Edgar peters fractal analysis of financial markets pdf. E
Year of issue : 2004
Genre : finance, 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, it is time to recognize the great heterogeneity that underlies markets. The participation of all investors is not due to the same reason, while investors do not use their strategies on the same investment horizons. The stability of the markets is inevitably linked to the heterogeneity of investors. The "mature" market is as heterogeneous as it is old. If all participants had the same investment horizon, if they reacted the same way to the same information, and invested for the same purpose, instability would rule everywhere. Mature markets, on the other hand, have been remarkably stable for a long time. A day trader can trade anonymously with a pension fund: the former trades frequently for short-term profits; the latter trades infrequently and for the sake of long-term financial security. The day trader reacts to technical trends; investments pension fund based on long-term potential economic growth. And yet, all act simultaneously, and each diversifies the other. The reductionist approach, with its rational investor, cannot deal with such heterogeneity without complex multi-element models that are reminiscent of Rube Goldberg's contraption. These models, characterized by numerous 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 purpose of this book is to present the fractal market hypothesis, 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 present new tools that analysts can add to their toolbox, as well as review existing tools.
This book is not a story, although the main emphasis, nevertheless, is on the conceptual aspects. Analytical methods are carefully studied within the framework of the conceptual framework. As in the previous book, I believe anyone with a sound knowledge of business statistics will find much to gain 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 is non-linear and often chaotic (unpredictable) in nature. This necessitates the search for alternative modeling methods using non-standard mathematical tools. To date, there are quite a few directions in this field of economic and mathematical science. In the analysis of socio-economic processes, such mathematical tools 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 the economy, while domestic researchers began to consider this theory relatively recently. The use of fractal analysis in economics is described in the works of such prominent researchers as B. Mandelbrot, E. Peters, V. Arnold, P. Berger, I. Pomo, C. Vidal, G. Schuster, R. Manten, 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 is most consistent with the systemic nature of social and economic processes occurring in a non-linear dynamics of many factors of external and internal environments.
In the real world, pure, ordered fractals, as a rule, do not exist, and one can only speak of fractal phenomena. They should only be considered as models that are approximately fractals in a statistical sense. However, a well-built statistical fractal model makes it possible to obtain fairly accurate and adequate forecasts.
An example of one of the most effective applications of fractal theory in modeling market processes is fractal model stock market . Due to the peculiarities of the functioning of the market valuable papers, it is quite 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. In favor of the effectiveness of this approach is the fact that many participants stock exchanges spend a lot of money to pay for the services of specialists in this field.
The fractal analysis of markets, in contrast to the theory of efficient markets, postulates the dependence of future prices on their past changes. Thus, the pricing process in the markets is globally determined, dependent on "initial conditions", that is, past values. Locally, the pricing process is random, that is, in each case, the price has two development options. Fractal market analysis comes directly from fractal theory and borrows the properties of fractals to make predictions.
The main properties of fractals in 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 certain structure with unique properties.
Market fractals have a "memory" of their "initial conditions".
The first practitioner who applied the 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 works as"Trade Chaos", "New dimensions in stock trading",
"Trade Chaos Second Edition". Over time, many inattentive traders and analysts considered that behind the beautiful name lies a competent PR move by the author rather than the actual use of fractals in the market. The main mistake that leads to a distortion of the analysis results is the misinterpretation of the concept of "overcoming the fractal". The ambiguity of the fractal analysis stops if the word "overcoming" is understood not as a puncture by the price of the fractal level, but as a breakdown confirmed by the closing of the price above or below the fractal level.
Description of the market using fractals.
At the moment, fractal market analysis is the most common in the market.
Even if we take the price movement within one minute, we will still get a line that connects the opening price and the closing price. The generator for the price movement is another common structure, well known to the trader, -
"momentum-correction-momentum", which looks like below:There can be an infinite number of these same generators on the market, and there may not be two turning points at all. What information can these figures give the trader? If you look at the movement of the price of an individual instrument, you can see that the structure of the generator is repeated on all time scales of the instrument (shows
fractal properties). Let's take it for granted 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 correction are replaced by the corresponding fractals (generators), we will get the following structure:Going deeper and deeper, we will reach the minute, and then the tick charts, on which the basic fractal will appear again and again. Tellingly, the ratios between the lines of the generator will remain fixed on any time frame. The angles between the lines of the generator on the minute and month
the graphics will match each other, the ratio of their lengths as well. This amazing discovery gives us a completely new look at the usual price movement.
