Fractal analysis of financial markets - E. Peters - Application of Chaos Theory in investment and economics

The first practitioner who applied fractal theory in the analysis of financial and commodity markets was Bill Williams . Subsequently, his method of fractal market analysis became widespread in many countries. This was facilitated by his works such as"Trading Chaos" "New dimensions in stock trading", "Trading Chaos second edition". Over time, many inattentive traders and analysts believed that behind the beautiful name lies more a clever PR move by the author than the actual use of fractals in the market. The main mistake that leads to distortion of the analysis results is the incorrect interpretation of the concept of “overcoming a fractal.” The ambiguity of fractal analysis ceases if the word “overcoming” is understood not as a puncture by the price of a 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 on the market. Forex . Let's try to explain in the simplest way how it works. The most basic graphical element of the market (here we mean price fluctuation charts) is a straight line directed from top to bottom or bottom to top. To everyone trader (stock trader) this is well understood - the price either rises or falls, this process occurs over time. Thus we have initiator , which looks like this:

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 price movement is another common structure, well known to the trader - "impulse-correction-impulse", which looks like below:

There may be an infinite number of these generators on the market, and there may not be two turning points. What information can these figures give a trader? If you look at the price movement of an individual instrument, you can see that the structure of the generator is repeated on all time 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 get the following structure:

Moving deeper and deeper, we will reach minute and then tick charts, on which the basic fractal will appear again and again. Typically, the relationships between the generator lines will remain fixed on any time structure. Angles between the generator lines on the minute and monthly
The graphics will correspond to each other, the ratio of their lengths will also correspond. This amazing discovery gives us a completely new look at the usual price movement.
Of course, this understanding is simplistic, and, in Mandelbrot’s own opinion, “caricatured.” It serves us to describe the general principle of the structure of price movement. A real market generator can be much more complex.
In modeling market behavior, Mandelbrot uses a more complex "multifractal" model, which uses three dimensions and the so-called "fractal cube". We will not dwell on it in detail. Instead, let's look at two other observations of fractal geometry that are easier to understand and give the trader
food for thought.

The market has a memory.

Benoit Mandelbrot's extensive research into the cotton market led him to the following conclusion: periods of high volatility or "turbulence" tend to cluster in"clusters" . This means that events, the probability of which, according to generally accepted financial models, is an insignificant fraction of a percent, in many cases occur in sequence - one after another. This is fundamentally inconsistent with the “random walk” model that is used throughout the world for risk management. According to it, all events in the market are independent of each other. Mandelbrot makes a convincing case. that this is not so. Market events tend to remain dependent on each other. He calls this effect -"The Joseph Effect", using as a metaphor the famous biblical parable of Pharaoh, who had a dream about seven fat and seven skinny cows (seven harvest years and seven lean years).

What does it represent "price cluster"? By price cluster we mean"trend" Trend in economics - the direction of preferential movement of indicators. Usually considered within the framework of technical analysis, which implies the direction of price movements or index values. Charles Dow noted that during an upward trend
the 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), downward (bearish) and side (flat) ) . A trend line is often drawn on the chart, which in an uptrend connects two or more price troughs (the line is located below the chart, visually supporting it and pushing it upward), and in a downward trend connects two or more price peaks (the line is located above the chart, visually limiting it and pressing down). Trend lines are lines of support (for an uptrend) and resistance (for a downtrend). An uptrend (an uptrend, a bullish trend) is a situation where each new local minimum and local maximum is higher than the previous one.

An example of a rising trend.

An example of a downtrend.

The Noah effect

And finally, Mandelbrot's third observation is the so-called"Noah" effect . From the Old Testament we know 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 booms. They never happen smoothly; almost always the market soars 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. Thus, in 1987, the Dow Jones Industrial Average fell by 22.6% in one day. After the crash, computer programs were blamed for everything, but Benoit Mandelbrot had 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 drives this dynamic. This hypothesis is also new and is not consistent with the efficient market hypothesis, which states that 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 at which the transaction was opened.

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

Concluding the article, we will also mention the use of fractals in time series modeling. In particular, such a characteristic of a time series as fractal dimension makes it possible to determine the moment at which the system becomes unstable and is ready to transition to a new state.

Example of a time series.

