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Fractal market analysis. Fractal market analysis from professionals Lever fractal analysis of financial markets

M.V. Prudsky. FRACTAL ANALYSIS OF FINANCIAL MARKETS.The basic properties and nature of fractals, the possibilities of their use in everyday life, as well as the advantages of the fractal approach in modeling financial markets are considered. The main stochastic time series models will be analyzed and, using the dollar exchange rate as an example, a fractal ARIMA of log increments will be constructed, the determination of which will be based on a fractal R/S analysis of the dimension of the dollar exchange rate chart. An interpretation of the Hurst exponent will also be given - the result of R/S analysis, which allows us to judge the possibility of predicting the financial instrument under study.

Presentation of the main material

In the modern world, financial markets attract quite a wide public interest. The circle of people who deal with financial analytics varies from ordinary traders to analysts of global corporations and government agencies. Humanity has long been interested in the laws of behavior of such practically unpredictable objects. Stock quotes, exchange rates, prices for futures, options and other financial instruments are just a small part of what a qualified person can make money from. There are many ways to analyze events occurring in the markets. This includes technical analysis, fundamental analysis, the Elliott wave theory, as well as many different lesser-known techniques. But one technique stands out among them for its simplicity and originality - fractal analysis. Many have heard about what a fractal is, studied it in schools and universities, seen the simplest one-dimensional and complex differential multidimensional fractals, but few know about their true benefits. Invented by Mandelbrot, they found application in almost all areas of everyday life. The shape of a mollusk shell, turbulent turbulence in the air, human vessels, the crown of a tree, the shape of a leaf, waves, coastline, cracks, lightning and many other familiar objects of the real world have a fractal nature. Charts of stock and currency quotes also have a fractal nature. If you calculate the fractal properties of time and space of financial instruments, it becomes possible to make point and interval forecasts of future values with high accuracy. Fractal (lat. fractus - crushed, broken, broken) is a geometric figure with the property of self-similarity, i.e. composed of several parts, each of which is similar to the entire figure. In fact, there is no precise definition of the term "fractal". Benoit Mandelbrot, the father of fractal geometry, also did not formulate a precise definition. Fractals have certain features that are measurable and properties that are desirable for modeling purposes. The first property is self-similarity. It means that the parts are in some way related to the whole. This property of self-similarity makes the fractal scale-invariant. Fractal dependencies look like a straight line on graphs, where both axes have a logarithmic scale. Models described in this way must use a power law (a real number raised to a power). This power-law scaling feature is the second property of fractals, the fractal dimension, which can describe either a physical structure such as a lung or a time series. The word "fractal" can be used not only as a mathematical term. In the press and popular science literature, a fractal can be called a figure that has any of the following properties:

1. Has a non-trivial structure at all scales. This is in contrast to regular figures (such as a circle, ellipse, graph of a smooth function): if we consider a small fragment of a regular figure on a very large scale, it will look like a fragment of a straight line. For a fractal, increasing the scale does not lead to a simplification of the structure; on all scales we will see an equally complex picture.
2. Is self-similar or approximately self-similar.
3. Has a fractional metric dimension or a metric dimension that exceeds the topological one.

Figure 1 – Example of a fractal

Fractals are certainly beautiful mathematical quirks of nature. If you look at the graph of the Weierstrass function, you can see similarities with graphs of exchange rates or stock quotes. This fractal is described by the function

where a is an odd number and b is a number less than one. This function is continuous and nowhere differentiable. Used for time series modeling using the Monte Carlo method.

Stochastic time series models There are several short-term memory processes that are commonly used to predict prices in financial markets. Among them:
1. AR.
2. MA.
3. ARMA.
4. ARIMA.
5. ARFIMA.
I will dwell in more detail on fractal autoregression.

ARIMA ARIMA (English: autoregressive integrated moving average) - an integrated autoregressive moving average model - a model and methodology for time series analysis, sometimes called Box-Jenkins models (or methodology). The ARIMA(p,d,q) model means that the order time series differences follow the ARMA(p,q) model.

Using the lag operator, the model can be written as follows:

ARFIMA models These models assume the use of fractional orders of differences, since theoretically the order of integration d of a time series may not be an integer value, but a fraction. In this case, we talk about fractionally integrated autoregressive moving average models (ARFIMA, AutoRegressive Fractional Integrated Moving Average). To understand the essence of fractional integration, it is necessary to consider the expansion of the operator of taking the d-th difference into a power series in powers of the lag operator for fractional d (Taylor series expansion):

In addition, the coefficient for the kth term of the Taylor series = Гk-dГk+1Г-d. The ARIMA model is applied to the resulting differences. Thus, the latter model is more accurate due to its fractal properties.

R/S analysis of the dollar to ruble exchange rate. Before modeling a series of dollar rates, it is necessary to calculate its fractal dimension. To do this, you should use the R/S analysis technique and calculate the Hurst exponent. To perform all the necessary calculations, the author used the statistical analysis package “R 2.5.1”, as well as the analytical complex “Forecast 5.26”. The first step will be to transform the original series into a log-increment series; in the future, all modeling operations will take place specifically in relation to the transformed series. In Fig. 2 you can see the transformed row.

This figure especially shows the chaotic nature of the indicator during the crisis and post-crisis periods. However, at this stage, when applying R/S analysis, you can encounter a serious difficulty - this technique requires data independence over time. It is a known fact that daily data on the rates of financial instruments have a very high first-order autocorrelation. There can be up to 7-10 values correlated. To eliminate this problem, a technique for calculating AR(1) differences is used. Of course, the first-order difference method does not eliminate all linear autoregressive dependence and does not take into account differences of higher orders, but it allows us to reduce it to a minimum sufficient to apply analysis with the initial condition of independence. Externally, the log-increment series, adjusted for AR(1) differences, is almost no different from the original series, but its autocorrelation is much lower. To calculate the fractal dimension of the series, 4500 values of the ruble/dollar exchange rate were used since the beginning of its publication by the Central Bank of Russia. There are several difficulties associated with the available range: 1. Until 2002 (inclusive), the Central Bank of the Russian Federation recorded exchange rate values only to the 2nd decimal place, which created rounding errors and inaccuracies. 2. The dollar exchange rate changes dynamically throughout the day and sometimes rounding creates a fixation at the same rate for several days. (especially relevant for the previous disadvantage). As a result of these problems, entire sequences of zero log-increases arise in a series of values. The largest such chain was discovered towards the end of the period under study – it amounted to 10 values. To carry out the analysis, it was necessary to split the adjusted log-increase series into several groups of series of shorter length. Next, calculate the R/S statistics in each group of rows and average them by the number of elements. The length must be constantly increased to half the initial row. The authors advise against taking a length less than 10, as it may distort the value of the RS statistic. The table presents the results of the R/S analysis of the dollar exchange rate.