Of course, this understanding is simplified, and, according to Mandelbrot himself, "cartoon". It serves us to describe the general principle of the structure of the price movement. A real market generator can be much more complex.
In modeling the behavior of the market, Mandelbrot uses a more complex "multi-fractal" model, which 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.
Extensive research on the cotton market led Benoit Mandelbrot to the following conclusion: periods of high volatility or "turbulence" tend to condense into
"clusters" . This means that events, the probability of which, according to generally accepted financial models, is negligible fractions of a percent, in many cases occur in succession - one after another. This is fundamentally inconsistent with the "random walk" model that is used worldwide for risk management. According to it, all events in the market are independent of each other. Mandelbrot convincingly shows. it is not so. Events in the market tend to remain dependent on each other. He calls this effect -"The Joseph Effect", using as a metaphor the famous biblical parable of the pharaoh, who had a dream about seven fat and seven skinny cows (seven productive and seven lean years).What does it represent
"price cluster"? The 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 Dow noted that in an uptrendthe subsequent peak on the chart should be higher than the previous ones, in a downtrend, subsequent declines on the chart should be lower than the previous ones (see Dow Theory). Highlight trends ascending (bullish), descending (bearish) and side (flat ) . A trend line is often drawn on the chart, which, on an uptrend, connects two or more price troughs (the line is below the chart, visually supporting it and pushing it up), and on a downtrend, it connects two or more price peaks (the line is above the chart, visually limiting it). and pressing down). Trend lines are support lines (for an uptrend) and resistance lines (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.
Uptrend example.
![](https://i1.wp.com/m-rush.ru/images/fractalsineconomic/downtrend2.gif)
Downtrend example.
Noah effect
And finally, Mandelbrot's third observation is the so-called Noah effect . We know from the Old Testament that the global flood began unexpectedly, and its destructive power turned out to be very great. The "Noah" effect is a metaphor that characterizes market reversals - stock market panic crashes and ups. They never happen smoothly, almost always the market shoots up or collapses with such force that none of the investors expected.
This always causes panic among the stock exchange public, which is shocked by such price movements. So, in 1987, the Dow Jones fell by 22.6% in one day. After the crash, everything was blamed on computer programs, but Benoit Mandelbrot has a completely different opinion - it's not about the programs at all, it's about the very nature of the market. It is the inherent nature of the market that causes such dynamics. This hypothesis is also new and is not consistent 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 transaction.
Mandelbrot's summary is this: the market is a very risky place, much more risky than is commonly believed. For traders, risk is not a source of danger, but a potential source of profit. If you use knowledge of price movements correctly and be on the “right” side of risk, it will be a boon, and
not a curse.
Finishing the article, we also mention the use of fractals in time series modeling. In particular, such a characteristic of the time series as fractal dimension makes it possible to determine the moment at which the system becomes unstable and is ready to move to a new state.
Time series example.
Thus, the theory of fractals provides a qualitatively new approach to economic modeling. However, its novelty and inconsistency with classical methods make it difficult to use it widely. One of the main limiting factors is the randomness of the fractal model, which is due to the exclusive interdependence of its input and output parameters. Even the slightest change in the input parameter or the smallest error in 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 verify (evaluate) the results obtained in fractal modeling. At the same time, this is indeed the most promising modern area of mathematics in terms 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 Trading Chaos relies on. In fact, Williams improved the Elliott wave theory, supplementing it with specific criteria for identifying the moment of completion and beginning of waves.
But to complete the picture, today we will continue to consider Bill's trading techniques, which significantly increase profits from speculation, and let's start, perhaps, with fractals.
