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Identifying Order and Disorder in Chaotic Market with Elliott Wave Trend

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Identifying Order and Disorder in Chaotic Market with Elliott Wave Trend

Young Ho Seo

www.algotrading-investment.com

7 October 2018

Article version: 1.0

As a trader, you will agree that scientific and engineering method is the only legitimate method to improve your profits. Whether you are using correlation, geometric patterns, or statistics, they are science. Even latency arbitrages are just a sort of computer science or engineering, one could dig deeper on how to fine-tune the algorithm, network and internet speed to beat others.  If any trading strategy has persistent dynamics towards profitability, we can formulate or describe their working mechanism using known science and engineering principle. Hence, we can reproduce the same mechanism repeatedly for our trading. Most winning trading strategies are based on some sort of precise science and engineering. Profit will never come blindly or by just a chance.

Topic in this article is partly about chaos theory. As one knows, chaos theory as in nonlinear dynamics is a hard-core math topic. However, we will not touch the complex theory because this article is for average trader. The focus of this article is to provide some intuition over this hard science, chaos theory. Chaos theory reveals several characteristic for some dynamic system like stock or currency market. Here is the list of four important characteristics:
1. Highly sensitive to initial condition (i.e. Butterfly effect)
2. Feedback loop
3. Order/Disorder
4. Fractals (i.e. self-similarity)
5. Etc

Firstly, scientist often uses the term butterfly effect. This term describes the situation in which a butterfly flapping its wings in New Mexico cause a hurricane in China. A different way to express this is that small changes in the initial condition can lead to drastic changes in the results. We can meet full of similar examples in our life. One example of butterfly effect is that the rejection of an art application lead to World War Two. This is probably the most widely known butterfly effect. In 1905, Adolf Hitler applied to the Academy of Fine Arts in Vienna. He was rejected twice. After his rejection, he was forced to live in the slums of the city and his anti-Semitism grew. He joined the German Army instead of fulfilling his dreams as an artist. World would been changed a lot if Academy of Fine Arts accepted him.
Secondly, feedback loop describe that systems often become chaotic when there is feedback present. A good and relevant example is the stock market. As the value of a stock rises or falls, stock traders tend to buy or sell that stock. This feedback from stock traders will further affect the price of stock going up and down chaotically. This feedback loop produces the fractal wave or equilibrium fractal wave in the market.
Thirdly, chaos is not necessarily disorder although people use the term chaos interchangeably with disorder. Instead, chaos suggests that a system can transition between order and disorder. In turn, order and disorder is connected to the predictability and unpredictability of the financial market. In some science, this is called entropy. In fact, this is the main topic of this article. We will get back to this later.
Fourthly, a chaotic system can exhibit fractal pattern. We have wrote the comprehensive description on how to use this characteristic in our trading. Please refer to the book: Financial Trading with Five Regularities of Nature: Scientific Guide to Price Action and Pattern Trading. We will not reintroduce this in this article.
We have briefly covered four main characteristics of a chaotic system. Stock and forex market are the live example of a chaotic system. Order and disorder in chaos theory describe that the financial market can transition from order to disorder or from disorder to order. Order and disorder are highly related to the predictability and unpredictability of market. Ordered market is much easier to predict whereas disordered market is hard to predict. Every trader will agree that sometimes market is easy to predict with some simple technical analysis. Sometimes, the market is almost unpredictable no matter what kind of information or analysis on your hands. If you felt this, then you are probably right. This is the corresponding behaviour of order and disorder described in chaos theory.
As a trader, your will be better off to avoid highly disorder market but to trade on highly ordered market. Unfortunately, order and disorder does not provide the rigid binary states, where you can pick up one or the other. We can only tell that degree of order and disorder. For educational purpose, I always like to present Hurst exponent. Hurst exponent will provide the some indication if the market is predictable or not predictable.
Even though I like to use Hurst exponent for an educational purpose, I do not use Hurst exponent for my trading. Some years ago, inspired from this order and disorder concept, me and my friend tried to use Hurst exponent for several month. Unfortunately, we did not gain much benefit from Hurst exponent for our trading. There can be few reasons for this. Firstly, Hurst exponent was not designed for trading. Secondly, Hurst exponent is not a timing tool for trading. You are fine to use it for some retrospective study in your theoretical research or some educational purpose. However, we do not recommend using it for live trading because the range analysis requires sufficient samples to derive the accurate result. You will experience the typical lagging from the averaging algorithm. Hence, it is more theoretical tool.
As a trader, you will never use lagging indicator because you will sell when market is going up and you will buy when market is going down. Then what could be alternative indicator or analysis to gauge the predictability of the financial market? Here we are talking about techniques we can use for our live trading.