Table 1 - Results of R/S-analysis/

Thus, the initial data for regression and determination of the fractal dimension will be the 2nd and 4th columns of the Table. In order to find out the dimension of the series, it is necessary to solve the equation by taking logarithm: RS=nH ec As a result, the required regression will have the form lnRS=c+H lnn. The solution to this regression will be the following values: C = -0.4617; H = 0.6294; R2 of the obtained regression is 0.997529, which indicates the high accuracy and credibility of the results obtained. In Fig. Figure 3 shows a graph of R/S statistics and regression on the y scale. The x scale shows the logarithm of the length of the subperiod (n).

Figure 3 – Result of R/S analysis

Based on the obtained value of the Hurst exponent, we can conclude that the series is persistent. Although the level of persistence of the series is low (the value of the indicator is closer to 0.5 than to 1, the log-growth of the dollar exchange rate is nevertheless amenable to modeling. They have long-term memory and are derived and predicted from their previous values. This turned out to be quite natural, since persistent time series are very distributed in nature.

Construction of the ARFIMA fractal model

The calculation of the Hurst exponent was required to determine the parameter of the fractional differentiation operator in the ARFIMA model. Fractional integrated autoregressive moving average models are fractal and therefore very suitable for modeling the dollar exchange rate. The parameter d for such a model will be equal to H-0.5 = 0.1294. To build such a model, it is first necessary to fractionally differentiate the original series of dollar exchange rates by degree d. Further modeling will take place relative to these differences.

First, you need to write the Taylor series expansion of the difference operator 1+L0.1294. This difference will take into account values in several previous periods. Before using the coefficients of the Taylor series, it is necessary to prove that for degree d the numerical sequence of coefficients for lag operators converges. To do this, we will use Leibniz’s test: 1) prove that a1>a2>a3>…>an; 2) we prove that an tends to 0.

Proof:

1. limk>?-1k+1 j=0kd-j k!-1k j=0k-1d-j k+1!=k-dk=1-dk. For all finite values of k, the ratio of the (k+1)th and kth terms of the series

2. Next, you need to compare it with the series 1k, which exceeds it in value, and at the same time tends to 0. Thus, we can conclude that the numerical sequence of coefficients for the Taylor series also tends to 0 according to the Leibniz criterion. Despite the fact that the dollar exchange rate has an infinite long-term memory, in my opinion, the most logical and optimal solution would be to limit the number of terms in the Taylor series for calculating the differences, since it would be incorrect to estimate tomorrow's exchange rate taking into account the exchange rate of ten years ago.

Thus, it was decided to limit ourselves to the previous 30 days for calculating each of the differences (month). The table shows the results of calculating the coefficient values for each lag

Table 2 - Coefficients for lags for fractal differences

Figure 4 shows the results of calculating the differences over the entire period under study.

This chart is almost the same as the original chart for dollar rates, but the model of these differences will be much more accurate than a simple or entire integrated model of the daily dollar rate. For modeling, it is preferable to take the last 40 difference values, since this does not exceed the monthly dynamics too much and at the same time makes the model meaningful. By lengthy search of several variations, the optimal form of the autoregressive moving average (ARMA) model for the differences was established. It turned out to be the ARMA(4.7) model. The table presents the main characteristics of the model.

Table 3 - Statistics of the ARMA(4,7) difference model

The coefficient of determination suggests that the model as a whole, despite some saw-toothing, well explains the dynamics of fractal differences over time. In Fig. Figure 5 shows a graph showing the model, original and forecast series

After modeling and forecasting the differences, a stage comes when it is necessary to restore the original series, having the values of the differences at our disposal.

The constructed model has the ability to make short-term forecasts of the dollar exchange rate.

conclusions

After the analysis and modeling, I would like to note the high prospects for using fractal analysis in the study of the properties of financial markets, since, despite the fact that these models are highly accurate and effective, they are not the pinnacle of achievements of fractal analysis. At the moment, there are multifractal models used not only for simulating and forecasting financial markets, but also in such areas as earthquake prediction. Such models are very common in scientific laboratories in Europe, since the meaning of such models involves penetration into the very essence and structure of the indicator that is being studied.

List of used literature

1. Kronover R. M. Fractals and chaos in dynamic systems. Basics of the theory. M.: Postmarket, 2000. 353 p.
2) Magnus Y.R., Katyshev P.K., Peresetsky A.A. Econometrics. Beginner course. M.: Delo, 2007. 504 p.
3) Mandelbrot Benoit, Richard L. Hudson (Dis)obedient Markets: A Fractal Revolution in Finance The Misbehavior of Markets. M.: Williams, 2006. 400 p.
4) Morozov A.D. Introduction to the theory of fractals. M.; Izhevsk: In. computer studies, 2002. 160 p.
5) Peters E. Fractal analysis of financial markets: application of chaos theory in investment and economics. M.: Internet trading, 2004. 304 p.
6) http://ru.wikipedia.org
7) http://www.nsu.ru/phpBB/viewtopic.php?t=19201
8) http://www.cbr.ru/
9) http://fraktals.ucoz.ru/publ/12-1-0-54

Price movement has a fractal nature because the actions and reactions of people in the market are repeated. The challenge is to recognize these repeating patterns on the price chart. In this article we will consider in detail one of the ways to find such models.

The laws of gravity, capacity, inertia and cyclicality are important driving forces in financial markets. All market patterns, behavior and dynamics can be seen as symptoms or results of these laws. These basic forces are easily understood and intuitively perceived. Their presence can be proven using simple, irrefutable logic based on empirical evidence. In this article we will look at the fractal structure of markets, its manifestations and consequences, and the opportunities it presents to the astute and ultimately successful trader.

Fractals in financial markets

Fractals are a natural phenomenon and at the same time mathematical sets. What they have in common is their repeating pattern, which can be observed on any scale of time and space. To put this into financial context, take a look at Figure 1, which shows three candlestick charts. One is a daily chart (one candle represents one day of trading), another is a 5-minute chart (one candle condenses 5 minutes of trading), and the third is a weekly chart (all movements for the week are compressed into one candle). Each chart represents a different type of financial asset - , index and commodity. Additionally, each one covers a different period of time.

Picture 1

But even taking all this into account, it is still impossible to tell which graph belongs to what. Without prices on the vertical axis and/or timestamps on the horizontal axis, it will be impossible to distinguish them. In fact, since these three graphs are shown next to each other, they can be mistaken for one continuous graph. For those who are interested, the left chart is the weekly time frame for gold, the middle chart is the daily time frame for the S&P 500, and the right chart is the 5-minute Google, Inc. (GOOG).

A good analogy here is the concept of numerical infinity. There are two approaches to numerical infinity. One is that for each number there are neighboring numbers - a smaller and a larger one, for which, in turn, there are also smaller and larger neighboring numbers; and so on ad infinitum; this is infinity of size. Another approach is that between any two numbers there is an infinite number of other numbers - this is an infinity of precision. The same can be said for data in financial markets. New quotes are constantly arriving, which can be viewed on timeframes of varying degrees of accuracy. The only exception to this comparison is that the scale (if we are talking about price movement) is not infinite. In practice, the smallest scale is a single operation. But the concept of infinity can still be used to see the fractal nature of price data in financial markets.