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 widely. One of the main limiting factors is the chaotic nature of the fractal model, which is due to the exceptional interdependence of its input and output parameters. Even the slightest change in the input parameter or the slightest 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 completely impossible to verify (evaluate) the results obtained from fractal modeling. At the same time, this is truly the most promising modern area of ​​mathematics from the point of view of applied research in economics.

Sources: fortrader.ru, Wikipedia and other materials from the Internet..

Fractals are quite popular among many traders. Interest in fractal analysis arose after the publication of several works by Bill Williams on this topic. Fractals were invented before him, but were referred to under a different name. Williams, studying financial markets, came to the conclusion that the movements of the rates of many financial instruments are chaotic. As a result of his research, he proved that the graph of changes in the value of cotton is similar to the coastline and the movement of blood in the human body.

In his research, Williams came to the conclusion that markets are chaotic, not linear systems, so using indicators based on linear functions on them is useless. In his opinion, stability in the markets is present only a small fraction of the time, and in all other cases chaos reigns in them.

A fractal is a repeating formation that is found in one form or another on any price charts. Coastline fractals, like stock market fractals, are of the same nature. A fractal consists of at least five bars.

Upper and lower fractals can be in the same group of bars. Sometimes an upper and lower fractal are formed simultaneously on the same bar. When a fractal is formed, it is endowed with all the properties.

When assessing the upper fractal, you need to pay attention to its maximum. When studying the lower, accordingly, the minimum. A fractal start is formed from two successive fractals directed in different directions. The fractal signal appears on the side opposite to the start. The fractal stop is located behind the far fractal. If an opposite signal appears, it cancels the previous ones.

This technique allows you to increase the percentage of profitable trades, but the average losing trade will be higher. Since stop losses when using such a strategy will be infrequent, you can ultimately count on good profits. Fractal market analysis does not always give 100% profitable trades. In this regard, it should not be used only in a trading system. It is recommended to use other tools to confirm signals or filter.

When using fractal analysis, it is also important to study data from different time frames. The system that Bill Williams described in his works is trendy. To use it correctly, you must first determine the dominant trend in the market by looking at the older period.

The system should also take into account the “fractal lever”. This is the name of the possible amplitude during rollbacks. You can evaluate “fractal leverage” using the standard Fibonacci lines that are available in MT4. Corrections up to 38% Fibonacci are evidence of a strong trend movement. In this case, the fractal lever is strong. The opposite is true if the rollbacks are 62% fib or more.

Fractals and wave theory

Fractals can also be used in conjunction with wave theory. After all, in its essence, a fractal is nothing more than the beginning or end of an impulse movement or wave. A certain complexity arises here, because different impulses are formed at different periods of the charts. Traders who have gained experience in using wave theory have no difficulty in accurately identifying a specific wave on a specific time frame.

If several groups of fractals are formed at the same level, then if this level is broken through, a long and powerful trend should be expected. Fractal market analysis gives very good results in the presence of trends. When the price stays in the channels for a long time, the strategy for breaking out the fractal brings losses. The difficulty is that recognizing an emerging flat can be quite difficult.

How to apply a fractal strategy in a flat?

You should trade for a breakout only in the direction of a pronounced trend. You should not worry about several losses in a row. The future profit will certainly cover all the losses that the strategy incurred during fluctuations in the corridor. A good effect is achieved when working on small time intervals. If a trader enters a fractal breakout, focusing on the daily chart, then a stop loss can be set based on H4. Usually, the more fractals are located at the same level and the longer the flat lasts, the stronger and more directional the future movement will be.

To reliably determine whether a fractal breakout is true or false, you can use breakout candlestick analysis. If the breakout candle is “strong”, that is, it has a large body and its closing level is located far from clusters of fractals, then there is a high probability that the movement will continue in the chosen direction. Using this conclusion, you can successfully trade on small charts in order to increase profits. For example, if yesterday there was a breakout on D1, then today we can consider breakouts on the four-hour chart.

If after the breakdown of a cluster of fractals a reversal candlestick pattern has formed, then in the future most likely a flat will reign in the market and new fractals will appear. In this regard, great attention should be paid to the analysis of the breakout candle. To increase efficiency, it is recommended to familiarize yourself with at least the basics of Price Action (candlestick analysis).