In one of the earlier publications, we have already touched upon the topic of identifying and constructing fractals (including with the help of indicators). Therefore, we will not repeat the theory again, we will only note that a fractal is a formation, the central extremum of which is above (below) the corresponding extrema of four neighboring bars.
The whole logic of fractal analysis in 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 strictly of three elements:
- Fractal start - the first extremum preceding the signal;
- Signal fractal - is formed in the opposite direction to the starting fractal;
- A fractal stop is the largest downtrend top (or uptrend bottom) of the last two fractals.
In this way, 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, in order 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 the floating profit on the first trade allows you to transfer the stop order on the totality of orders to breakeven, while the volume of each new trade is either equal to a constant or divided by a certain coefficient.
For example, suppose that the third wave has begun in the market, for which all the wavers are hunting, in this case, the algorithm of the trader's actions will be as follows:
Besides, without fractal analysis any attempt to look for wave structures is doomed to failure - this is a fact that has been verified by several generations of traders, although Bill warned about this. In his "five bullets" outlined in the ninth chapter of Trading Chaos, Williams listed the main signs of the end of a trend:
- There was a divergence 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 marking, should already begin (but not the fact that it will form 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 one;
- Among the three maximum (minimum) bars, a “squat” appeared (see previous publication);
- MACD histogram bars crossed the signal line in the opposite direction to the last trend.
In conclusion, we note that, despite the universality and good practical results, Williams' theory has something to complain about. For example, Bill claims that the market does not follow 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 Trading Chaos, Williams was simply able to describe the behavior of the market crowd for the first time with technical analysis tools, that is, roughly speaking, mathematics, which deserves respect in any case.
Currently, many methods of analyzing financial markets are used, on the basis of which traders create their trading strategies.
Among the various tools for analysis and forecasting, fractal analysis stands a little apart. This is a separate versatile and interesting theory for discussion and study. The first impression speaks of the simplicity of the theme, but dig deeper and you will see many hidden nuances.
Understanding fractals is the key to seeing the hidden information about the market. But it is she who is one of the key factors in the market success of the speculator and the key to a large stable profit.
pyramids
It can be seen that these pyramids consist of several similar pyramids of a smaller scale. Those in turn consist of even smaller objects of the same shape. This image clearly demonstrates the essence of fractals.
The whole is made up 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 forms, are characterized by one or two parameters.
The size of a circle directly depends on the size of the diameter, the size of a square is given by one of its sides, the size of a triangle is based on the lengths of all its sides. But in the 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 the process of researching economics, Mandelbrot discovered that seemingly 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 achievements to financial markets. Studying the statistics of cotton prices over a very long period of time (more than 100 years), Mandelbrot was able to identify the trend in their changes. With this, he made an important discovery for economists and became very famous in the economic environment.
Market
What is a financial market? What does he look like?
The market is presented to us in the form of charts 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 makes constant fluctuations, thus forming a structure of a repetitive nature. It is visible in all markets, regardless of the time scale.
What happens if you compare 4 different price charts?
In picture A EUR/USD 1-hour chart is presented;
Picture B- this is a 4-hour gold price chart;
Picture C– USD/JPY on a scale of 5 minutes;
Picture D shows the price changes for IBM shares on a daily basis.
Without signatures and explanations, it is unlikely that anyone will be able to distinguish the 4-hour gold chart (B) from the 5-minute USD/JPY chart (C).
These 4 graphs are not quite similar to each other, but share some common patterns. At a given time interval, the price moves in one direction, then changes its direction to the opposite and partially restores the previous movement, after which it reverses again.
It doesn't matter what timeframe is used for the charts - they all look pretty much the same (constant fluctuations), just like fractals.
Fluctuations form the waves of the market. 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 that form waves. Several of these waves form a large wave of a similar shape (impulse-correction). Several small waves form one medium-sized wave.
Waves
Waves of medium size 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 a higher temporal order. As in the case of waves, small movements form one average, and so on. So distinguish between short-term trends, medium-term and long-term.
The essence of fractal theory is simple and clear. How to use it for your own purposes, to make a profit?
The use of fractals in financial markets was described in his book "Trading Chaos" by a well-known trader Bill Williams.