Figure 1 is an example of the never-ending stream of empirical evidence. Is it possible to put forward a common sense explanation, or a universal law, that would take this phenomenon into account? If so, that might explain how . We believe that it is possible to formulate a universal law. Any chart depicting the behavior of financial markets, regardless of its time frame or location in time, is the result of past transactions. We mean operations performed by people in response to various impulses. The diagram in Figure 2 provides an external view of the financial market. The financial market consists of external impulses new to the system (news, reports and other fundamental data), as well as an output signal that is internally returned to the system (people reacting to price movements).

Figure 2

Charts are nothing more than the cumulative result of the past actions of all traders or executed orders. Because people act and react to what the market does in the same way and in the same way across all time frames, their behavior ultimately manifests itself in the same patterns, regardless of scale.

Human emotions are constant, no matter what time frame we consider. The same applies to the behavior resulting from these emotions.

Focal points

Traders use the same methods and indicators to search for the same type of signals, regardless of the timeframe on which they work. Knowing this, it is worth monitoring several time frames during the trading process. Something similar was done by Alexander Elder, who developed his three-screen trading system, which suggests that the trader needs to look at one time frame below and one time frame above the one on which he is trading.

Just as a perfect storm begins as an innocent breeze that eventually develops into a hurricane, so too can one try to profitably pick up on the points where signals on different time frames begin to agree. The greater the number of signals (different or identical) on all timeframes, the greater the importance of this particular point in time.

The number of charts that simultaneously contain similar signals determines the importance and depth of understanding market dynamics. Think about how many people are watching this chart and this signal at this moment, looking at different time frames. The computer is an ideal tool for processing such a large amount of information. For example, you could look at 50 possible formations or signals on 20 different time frames for a particular stock, and then repeat this for several thousand more stocks.

We will then come to understand that the future of any chart is determined by the cumulative execution of orders that have not even been placed yet. It is impossible to know in advance whether a given intraday trade will be short-term, lasting a few days or weeks, or will become a long-term trade that you will hold for several weeks to several months. Each transaction develops from the embryonic stage - this is the smallest form on the smallest time scale. This is why fractals play an important role in trading.

Atoms of Trade

Every trend, regardless of its length, starts from the lowest Low (in the case of an uptrend) or from the highest High (in the case of a downtrend). Each bottom, when close enough, has a V-shape consisting of three bars. Likewise, each vertex should look like an inverted V when viewed at its highest point at sufficient magnification. This means that at the most basic level, regardless of the time frame in question, there are always three bars that make up this atom - the building block of any chart. Trends and reversals will always end or begin with three bars, the middle of which represents the extreme high or extreme low. Take a look at Figure 3. On the left chart you can see a three-bar pattern called a single-bar down fractal. "With one bar" means that on each side of the middle bar there is one bar with higher Highs.

Figure 3

Next to this model in the diagram there is an up fractal with two bars, i.e. there are two bars on each side of the middle bar. It is necessary to be aware of some of the nuances of these definitions found in the trading literature. For example, for an up fractal with five bars, most sources require that there must be at least two bars on each side of the top or bottom for the formation to be called a fractal. There is a difference of opinion as some believe that surrounding bars do not necessarily have to show a sustained upward or downward trend, and some believe otherwise. You can see an example of this situation in the third diagram in Figure 3. The red bar is an up fractal with three bars, because to the right of the red bar there are actually three bars with lower Highs, despite the fact that the third is higher than the second. In some literature this is called a three-bar up fractal because the fourth bar from the right again has a lower High. Likewise, if you look at the bars to the left of the green, you will notice that the third bar from the left has a higher Low than the green bar, although its Low is lower than the second bar to the left of the green. There is quite a bit of confusion in the literature regarding the definitions of fractal patterns and how to use them. Therefore, in this matter we need to go one step further.

Fractal continuum

In addition to all classifications that take into account neighboring bars, each bar can be assigned a set of four numbers. The number of bars to the left and right of the bar in question that exhibit higher Lows than the bar in question is called the Chartmill number of left/right support for that bar (CLS and CRS, respectively). Likewise, a given bar's Chartmill Left/Right Resistance number (CLR and CRR, respectively) takes into account the number of bars to the left and right of a given bar with lower Highs. These numbers are clear and avoid confusion. The time frame you use for your analysis should not affect how you define and analyze the fractal nature of the market. It is important to have objective indicators and signals. Moreover, these indicators and signals must ignore any characteristics of visual perception, for example: time scale on the horizontal axis or linearity/logarithmicity of the axis. Only then can objective, chart-independent indicators be created that can be applied algorithmically, scanning for focal points.

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Fractal theory of the Forex market is the author's attempt to look at the foundations of financial markets through the prism of such concepts as synergetics, chaos, Mandelbrot set, Hurst exponent and Brownian motion. The book will help the reader to understand these concepts as they apply to the Forex market and will change the reader's perception of such things as quotes and prices. The theory of fractals can be successfully used in combination with both technical and fundamental analysis. And the information that the reader will receive while 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

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 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 information sources influence the course of quotes every day. Another condition for self-organization is the initial deviation from equilibrium. Such a deviation may be a consequence of a directed influence from the outside, but it can also arise in the system itself randomly, stochastically. The third condition: all processes in the system (processes amenable to statistical analysis) occur cooperatively, self-consistently. In the market, this condition is met in coordination with time scales. 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 own dynamics of a nonlinear system - its property of exponentially quickly spreading arbitrarily close trajectories. As a result, the shape of the trajectories depends very much 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 work “Science and Method” (1908), says: “In unstable systems, “a completely insignificant cause, eluding us due to its smallness, causes a significant effect that we cannot foresee. (…) Prediction becomes impossible.”

Mandelbrot, Prigogine and others discovered that on the border between conflicts of opposing forces there is not the birth of chaotic, disorderly structures, as previously thought, but the spontaneous emergence of self-organization of a higher level of 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 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 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 one market participant can have an advantage over another, thereby making excess profits. However, Robert Shiller has shown that some price changes occur due to changes in fundamental information and uncertainty about future cash flow trends. Only 27% of the volatility of earnings in 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, without taking into account the psychological characteristics of the participants. The linear paradigm postulates that exchange rates follow random walk trajectories (Brownian motion), and their distribution is normal and bell-shaped. The width of the bell (its sigma, or standard deviation) represents how far price changes deviate from the average; events at the edge 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 volatility can increase for a long time. Large price hikes have become common. The Gaussian normal distribution model does not reflect the reality of what is happening in financial markets. The value of currencies is not postulated by efficient market theory.

Old methods must be replaced by new ones that do not imply independence or normality. New methods should include fractals and nonlinear dynamics. The nonlinear paradigm must allow 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 degrees of freedom, which significantly limits their adaptive capabilities; they are forced to yield to more adaptive competitors in the development process.

Peters proposed a very interesting theory in his book, according to it, markets remain stable when many investors participate in them and have different investment horizons (investment periods).