Bill Williams recommended looking not only at the reversal candle, but also analyzing the volume. If the candle has a large body, but the volume is small, then the signal is weak. Signals that come from fractal clusters are strong when they form on longer-term charts (as is the case with candlestick analysis). Williams himself recommended watching D1. At the same time, it is necessary to analyze other timeframes. As mentioned in this article, fractal analysis is best combined with something else to increase the profitability of the strategy, because no tool can boast of 100% accurate signals.

The video contains useful information on the topic under consideration.

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In the previous article, we briefly reviewed the basic principles on which Trading Chaos is based. 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 make the picture complete, Today we will continue to look at Bill's trading techniques, significantly increasing profits from speculation, and let’s start with, perhaps, fractals.

In one of our earlier publications, we already touched upon the topic of identifying and constructing fractals (including using indicators). Therefore, we will not repeat the theory again; we will only note that a fractal is a formation whose central extremum is located above (below) the corresponding extrema of four neighboring bars.

The entire logic of fractal analysis in Trading Chaos is based on the search for breakouts of extreme points, but unlike later trading strategies developed by other traders, the original model according to Williams consists of strictly three elements:

1. Fractal start – the first extremum preceding the signal;
2. Signal fractal – formed in the opposite direction to the starting fractal;
3. Fractal stop is the largest top on a downtrend (or bottom on an uptrend) of the last two fractals.

Speaking of trends, in the Chaos Williams theory this issue is resolved by itself, since fractals become an integral part of wave analysis, and a wave (or wave structure) is a trend. At the same time, to maximize profits and further create a pyramid, it is permissible to switch to a lower timeframe after the start of the wave.

Pyramiding is an increase in a position in the direction of the trend after the floating profit on the first transaction allows the stop order for a set of orders to be transferred to breakeven, while the volume of each new transaction is either equal to a constant or divided by a certain coefficient.

For example, suppose that the third wave has begun in the market, which all wave traders are hunting for, in this case the trader’s action algorithm will be as follows:

1. A divergence appeared on the MACD between the third and fifth waves;
2. The current price is located in the target zone, i.e. the fifth wave, according to the approximate markings, should already begin (but it is not a fact that it will be fully formed), as a rule, beginners use Fibonacci levels to build zones, but much more often the situation is assessed visually;
3. A fractal has formed at the next top during a bullish trend and at the bottom during a bearish one;
4. Among the three maximum (minimum) bars, a “squat” appeared (see previous publication);
5. The MACD histogram bars crossed the signal line in the opposite direction to the latest trend.

Therefore, one should not look for a hidden meaning in Trading Chaos; Williams was simply able to describe for the first time the behavior of the market crowd using the tools of technical analysis, i.e., roughly speaking, mathematics, which deserves respect in any case.

Many traders use fractal analysis in their trading. Fractals help a trader identify market reversals.

By the way, the first person to look at the market as a nonlinear system was Bill Williams. In his scientific work, he discovered that the market moves chaotically. Bill Williams saw the harmony of chaos in cotton prices, human bleeding, and coastal surf.

Chaos in the work on the foreign exchange market, according to B. Williams, is its basis, therefore the exchange rate cannot be predicted using linear calculations.

Fractal, translated in Latin, means broken, crushed, shattered.

A fractal is a mathematical model that is self-similar. That is, each part of the fractal is similar to the fractal system as a whole. We can also observe the repetition of graphic figures in the foreign exchange market. By looking at the daily, hourly, five-minute and other charts of any currency pair, we will see repeating chart patterns.

Where do fractals come from?

The very concept of fractals appeared in the 19th century, when mathematicians were looking for examples of self-similar patterns in nature.

The word fractal was coined by Benoit Mandelbrot back in 1975, although he wrote the book “Fractal Geometry of Nature” only in 1977.

An interesting fact is that scientists have found fractals in corals, broccoli, sea shells, pineapples, human bronchi, as well as in snowflakes, clouds, lightning, crystals, etc.

However, fractals burst into trading when scientists began to use computer modeling. At this time, scientists A. Almazov in his work “Fractal Theory. How to change your view of the market" and B. Williams in "Chaos Theory..." suggested using fractal analysis in the foreign exchange market.

How is a fractal constructed?

To build a fractal in the foreign exchange market, you need to find five candles, among which the middle candle must be the maximum or minimum.