He wrote that the definition of the fractal structure of the market allows you to find a way to understand the behavior of prices. 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 emphasized such a property as self-similarity. What is a Williams fractal?
Fractals B. Williams
This is a combination of 5 bars. The central bar, on both sides of which there are 2 bars with rising and falling extreme points. There are 2 types of fractals - buy and sell.
On the left of the image, a sell fractal is shown - the central bar that forms a low. On the left side of it there are 2 bars with lower lows, on the right side there are 2 bars with higher lows.
The 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 considers 4 existing fractal formations:
- true buy fractal;
- false fractal to buy;
- true sell fractal;
- false sell fractal.
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 show this mood.
What does the formation of a fractal in the market mean?
True buy fractal
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 be true. And this will indicate the prevailing bullish mood of market participants.
False buy fractal
But this image shows a false buy fractal. Those. a fractal with no potential. As in the previous example, the price, moving up, met resistance, which the bulls could not overcome. A buy fractal has formed. However, in this example, it is clear that the forces of buyers have dried up, and the bulls have not been able to 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 mood of market participants. Lies to the contrary.
As you can see, fractals show the behavior of market traders and their mood.
The fractal theory of the Forex market is an attempt by the author to look at the foundations of financial markets through the prism of such concepts as synergy, chaos, the Mandelbrot set, the Hurst exponent and Brownian motion. The book will help the reader to understand these concepts in the application to the Forex market and change his perception of such things as 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 charts of exchange rates and securities.
Alexey Almazov. Fractal theory of the Forex market. - St. Petersburg: Admiral Markets, 2009. - 296 p.
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Chapter 1. Synergetics
At present, when it becomes clear that financial markets are non-linear systems, synergetics has made it possible to extend non-linear concepts to the economic analysis of markets and 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 affect the course of quotes every day. Another condition for self-organization is the initial deviation from equilibrium. Such a deviation may be the result of a directed influence from the outside, but it may also occur in the system itself randomly, stochastically. The third condition: all processes in the system (processes that can be statistically analyzed) occur cooperatively, self-consistently. In the market, this condition is met in time scale matching. Graphs at different 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 non-linear system, i.e., by its property to exponentially quickly separate 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 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 in 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 is not the birth of chaotic, disorderly structures, as previously thought, but the spontaneous emergence of self-organization of a higher level order occurs (see, for example,).
Theory of wave synergetics. The essence of the theory is that the price chart has a certain structure of behavior. The model of Brownian motion acts as a structure.
Chapter 2. Linear and non-linear paradigms in the Forex market
If a system or an organism wants to survive, it must evolve and be far from equilibrium. Therefore, a healthy economy and market do not strive for balance, they strive for growth and development.
The linear paradigm states that markets are efficient. This theory states that since current prices reflect all public information, no market participant can have an advantage over another, thereby extracting excess profits. However, Robert Schiller has shown that some price changes are due to changes in fundamental information and the uncertainty of future cash flow trends. Only 27% of the volatility in the volume of profits in the US stock markets is explained in terms of fundamental information (see).
It is believed that traders are rational and obey the assessments of exchange rates made, while not taking into account the psychological characteristics of the participants. The linear paradigm postulates that exchange rates describe the trajectories of a random walk (Brownian motion), and their distribution is normal and has the shape of a bell. The width of the bell (its sigma, or standard deviation) represents how far price changes deviate from the mean; edge events are considered extremely rare.
However, the financial data does not match such assumptions. The magnitude of price movements may remain roughly constant for a year, and then suddenly the volatility may increase for a long time. Large price hikes have become commonplace. 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.
Old methods must be replaced by new ones that do not assume independence or normality. New methods should include fractals and non-linear dynamics. The non-linear paradigm must allow for the concept of long-term memory in market theory: an event can affect markets for a long time. The impossibility of development in linear systems is due to the fact that deterministic statistical systems have a small number of power-law freedom, which significantly limits their adaptive capabilities, they are forced to yield to more adaptive competitors in the process of development.
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 (terms of investment).