Harold Edwin Hurst (1880–1978) was an English physicist, 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 was called R/S analysis. Hirst's discovery is that R/S diagrams related to empirical chronicles generally consist of curves tightly wrapped around a certain straight line, but the angle of inclination N this straight 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, N, 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 being studied over time) of prices. The straight line corresponds to the indicator N(Hurst). When H = 0.5 the graph will correspond to 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 a part of it on an enlarged scale, then at a part of this part, etc., then it is not difficult 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 an enlarged copy of the fragment from the original photograph. Self-similar objects have infinitely many scales to suit all tastes.

In finance, the movements of stocks or currencies are similar in appearance, regardless of time scale and price. An observer cannot tell from the appearance of a graph whether the data represents weekly, daily, or hourly changes.

The third property of fractals is that fractal objects have a dimension different from Euclidean. It is called the Hausdorff-Besikovich dimension. This dimension increases as the tortuosity increases, while the topological dimension stubbornly ignores all changes occurring with the line. If this is a curve with a topological dimension equal to 1 (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 can fill a plane; (b) transition from dimension 1 to dimension 1.5

In multifractals, the role of the dimension indicator is the value N.

With a fractal dimension of 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 significantly more than 1.6, the system becomes unstable and is ready to transition to a new state. From this we can conclude that the more complex the structure we observe, the more the probability of powerful movement increases (Fig. 3).

Rice. 3. Simulated lines with different dimensions

When we apply classical models (for example, trend, regression, etc.), we say that the future of the object is uniquely determined, i.e. completely depends on the initial conditions and can be clearly predicted. You can run one of these models yourself in Excel. And fractals are used in the case when an object has several development options and the state of the system is determined by the position in which it is currently located. That is, we are trying to simulate chaotic development. The interbank foreign exchange market is precisely such a system.

Chapter 4. Elliott wave theory as the founder of the theory of fractals

Technical analysis of markets is a method of predicting the further behavior of a price trend, based on knowledge of the background history of price development. Technical analysis uses the mathematical properties of trends for forecasting, rather than the economic indicators of the various countries to which a particular currency pair belongs. Technical analysis is based on 3 postulates:

The market takes everything into account.
Price movements are subject to trends. A bullish trend is an upward price direction. A bearish trend is a downward price direction. Flat is a sideways (horizontal) movement of the market.
History repeats itself.

In Elliott theory, numbers are used to indicate a five-wave trend, and letters are used to indicate the opposite three-wave trend. If a 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. 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, rather than the simple pattern detailed in wave theory, many are disappointed that we do not find 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 actually happens in financial markets.

In the foreign exchange market, time is multifractal, and in the role of price we observe Brownian movement, generalized or fractional!

Elliott just laid the foundation and proposed a simplified form of price behavior.

Chapter 5. Benoit Mandelbrot's model

Benoit Mandelbrot proposed a fractal model, which has already become a classic and is often used to demonstrate both a typical example of a fractal itself and to demonstrate the beauty of fractals, which also attracts researchers, artists, and simply interested people. 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). In relation to the market, you can see that the weekly price scale has the most detailed data, which makes its structure clearer compared to 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 selects 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 its infinite dispersion.

When analyzing Forex charts, the golden ratio and .

Chapter 6. Generator – The Holy Grail in the Forex market

By model we will mean a naturally structured price structure formed in a completed cycle. Forex charts can be generated using diagonal self-affinity. Transformations are called affine when they use 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 in 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 easily correlated with the volatility of exchange 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 price volatility and model dimensions 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 demarcated 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 (shown by an arrow in the figure).
The pullback from the origin wave should not cross its base.
The key level for canceling this structure will be a breakdown of the 23.6 Fibonacci level.

Main features of the trident wave:

Unlike origin, it does not start from the next low of a downward trend.
The beta point should 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 not go at an angle, but in a horizontal direction.

Features of the impulse wave:

It is the most noticeable of all waves, which is expressed in the duration and speed of its movement.
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.

Characteristic 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 may be significantly lower than alpha, which does not mean the upward movement is canceled.
In this structure, it is very important to ensure that the revival maximums do not become the maximum levels of the entire model as a whole.

In order to learn how to correctly define a cycle, we must be able to vary 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 a model at a certain price scale, then we are faced with part of a larger cycle. That is why we will not observe a whole pattern after every downward (upward) movement. Cycles that develop on their characteristic price scale have a higher dimensionality than part of a larger cycle located on the same scale.

Chapter 8. Determining 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 length of the sample and the judgment of the economist or hydrologist. The first of these factors 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 proposed using the Hurst exponent to determine 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 there is in the graphs. minute and hour charts. The shadows we observe at different time scales determine how noisy the time series is! The longer the shadows of the candles, the more noisy the pair is, which is expressed in the deviation of the price at the time of receipt of new information from the true value (structure). From this we can conclude that the scale in the foreign exchange market represents a kind of filter that filters out all unnecessary information and determines the more important one (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 memory of the initial conditions is lost.” Peters attempts to determine cycle length using R/S analysis. He found that the average cycle length for the S&P500 index is 4 years. However, the author makes an allowance for the fact that this is a kind of statistical cycle and that for practical trading it is of absolutely no interest.

Fractal theory is applicable to the market, and those irregular curves that we see on our monitors every day are nothing more than a fractal time series. This is where the concept of multifractal stock exchange time comes from. Einstein found that the average square of the distance at which a randomly walking particle moves away from the starting point is proportional to time. The average square of the distance for a fractal environment turns out to be proportional to some fractional power of time, the indicator of which is related to the fractal dimension of the environment α :

(4) α = 1/N

(5) dP ~ (dt) N

If H = ½, the model is characteristic of an efficient market. Where it is postulated that the price distribution process corresponds to a Gaussian one. Here dP– price change corresponding to the time interval dt. The most common value of the Hurst exponent for foreign exchange markets ranges from 0.58 to 0.6, which corresponds to α = 1,7 ("model 1.7") Because the N is constantly changing, it will be 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 maximum or just another spike? This is done by determining the high and low levels that the price has made in the past. Fractal market theory suggests that past price values correlate with future price 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 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 a huge opportunity by not using the structure of how different currency pairs behave. I do not encourage you to open trades on 2, 3, and even less so on 10 pairs at the same time. You can always open trades on only one currency pair, but do this 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 therefore different key levels. One of them reaches levels much faster than similar currencies. All we have to do is set the key levels and wait for their breakdown. As soon as they are broken through, we can assume with complete confidence that for a currency where the price has not even approached them yet, a breakout 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 a high risk of deviating from the correct forecast. I am not at all an opponent of indicators and fundamental analysis; I am only calling for the fact that when using these tools, we should not forget about the subject itself. The use of fractal theory helps determine the price direction, however, taking into account differences in the volatility of each pair separately, indicators will help us most accurately orient ourselves in the current situation. Yes, they are more convenient in the sense that with their help you can find the most key points for entering or exiting the market. But indicators show very poorly the general direction of the price, which is their significant disadvantage.