In the Forex currency market, everything is the same as in nature: a fractal is somewhat reminiscent of the fingers of a human hand. Put your hand on the monitor, and just as in the definition of a fractal, you will see that your middle finger will show the tip, where the fractal will be outlined.

However, not everything is so simple. Fractal calculations use a continuous sequence of candles. Therefore, it happens that one middle candle will be both the maximum and minimum for a different group of fractals.

Elements of fractal analysis

In the foreign exchange market, there are fractal models that describe the beginning of the movement of exchange rates. For example, a fractal start characterizes the movement of a currency pair, in which an upper fractal is formed first, for example, and the end of the movement closes the lower fractal.

Fractal model: a fractal signal, on the contrary, describes the end of the exchange rate movement, as opposed to the fractal start.

Thus, the impulse movement of the exchange rate begins with a fractal start and ends with a fractal signal.

Fractal stop

A fractal stop is the previous fractal on which an insuring Stop-loss order can be placed.

From the practice of using a fractal stop, it is important to note that sometimes the fractal stop is too close to our entry point. Therefore, it is better to place a fractal stop on a fractal that formed at a key low or high. It’s both safer and more effective.

Fractal analysis is used by traders mainly in the trend area of ​​the foreign exchange market. As you understand, a combination of five candles appears constantly. Therefore, in the case of a flat movement of the foreign exchange market, fractals give many false entry signals.

To effectively apply fractal analysis, traders use time frame overlay. How does this work?

We apply the fractal indicator in the MT4 platform through the menu Insert – Indicators – Bill Williams – Fractals.

Thus, we see the fractal indicator on the D1 chart of the EUR/USD currency pair.

From the figure above, we can see that the maximum of the fractal start was formed on May 8, 2014, when the daily candle closed at a price of 1.3839.

To enter the market more accurately, we use the tactics of adjusting time frames and look for an entry on the lower time frame – H1.

As can be seen from the figure, on the H1 chart of the euro/dollar, a flat formed at the price level of 1.3744-1.3774, which lasted until May 13, 2014. We marked the fractal level with a red line, after breaking through which, we expect to sell the pair euro/dollar At 9:00 on May 13, 2014, the fractal level was broken down and we entered a Sell position on the euro/dollar. Naturally, we place a stop-loss order slightly above the upper border of the flat, beyond the level of 1.3774.

In the terminology of fractal analysis, there is another important concept, the fractal lever.

In trading, fractal leverage is the size of the correction. To identify retracements in fractal leverage, traders use the Fibonacci grid.

There is an observation that if the correction reaches 38% on the Fibonacci grid, it means there will be good momentum and the fractal leverage is strong. In the case when the correction reaches up to 61.8% on the Fibonacci grid, then the fractal leverage is weak and you can ignore entering the market.

Let's take a closer look at the fractal level that we used in our example of entering the market above.

As you understand, the foreign exchange market moves in impulses and corrections. So, the fractal level is formed due to the accumulation of fractals, combinations of five candles, during correction.

In fractal analysis, it is important to understand that entering a breakout of a fractal level is safer and more effective than entering at the edges or turning points of the market. Also, you need to take into account the fact that it is better to take breakouts of fractal levels along the trend. Trend is our friend. Everyone who is against the trend is not our friend and, as a rule, quickly loses their deposit and then criticizes the foreign exchange market.

There is an observation that the longer a flat and fractal level is formed, the greater the likelihood of its breakthrough, and the greater the opportunity to make money from it.

Fractal level breakthrough

In the practice of using fractal analysis, difficulties arise in determining a true or false breakthrough of a fractal level. After all, even from our example above it is clear that the fractal level was broken through and tested several times. How to understand where the true breakthrough of the fractal level is and where it is not.

Traders in their work, when determining a breakthrough of a fractal level, use the search for a breakout candle. A breakout candle is a candle that confidently closed outside the fractal level. It can be assumed that the next candle, after the breakout, will also continue its movement towards the breakout.

But it is possible that after a breakout candle there is a return to the fractal level, something similar to the expansion of a flat. To enter the market more accurately in such cases, you should use other methods of analysis, possibly using additional indicators.

Thus, in our short review of fractal analysis in the foreign exchange market, we learned that fractals exist in nature, that human fingers help determine the fractals of the foreign exchange market on your monitor. Also, you already understand what a fractal start, signal and lever are.