gains, for them only one goal begins to exist - to earn more and faster. This syndrome manifests itself in the following:
- Implementation of a large number of transactions in a short period.
- Lack of concept of risk.
- After a successfully completed deal, this player opens another one.
- There is a rejection and misunderstanding of the theory. Market operation takes place without preheating.
- Inability to accept defeat.
Everyone knows this type of order as stop loss and take profit. You must understand that if you do not know where to limit your losses and how much profit you need to take away, before you want to make a trade, you have already lost.
After you carry out a successful deal, you are in the state of a winner who can handle everything, you celebrate, and during the holiday it is not typical for a person to soberly assess the situation. Therefore, do not rush to immediately open another deal, take a rest, gather your strength, and your trade will become truly professional.
Literature
David Ruel. ... - Izhevsk: RKhD, 2001 .-- 192 p.
Benoit Mandelbrot. : a fractal revolution in finance. - M .: Williams, 2006 .-- 408 p.
Benoit Mandelbrot. Fractal geometry of nature. - Moscow – Izhevsk: IKI, 2002. - 656 p.
Benoit Mandelbrot. ... - Izhevsk: RKhD, 2004 .-- 256 p.
Edgar Peters. ... - M .: Internet-trading, 2004 .-- 304 p.
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