Chapter 11. Psychology of trading

When working on real accounts, a person no longer thinks, he works. Traders, imperceptibly for themselves, stop thinking intelligently about their losses and winnings; for them, only one goal begins to exist - to earn more and faster. This syndrome manifests itself in the following:

Carrying out a large number of transactions in a short period.
Lack of the concept of risk.
After a successfully completed transaction, this player opens another one.
There is a rejection and misunderstanding of the theory. Work on the market takes place without pre-heating.
Inability to accept defeat.

Everyone knows this type of orders as stop loss and take profit. You must understand that if you, before you want to make a trade, do not know where to limit your losses and how much profit you need to take, you have already lost.

After you make a successful transaction, you are in the state of a winner who can handle everything, you celebrate, and during a holiday it is not typical for a person to soberly assess the situation. Therefore, do not rush to immediately open another deal, rest, gather your strength, and your trading will become truly professional.

Literature

David Ruel. . – Izhevsk: RHD, 2001. – 192 p.

Benoit Mandelbrot. : 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.

Fractals occupy me more and more. 🙂 I came across the first mention of them in the books of Nasim Taleb and. And then I immersed myself in the subject by reading the book by Benoit Mandelbrot. Inspired by these works, I even undertook some small research:

I present to you another book on this topic. There is more mathematics here than in previous works, but I will try not to overdo it...

Download a short summary in format

Chapter 1. Introduction to Fractal Time Series

Western culture has long been obsessed with sleek and symmetrical. Fractal geometry – the geometry of the Demiurge. Unlike Euclidean geometry, it is based on roughness and asymmetry. “Self-similarity” is a defining property of fractals. Most natural structures, especially living things, have this property. The second problem that arises when applying Euclidean geometry to our world is the problem of dimension. …the perception of dimension can change depending on our distance from the object. We will see the difference between the smoothness of the Euclidean world and the roughness of our world, which limits the usefulness of Euclidean geometry as a method of description.

The conflict between the symmetry of Euclidean geometry and the asymmetry of the real world can be further extended to our concept time. Traditionally, events are viewed as either random or deterministic. In fractal time, randomness and determinism, chaos and order coexist. It seems that even great events depend on chance. However, similar theories are developed in parallel by several scientists. This implies that these discoveries were meant to happen. History demanded this.

Time had no meaning in Newtonian mechanics; theoretically, time could be turned back because Newton's equations worked equally well whether time went forward or backward. At the same time, such a process as mixing liquids is a process that depends on time and irreversible. In thermodynamics, the hand of time points only to the future. The first blow was dealt to the idea of the universe as a clockwork mechanism.

The second blow came with the advent of quantum mechanics. The realization that the molecular structure of the universe could only be described by states of probability further undermined the deterministic view. But there was still doubt. Is the universe deterministic or random? It gradually became apparent that most natural systems are characterized by local randomness and global determinism. These opposite states must coexist. Determinism will give us the law of nature. Randomness brings innovation and variety. A healthy, growing system is one that not only can survive occasional shocks, but can also absorb such shocks to improve the entire system when it makes sense to do so.

We have come to the third blow to Newton's determinism: the science of chaos and fractals, where chance and necessity coexist. In these systems, entropy is high, but never reaches a maximum state of disorder due to global determinism. Chaotic systems export or "dissipate" their entropy, similar to how mechanical devices dissipate some of their energy as friction.

The play of chaos shows that local randomness and global determinism can coexist to create a stable, self-similar structure, which we call a fractal.

In fact, there is no precise definition of the term "fractal". Benoit Mandelbrot, the father of fractal geometry, also did not formulate a precise definition. Fractals have certain features that are measurable and properties that are desirable for modeling purposes. The first property is self-similarity. It means that the parts are in some way related to the whole. This property of self-similarity makes a fractal scale-invariant. Fractal dependencies look like a straight line on graphs, where both axes have a logarithmic scale. Models described in this way must use a power law (a real number raised to a power). This feature power law scaling, is the second property of fractals, fractal dimension, which can describe either a physical structure, such as a lung, or a time series.

Fractal dimension characterizes how an object fills space. The fractal dimension of a time series measures how jagged the time series is. According to expectations, a straight line should have a fractal dimension of 1 equal to its Euclidean dimension (the fractal dimension of a plane is 2). The fractal dimension of a random time series is 1.5. ...fractal dimension can be solved as the slope of a graph on a logarithmic scale on both axes.

The fractal dimension of a time series is important because it recognizes that a process can be somewhere between deterministic (a line with fractal dimension 1) and random (fractal dimension 1.5). In fact, the fractal dimension of a line can range from 1 to 2. With values of 1.5< d < 2 временной ряд более зазубрен, чем случайная последовательность, или имеет больше инверсий. It goes without saying that time series statistics with fractal dimensions other than 1.5 would be very different from Gaussian statistics and would not necessarily fall within the normal distribution.

Chapter 2. Failure of the Gaussian hypothesis

The fractal nature of the stock market is reflected in the fact that

a) profit distribution curves differ significantly from the Gaussian bell (Fig. 1).

b) the 1-, 5-, 10-, 20-, 30- and 90-day profit curves look the same: scale-invariant (Fig. 2); it can be seen that all curves are characterized by a higher peak (the probability of average values is higher than in the normal distribution), a dip slightly further from the average value (in the region of 1–2 sigma), thick tails - a high probability of extremely large deviations (more than 3 sigma) .

Rice. 1. Dow Jones Industrial Average, frequency distribution of returns: 1888-1991; the abscissa axis is the number of standard deviations, the ordinate axis is the frequency

Rice. 2. Dow Jones Industrial Average, N-day returns minus normal frequency; N = 1 (a), 10 (b), 20 (c), 30 (d)

What does this mean? First, the risk of a large outlier (black swan) occurring is much higher than implied by a normal distribution. The normal distribution says that the probability of an event occurring over three standard deviations is 0.5% or 5 in 1000. The figures show that the actual probability is 2.4% or 24 in 1000. Second, day traders face the same number six-sigma events on their time frames that 90-day investors face on their time frames.

Term structure of volatility. Typically, we use standard deviation to measure volatility and assume that it is scaled by the square root of time. For example, we "annualize" the standard deviation of monthly returns by multiplying it by the square root of 12. This practice comes from Einstein's observation that the distance a particle travels in Brownian motion increases in proportion to the square root of the time it takes to measure it.

The square root of time is shown as a solid 45-degree line in Fig. 3. Volatility initially increases at a faster rate than the square root of time, and for N > 1000 days the slope drops sharply to 0.25. If we think of risk as a standard deviation, investors bear more risk than is implied by the standard deviation for investment horizons of less than four years. However, investors bear increasingly less risk over investment horizons longer than four years. As has always been known, long-term investors bear less risk than short-term investors.

Rice. 3. Dow Jones Industrial Average, volatility time structure: 1888-1990.