Harold Edwin Hurst (1880–1978) was an English physicist who became famous for his studies of the Nile floods. Hurst introduced a new statistical technique based on the expression R(t, d)/S(t, d). This method has been called R/S analysis. Hurst's discovery is that the R/S diagrams related to empirical chronicles generally consist of curves closely wrapping around a certain straight line, but the angle of inclination H this line varies from case to case. Different curves behave very differently, they are located near some straight line, the slope of which, H, often exceeds 0.5 (i.e. does not correspond to a normal distribution; Fig. 1).
Rice. 1. Estimation of the Hurst exponent
The wavy line depicts a time series (a set of observed parameters of the system under study over time) of prices. The straight line corresponds to the indicator H(Hirst). When H = 0.5 the graph will follow a 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 its part on an enlarged 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.
Fractal properties.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 at different 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 an infinite number of scales for all tastes.
In finance, the movements of a stock or currency are outwardly similar, regardless of the time frame and price. The observer cannot tell by the appearance of the graph whether the data is for weekly, daily, or hourly changes.
The third property of fractals is that fractal objects have a dimension other than Euclidean. It is called the Hausdorff-Besikovich dimension. This dimension increases as the tortuosity increases, while the topological dimension stubbornly ignores all changes that occur with the line. If it is a curve with a topological dimension equal to 1 (a straight line), then the curve can be complicated by an infinite number of bends 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; (b) transition from dimension 1 to dimension 1.5
In multifractals, the dimension indicator is the value H.
When the fractal dimension is less than 1.4, the system is affected 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. From this we can conclude that the more complex the structure we observe, the more the probability of a 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 an object is uniquely determined, i.e. depends entirely on the initial conditions and is amenable to a clear forecast. You can independently perform one of these models in Excel. And fractals are used in the case when the object has several options for development and the state of the system is determined by the position in which it is currently located. That is, we are trying to simulate a chaotic development. This system is the interbank foreign exchange market.
Chapter 4
Technical analysis of the markets is a method of predicting the further behavior of the price trend, 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 the 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. A bullish trend is an upward price direction. A bearish trend is a downward price trend. Flat - lateral (horizontal) movement of the market.
- History repeats itself.
In Elliot's theory, numbers are used to designate the five wave trend, and letters are used for the opposite three wave trend. If the wave is directed towards the main trend and consists of five wave movements, then it is called impulse. 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. Elliot 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 confronted with the reality of the data, rather than the simple pattern detailed in wave theory, many become frustrated at not finding the 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 is actually happening in the financial markets.
In the currency market, time is multifractal, and in the role of price we observe a Brownian movement, generalized or fractional!
Elliot only laid the foundation and proposed a simplified form of price action.
Chapter 5. The Benoit Mandelbrot Model
Benoit Mandelbrot proposed a fractal model, which has already become a classic 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, and people who are simply interested. This model, which was called the "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). As applied to the market, one can see that the weekly price scale has the most detailed data, which makes its structure clearer compared to minute charts (Fig. 6).
Rice. Fig. 6. Dimension (detailing) 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. Usually this point is called the bifurcation point. The most amazing property of the Mandelbrot set is its infinite variance.
When analyzing Forex charts, the golden ratio and can be used.
Chapter 6. The Generator - The Holy Grail in the Forex Market
Under the model, we will mean a regularly built price structure, formed in a complete cycle. Forex charts can be generated using diagonal self-affinity. Transformations are called affine when it uses the operations of transfer and reduction.
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 within a given interval. Parameter D takes 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 concept of dimension can be quite correlated with the volatility of stock prices. That is, using the parameter D, we, by matching our model to a similar one on the market, can adjust it in such a way that the 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 market realities.
Chapter 7. Initial conditions and main stages of model development
For example, we will use model 1.9 (Fig. 8). Waves are separated by vertical lines. Model 1.9 includes the most complete and standard list of elements. All other models are derived from this structure.
Rice. 8. Model structure 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 (indicated by an arrow in the figure).
- The rollback from the origin wave should not cross its bottom.
- The key level of cancellation of this structure will be a 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 a downtrend.
- The beta point should never cross the bottom of the origin wave.