On the other hand, the return-to-risk ratio or “Sharpe ratio,” named after its creator, Nobel laureate William Sharpe, measures how much return is earned per unit of risk, or standard deviation. Over periods of less than 1,000 days or four years, the Sharpe ratio decreases continuously; at the 1200 day mark it increases sharply. This means that long-term investors are rewarded more per unit of risk than short-term investors.

Bonds behave similarly. And here is the currency! …long-term currency holders face ever-increasing levels of risk as their investment horizon expands. Unlike stocks and bonds, currencies do not offer any investment incentive for a strategy of buying and holding for the long term. In the short term, speculators in stocks, bonds and foreign exchange rates face similar risks, but in the long term, the risk of investors investing in stocks and bonds is reduced.

The emergence of boundaries for stocks and bonds, but not for currencies, is initially puzzling. Why is currency a security different from stocks and bonds? This question already contains an answer. The currency is called a “security”. Currency is an object that is traded, but it is not a security. It has no investment value. Profits from a currency can only be made by speculating on its value against the value of another currency. Currency is thus equivalent to purely speculative funds, which are usually equated to stocks and bonds. Stocks and bonds are not like that. They have investment value. Bonds earn interest, and the value of a stock is tied to the growth in its earnings due to economic activity. The aggregate stock market is tied to the aggregate economy. The currency is not tied to the economic cycle. In the 1950s and 1960s, we had a growing economy and a strong dollar. In the 1980s we had a growing economy and a falling dollar. A currency does not have a "fundamental" value that is necessarily tied to economic activity, although it can be tied to economic variables such as interest rates.

Chapter 3. Fractal Market Hypothesis

which is the formula for standard deviation. The normalized range was calculated by first rescaling or "normalizing" the data by subtracting the sample mean:

(4.4) Z r = (x r – x m); r = 1, … n

The resulting Z series now has a mean of zero.

The next step creates a cumulative time series Y:

That is, the r-th term of the series Y is equal to the sum of all terms of the series Z, starting from the first and ending with the r-th. Note that, by definition, the last value of Y(Yn) will always be zero because Z has a mean of zero. The adjusted range R n is the difference between the maximum and minimum values of the Y series:

(4.6) R n = max(Y 1 , …, Y n) – min(Y 1 , …, Y n)

The subscript n for R n now indicates that it is the adjusted range for x 1 , ..., x n . Because Y has been adjusted to mean zero, the maximum value of Y will always be greater than or equal to zero, and the minimum value will always be less than or equal to zero. Therefore, the adjusted range R n will always be non-negative.

This adjusted range R n is the distance the system moves in time n. If we install n= T, we can apply equation (4.1) provided that the time series X independent for increasing values n. However, equation (4.1) only applies to a time series that is in Brownian motion: it has zero mean and variance equal to one. To apply this concept to a time series that is not in Brownian motion, we need to generalize equation (4.1) to take into account systems that are not independent. Hurst discovered the following more general form of equation (4.1):

(4.7) (R/S) n = c*n H

Subscript n for (R/S) n, refers to the value of R/S for x 1, ..., x n; c = constant.

The R/S value of equation (4.7) is called the normalized range because it has a mean of zero and is expressed in terms of the local standard deviation. In general, the R/S value changes scale as we increase the time increments n according to the power law exponent equal to H, which is usually called the Hurst exponent. This is called scaling with power law. Again, this is a characteristic, although not exclusive, feature of fractals.

The Hurst exponent can be found by plotting log(R/S) n against log(n) and calculating the slope through simple least squares regression. Specifically, we work based on the following equation:

(4.8) log(R/S) n = log(c) + H*log(n)

If the system were independently distributed, then H = 0.5. Hearst will first study the Nile River. He found that H = 0.91! The normalized range increased faster than the square root of time. It increased as 0.91 roots of time, implying that the system (in this case the Nile height range) had traveled a greater distance than a random probabilistic process would have traveled. To travel a greater distance, it was necessary for changes in the annual Nile floods to influence each other.

According to the original theory, H = 0.5 would imply an independent process. It is important to understand that R/S analysis does not require that the underlying process be Gaussian, it only requires that it be independent. This would of course include the normal distribution, but also non-Gaussian independent processes like Student's t-test, gamma, or any other form. R/S analysis is nonparametric, so it does not require the shape of the underlying distribution.

0,5 < Н < 1,0 подразумевает persistent time series that is characterized by long-term memory effects. Theoretically, what happens today affects the future. In terms of chaotic dynamics, there is a sensitive dependence on the initial conditions. Such long-term memory occurs regardless of the time scale. All daily changes are relative to all future daily changes; all weekly changes are relative to all future weekly changes.

0 < Н < 0,5 означаетantipersistence. Such a system travels a shorter distance than a random system. In order for a system to travel a shorter distance, it must change more often than a probabilistic process.

As we have seen, persistent time series are the most common type found in nature. It is also the most common type in capital markets and economics.

Chapter 5. Checking R/S Analysis

When analyzing any process, we are always faced with one important question: “How do we know that we did not get our results by chance?” Significance testing in relation to probabilistic confidence intervals has become a major topic in statistics. …the initial guess is called the null hypothesis. We chose the Gaussian case as the null hypothesis because it is mathematically easier to test whether a process is a random walk and be able to say that it is not, than to prove the existence of some other process with long-term memory. Why? The Gaussian case allows one to find optimal solutions and is easy to model. In addition, the efficient market hypothesis (EMH) is based on the Gaussian case, which by default becomes the null hypothesis. Monte Carlo simulations showed that the null hypothesis was unfounded.

Chapter 6. Finding cycles: periodic and non-periodic

R/S analysis can not only reveal persistence, or long-term memory, in a time series, but can also estimate the length of periodic or non-periodic cycles. It is also noise-resistant. This makes R/S analysis particularly attractive for studying natural time series and, in particular, market time series.

The following presentation is too technical and, in my opinion, will only be useful to professional researchers of the stock, bond and currency markets. I will give only interesting conclusions from one of the chapters and the final chapter.

Chapter 12. Currency: The True Hearst Trial

Currency has interesting statistical and fundamental characteristics that distinguish it from other processes. As such, a currency is not a security, although it is actively traded. The major players, central banks, are not return maximizers; their goals are not necessarily those of rational investors. At the same time, there is little evidence of cycles in currency markets, although they do have strong trends.

Based on these characteristics taken together, we believe that currency is a true Hearst process. That is, it is characterized by processes of infinite memory. Long-term investors should be wary of treating currencies the same way they treat other assets. In particular, they should not assume that a buy-and-hold strategy will be beneficial in the long term. The risk increases over time and does not decrease over time. A long-term investor who must have currency risk should consider active trading in such assets. They offer no benefit in the long term.

Chapter 18 Understanding Markets

This book had two purposes. First, I intended it to be a guide to applying R/S analysis to capital market, economic, and other time series data. R/S analysis has been around for over 40 years. Despite its robustness and general applicability, it remains largely unknown. It deserves a place in any analyst's toolbox along with other tools that have been developed in traditional and chaos analysis.

My second goal was to describe a general hypothesis for synthesizing the various models into a coherent whole. This hypothesis had to be consistent with empirical facts using a minimum number of underlying assumptions. I call my model the Fractal Market Hypothesis (FMH). I believe that this hypothesis is the first attempt to understand the global structure of markets. FMH will undoubtedly be modified and improved over time if it stands up to the scrutiny of the investment community. I have used several different methods to test FMH; the outstanding tool was R/S analysis, used in combination with other methods.

A convincing picture began to emerge. R/S analysis and the fractal market hypothesis came together under the general heading of “fractal market analysis.” Fractal market analysis used self-similar probability distributions called robust Lévy distributions in combination with R/S analysis to study and classify the long-term behavior of markets. We have learned a lot, but there is still much to explore. I am convinced that markets have a fractal structure. As with any fractal, temporal or spatial structure, the more closely we examine the structure, the more detail we see. As soon as we begin to explain some mysteries, new unknowns are discovered. Here is a classic example of the fact that the more we know, the more we realize that we know nothing.

INFORMATION AND INVESTMENT HORIZONS

We discussed the impact of information on investor behavior. In traditional theory, information is considered as a generic concept. To a greater or lesser extent, it represents everything that can affect the perceived value of a security. Investor is also a generic concept. Basically, an investor is anyone who wants to buy, sell or hold a security based on available information. An investor is also considered to be rational - that is, someone who always wants to maximize profits and knows how to evaluate current information. The aggregate market is the equivalent of such an original rational investor, so that the market can immediately evaluate information. This generic approach, where information and investors are common cases, also implies that all types of information affect all investors equally. This is where this approach fails.

The market consists of many individuals with many different investment horizons. The behavior of a day trader is significantly different from the behavior of a pension fund. In the first case, the investment horizon is measured in minutes; in the latter case – in years. Information has different effects on different investment horizons. The main activity of day traders is trading. Trading usually involves crowd behavior and looking at short-term trends. A day trader will be more interested in technical information, which explains why many technicians say that “the market has its own language.” There is also a good chance that technicians will say that fundamental information is of little value. Most technicians have short investment horizons and, within their time frame, fundamental information is of little value. In this respect they are right. Technical trends matter most over short horizons.

Most fundamental analysts and economists who also work in the markets have long investment horizons. They are more inclined to deal with the economic cycle. Fundamental analysts will be inclined to think that technical trends are illusions that are of no benefit to long-term investors. True investment returns can only be achieved through valuation.

In this framework, both technicians and fundamentalists are correct for their specific investment horizons, since the impact of information depends largely on each individual's investment horizon.

STABILITY

Market stability is largely a matter of liquidity. Liquidity is available when the market consists of many investors with many different investment horizons. Thus, if a piece of information arrives that causes a major decline in price over a short investment horizon, long-term investors will come into the market to buy because they do not value the information as highly. However, when the market loses this structure and all investors have the same investment horizon, the market becomes unstable because there is no liquidity. Liquidity is not the same as trading volume. On the contrary, it is balancing supply and demand. The loss of long-term investors forces the entire market to trade based on the same information set, which is primarily technical, or the phenomenon of crowd behavior. Typically, a market horizon becomes short-term when the long-term outlook becomes very uncertain—that is, when some event (often political) occurs that makes the current long-term information set unreliable or perceived as useless. Long-term investors either stop participating or become short-term investors and start trading based on technical information as well.

Market stability relies on the diversity of participants' investment horizons. A stable market is one in which many investors with different investment horizons trade simultaneously. The market is resilient because different horizons value information flow differently and can provide liquidity if a crash or run occurs in one of many investment horizons.

Each investment horizon is like a generation of tree branches. The diameter of any branch is a random function with finite variance. However, each branch, taken in the context of the entire tree, is part of a global structure with unknown variance because the dimension of each tree is different. This depends on many variables, such as its type and size.

Each investment horizon is also a random function with finite variance depending on the previous variance. Since risk must be the same over each investment horizon, the shape of the frequency distribution of returns is the same when adjusting for scale. However, the overall, global statistical structure of the market has infinite variance; the long-term dispersion does not tend to a stable value.

The global statistical structure is fractal because it has a self-similar structure, and its characteristic exponent a (which also represents the fractal dimension) is fractional, ranging from 0 to 2. A random walk, which is characterized by a normal distribution, is self-similar. However, it is not fractal; its fractal dimension is an integer: a = 2.0.

The shape of these fractal distributions, compared to the normal distribution, is characterized by a high peak and thick tails. Fat tails occur because a large event occurs as a result of an amplification process. The same process causes infinite variance. Tails never tend to an asymptote y= 0.0, even at infinity. Also, when big events happen, they tend to be abrupt and intermittent. Thus, fractal distributions have another fractal characteristic: discontinuity. The tendency towards "catastrophes" was called by Mandelbrot the Noah effect, or more formally the syndrome of infinite variance. In markets, fat tails are caused by crashes and stampedes that tend to be sharp and intermittent, as predicted by the model.

LONG TERM MEMORY

In the ideal world of traditional time series analysis, all systems are random walks or can be converted to random walks. In such a case, the “supreme law of Unreason” can be applied and answers can be found. Because of this imposition of order on disorder, natural systems can be reduced to a few solvable equations and one basic frequency distribution, the normal distribution.

Real life is not that simple. Children of the Demiurge are complex and cannot be classified according to a few simple characteristics. We find that in capital markets, most series are characterized by long-term memory effects, or biases; Today's market activity offsets future activity for a very long time. Like Joseph effect can cause serious problems for traditional time series analysis. Long-term memory causes trends and cycles to occur. These cycles may be spurious because they are simply a function of long-term memory effects and random changes in market bias.

Through R/S analysis, it was shown that such a long-term memory effect exists and is a black noise process. The color of the noise that causes the Joseph effect will be important later when we discuss volatility.

It has long been suspected that markets have cycles, but no conclusive evidence has been found. The methods used looked for regular, periodic cycles - that is, cycles created by the Good. The Demiurge created non-periodic cycles - cycles that have an average, but not an exact period. Using R/S analysis, we were able to show that non-periodic cycles are likely in markets. Such non-periodic cycles last for many years, so there is a possibility that they are a consequence of long-term economic information. We found that similar non-periodic cycles exist for nonlinear dynamical systems, or deterministic chaos.

We did not find convincing evidence of short-term non-periodic cycles. Most of the shorter cycles that are popular among technicians are probably due to the Joseph effect. Cycles have no average length, and the mixing that causes them can change at any time - most likely in an abrupt and intermittent manner.

Among the more interesting results is the fact that currencies do not have a long-term cycle. This implies that it represents a fractional noise process in both the short and long term. Stocks and bonds, on the other hand, are fractional noise in the short term (hence the self-similar frequency distributions) but chaotic in the long term.