- If the slope between the points alpha and beta is steep, then the trend will be quite powerful and impulsive. If it is gentle, then the trend will not go at an angle, but in a horizontal direction.
Wave impulse features:
- It is the most noticeable of all waves, which is expressed in the duration and speed of its course.
- It almost always reaches the level of 161.8 from the origin wave.
- All indicators show the maximum value at the end of the impulse wave.
- This wave can consist of two cycles.
Salient features of the revival wave:
- Unlike the trident wave, alpha and beta points are not of particular importance in this structure, since the beta level can be significantly lower than alpha, which does not mean the upward movement is cancelled.
- In this structure, it is very important to ensure that the revival highs do not become the maximum levels from the entire model as a whole.
In order to learn how to correctly determine 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 on a certain price scale, then we have a part of a larger cycle. That is why not after each downward (upward) movement we will observe the whole pattern. Cycles that develop on their characteristic price scale have a higher dimension than a part of a larger cycle located on the same scale.
Chapter 8. Determination of cycles in the foreign exchange market
Is there a cycle in the market? To this day, there is no concrete answer to this question. This is how Mandelbrot describes the presence of cycles in financial markets: “... all periodicities are “artifacts”, not a characteristic of the process, but rather the cumulative result depending on the process itself, the length of the sample and the judgment of an economist or hydrologist. The first of these factors is external to the observer, the second (depending on the particular case) may 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 a 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 an object fills space. The higher D, the more noise on the graphs. minute and hour charts. The shadows that we observe on different time scales determine how noisy the time series is! The longer the shadows of the candlesticks, the more noisy the pair is, which is expressed in the deviation of the price from the true value (structure) at the time new information arrives. From this 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 most 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 loop has no defined 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 makes an attempt to determine the cycle length using R/S analysis. He found that the average cycle length for the S&P500 is 4 years. However, the author makes an amendment to 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 observe daily on the screens of our monitors are nothing but a fractal time series. This is where the concept of multifractal exchange time comes from. Einstein found that the average square of the distance that a randomly wandering particle moves away from the starting point is proportional to time. The mean square 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/Н
(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"). Insofar as 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 price highs and lows. The distance between the level and the shadows should be at least 1-2 points. How do you know if this is really a price high or just another surge? This is done by identifying the highs and lows the price has made in the past. The fractal theory of the market assumes that the past values of the price correlate with its future values.
When trading on 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 currency market is a system divided into many separate elements (currency pairs) that are essentially unrelated to each other! You are missing out on a huge opportunity by not using the structure of the behavior of different currency pairs. I do not urge you to open deals on 2, 3, and even more so on 10 pairs at the same time. You can always open deals on only one currency pair, but do it in such a way 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 a similar price movement structure. 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 through, we can assume with full confidence 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, is at great risk of getting away from the correct forecast. I am not at all against indicators and fundamental analysis, I only call for the fact that, using these tools, we should not forget about the subject itself. The use of fractal theory helps to determine the direction of the price, however, given the fluctuations in the volatility of each pair individually, the indicators will help us 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 indicators show the general direction of the price very poorly, which is their significant disadvantage.
Chapter 11
Working on real accounts, a person no longer thinks, he works. Traders imperceptibly cease to think rationally about their losses and gains, for them only one goal begins to exist - to earn more and quickly. This syndrome manifests itself in the following:
- Implementation of a large number of transactions in a short period.
- No concept of risk.
- After a successful transaction, this player opens another one.
- There is a rejection and misunderstanding of the theory. Work in the market occurs without preheating.
- Inability to accept defeat.
Everyone knows this type of orders, like 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 before you want to make a trade, you have already lost.
After you make a successful deal, you are in the state of a winner who can do everything, you celebrate, and during a holiday it is not common for a person to soberly assess the situation. Therefore, do not rush to immediately open another deal, take a break, gather your strength, and your trading will become truly professional.
Literature
David Ruel. . - Izhevsk: RHD, 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: RHD, 2004. - 256 p.
Edgar Peters. . - M.: Internet trading, 2004. - 304 p.