VOLATILITY

Volatility has been shown to be antipersistent—a frequently changing pink noise process. However, it is not mean reverting. Mean reversion implies that volatility has a stable expected value to which it ultimately tends. We have seen evidence that this is not the case. This evidence is consistent with the theory, since the derivative of the black noise process is pink noise. Market returns are black noise, so it's not surprising that volatility (which is the second momentum of stock prices) is pink noise.

The pink noise process is characterized by probability functions that not only have infinite variance but also infinite mean; that is, there is no mathematical expectation to which one can return. In the context of the notion that market returns are black noise, this makes sense. If market returns have infinite variance, then the average variance of stock prices must itself be infinite. It's all part of one big structure, and that structure has profound implications for options traders and others who buy and sell volatility.

TOWARDS A MORE COMPLETE MARKET THEORY

Much of the discussion in this book has been an attempt to reconcile the rational approach of traditional quantitative management with the practical experience of actually interacting with markets. For some time we could not bring them into line. Practicing monetary managers who have quantitative backgrounds are forced to infuse practical experience with theory. When practice does not correspond to theory, we simply recognized that at that point the theory fails. Our view was similar to physicists' acceptance of "singularities", that is, events where a theory fails. The Big Bang is one such singularity. At the moment of the Big Bang, the laws of physics fail and cannot explain the event. We have been forced to think of market crashes as singularities in capital market theory. They represent periods when no generalization of the efficient market hypothesis (EMH) applies.

Chaos theory and fractal statistics offer us a model that can explain such features. Even if events such as accidents turn out to be unpredictable, they are not unexpected. They do not become "outliers" in theory. On the contrary, they are part of the system. In many ways they are the price we pay for being capitalists. In my previous book, I noted that in order to remain vibrant, markets must be far from equilibrium. What I was trying to say is that the capitalist system (either the capital market or the entire economy) must develop dynamically. Random events must occur to stimulate innovation. If we knew exactly what was going to happen, we would stop experimenting. We would stop learning. We would stop innovating. So we must have cycles, and cycles mean that there will always be a period of upswing and a period of decline.

It has become common for researchers to look for anomalies, or pockets of inefficiency, where profits can be made with little risk. It has been rightly pointed out that a large market will correct such anomalies as soon as they become publicly known. FMH is not like that. She doesn't find a pocket of inefficiency where few can profit. Instead, it says that because information is processed differently at different frequencies, there will be trends and cycles across all investment horizons. Some will be stochastic, some will be nonlinear deterministic. In both cases, the exact structure of the trends changes over time. It is predictable, but it will never be completely predictable and that is what keeps markets stable. Chaos theory and fractal statistics offer us a new way of understanding how markets and economies function. There is no guarantee that they will make it easier for us to make money. However, we will be better equipped to develop strategies and assess risks.

Effective market hypothesis (EMH)

The relevance of a theory means that it must correspond to modern trends and must be potentially in demand, because it is an attempt to solve a scientific problem that is being studied. The relevance of fractal theory is beyond doubt.

Look

There is always a need to offer the trader an approach to analyze the market situation that is suitable for him. He gave the world the opportunity to apply the fractal analysis method.

History of the approach. Origin of the concept of fractal.

Benoit Mandelbrot is considered one of the founders of fractal geometry. It is he who has the right to be considered founder of fractal analysis. Which has been gaining momentum lately. So far it has not received widespread use.

The scientist was engaged in economic research. One day he noticed and studied the nature of price fluctuations in the market. It turned out that they are not of an arbitrary nature. However, they cannot be described by standard curves. But they may lend themselves to a different mathematical description that is distorted over time.

Having made this discovery, Mandelbrot began studying cotton price statistics over a hundred-year period. Then he made a discovery that spoke of the symmetry of long-term and short-term currency fluctuations. This achievement played an important role in the development of fractal analysis.

Modern observations show that fractals can be found anywhere around us. These are the outlines of the mountains and the winding line of the sea coast.

Sometimes we can observe continuously changing fractals, for example, in moving clouds or flickering flames. At this time, others retain their structure, which they acquired in the process of evolution. This applies to trees or the vascular system.

The concept of a fractal was introduced in 1975 by the French scientist Benoit Mandelbrot to refer to the irregular but self-similar structures with which he was concerned.

His work uses the results of research by other scientists who studied related issues in the years 1875 - 1925. This is Poincaré, Cantor, Julie, Hausdorff.

And these days, all the research results have been combined together. As a result, a definition of a fractal was given as a structure that consists of parts. These parts are in some sense similar to the whole.

The term fractal is derived from the Latin participle fractus. It corresponds to the verb frangere, which can be translated as breaking, breaking. This means creating fragments with irregular shapes.

Basics of fractal analysis or properties of fractals in the market.

Fractal properties help us distinguish and predict features of the surrounding reality. Before the advent of fractal theory, these features were estimated approximately or by eye.

Today, the fractal analysis method helps doctors analyze the fractal dimensions of complex signals, such as encephalograms or heart murmurs, and help diagnose serious diseases at the initial stage. This helps to cure the patient before the disease becomes incurable. Also, by comparing price behavior, analysts can predict future developments without making a gross forecasting error.

Irregularity of fractals.

The first property of fractals is irregularity. If a fractal is described by a function, then irregularity describes its property of not being smooth at any point.

We can easily confirm the presence of this property of fractals in the market, because price fluctuations, at times, are subject to such sudden changes that it leads most traders to confusion. Our task is to understand this chaos, bringing it to order.

Self-similarity of fractals.

The second property is the self-similarity of fractals as objects. The fractal model is said to be recursive. Each part of it follows the development of the entire model as a whole and reproduces it on different scales without any special changes. But changes are still present, and this means that we can perceive the object itself differently.

Self-similarity shows that an object is not characterized by scale. Because if there was one, we would immediately distinguish the enlarged part of the fragment from the source. Such objects can have an infinite number of scales to suit every taste.

Fractal analysis and theory of market efficiency.

If we use fractal analysis to work in the market, we make our work much easier. Fractal analysis, being a rather complex subject, can nevertheless help predict the rise or fall of prices, which means that pricing and market movements will not be completely unpredictable for you.

But to master fractal analysis, it is necessary to doubt the theory of market efficiency.

This theory states that the market price accurately and practically without delay reflects all known information and all the expectations of market participants.

This theory says that it is impossible to beat the market. Because new information arrives randomly, and the market reaction is instantaneous. Therefore, the price of securities at any given time is completely fair. This means that securities can neither be overvalued nor undervalued. Therefore, it is not possible to make a profit.

It follows from the theory that no analysis, fundamental or technical, can help increase the profitability of an operation, since the price has already absorbed the entire flow of information and cannot react differently. Also, the theory believes that past data does not affect future data.

This means that researching historical data is technical or pointless.

However, the theory does not specify that many investors' expectations are based on past prices. It does not say that they also used information about the company’s successes to make investment decisions. Since prices were determined by expectations, it is obvious that past prices influence future ones.

This is where the theory of fractals differs from the theory of market efficiency. The theory of fractals makes it possible to grasp the connection in acceptable graphic structures.

You can test the theory of fractals using the services.