The discovery really came to me by chance. I found that density of specific geometric patterns could be used to effectively spot the predictability of the financial market. Now let us disclose what the specific geometric patterns are first. Now think about the trend with feedback loop in a chaotic market. For uptrend, the price series will move in zigzag form as in Figure 1. While we are reaching the top of our trend, the price will go through several patterns as in Figure 2. Initially it will form 1 triangle, then price will expand to form 2 triangles and 3 triangles. Finally, it will form the 4 triangles (Figure 2). R. N. Elliott saw this much earlier than myself in 1930s in his Elliott Wave patterns. One thing he did not see was that the density of these geometric patterns can indicates the predictability of the market. The “density of these geometric patterns” is simply the count of these geometric patterns at specific time. We use the density or count as to gauge the degree of predictability in the financial market. Probably not every trader would be able to see this in the past. The idea is simple but producing such a tool to count these geometric patterns is a complicated business. I only discovered this while I was developing the template and pattern approach for the scientific Elliott wave counting.
Let us think about how this works. In my opinion, each pattern in Figure 2 represents some degree of order in the financial market. Especially when we observe two, three, or four triangles moving in the path of trend, we can tell that there is high order. At the same time, the fractal nature of financial market exhibits these patterns in many different scales. At the specific time, small and large scale of these patterns can be overlapping to form more solid evidence of order as shown in Figure 4.
What is the practical implication of this for our trading? The density of these geometric patterns tells us when the financial market is highly predictable and when not. As we have mentioned, you will be better off to trade when the market is more predictable of course.
Trader might ask question if the ratio in each pattern in Figure 2 is matter because they trade often with Fibonacci ratios. The answer is no in this case because we are only concerning the trend movement. In the trend movement, the direction of triangle is matter but the triangle can be formed in any ratios. However, if you are counting the specific Elliott Wave pattern, then you have to use a specific ratio to predict the trading direction. This is a separate concern from identifying high order or highly predictable market state. Elliott Wave trading principle is well built inside our Elliott Wave Trend. There is no need to worry about this.
In addition, be precise that the density of these geometric patterns concerns the order and predictability of the financial market. Knowing the predictability will not cover your entire trading setup. For detailed trading entry or exit, you will need to use the specific entry and exit tool.
For example, of course, you can use Elliott wave pattern to make your trading entry and exit complete. That is already built inside Elliott Wave Trend. For example, in Figure 5, we have shown that the turning point is coincided with the Wave 4 location of Elliott Wave 1234 pattern. You can setup buy entry with this logic.
However, Elliott Wave pattern is just one of the trading strategy you can use. Once you have identified that market is in high order or highly predictable state, you can always use other proven trading strategies. For example, in Figure 6, we can see that AB=CD pattern gave us the identical buy signal too. What trading strategy you are using is really depending on your skills and discipline.
If you still want even simpler explanation, then use this two-step trading process.
1. Identify the high order or highly predictable timing in the market.
2. Apply your favourite trading strategy for the detailed market entry and exits including Elliott wave, harmonic pattern, supply demand or price breakout patterns, etc.

Finally, one might ask if they can go beyond 4 triangles like or 5 triangles or 8 triangles to calculate the density. From my empirical study, it is rare to see anything above 4 triangles. In this point, my empirical study and R. N. Elliott observation agrees. That is why Elliott comes up with Elliott Wave 12345 pattern and not Elliott Wave 12345678 pattern.

Final Note:
We hope this is useful for your profitable trading. Elliott Wave Trend has the following features:
1. Detect the density of geometric patterns (identifying order and disorder of financial market.)
2. Automatic Elliott Wave Counting for trading.
3. Manual Elliott Wave Counting for trading.
4. Automatic Wave Structure Score calculation.
Here is the links to our Elliott Wave Trend:
https://www.mql5.com/en/market/product/16479
https://www.mql5.com/en/market/product/16472
https://algotrading-investment.com/portfolio-item/elliott-wave-trend/

Figure 1: Zig Zag price movement for uptrend.

Figure 2: Four geometric patterns in the course of trend movement.
 
Figure 3: Count of the geometric patterns in chart.
 
Figure 4: Count of the geometric patterns in chart.

 
Figure 5: Elliott Wave 1234 identification.

 
Figure 6: AB=CD pattern at highly predictable time.

Scientific Guide to Equilibrium Fractal Wave

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Scientific Guide to Equilibrium Fractal Wave

Written Date: 22 Sep 2018

Article Version 1.2

By Young Ho Seo
Quantitative Developer and Financial Engineer
 admin@algotrading-investment.com

www.algotrading-investment.com

1. Introduction to Equilibrium Fractal Wave

The concept of Equilibrium Fractal Wave was first introduced in the book: Financial Trading with Five Regularities of Nature: Scientific Guide to Price Action and Pattern Trading (Seo, 2017). At that time, the book was written for the pure motivation to identify the important market dynamics for financial traders. The concept of Equilibrium Fractal Wave was born by combining two scientific areas including time series and fractal analysis. The main propositions in the Equilibrium Fractal Wave include:

1. The separate or combined analysis of trend and Fractal wave is possible.
2. The repeating patterns in Equilibrium Fractal Wave are equivalent to the infinite number of distinctive cycles because the scale of the repeating pattern varies infinitely.
3. Equilibrium Fractal Wave is just a superclass of all the periodic wave patterns we know.

First, let us demonstrate the equilibrium fractal wave for readers. The easiest way to demonstrate the equilibrium fractal wave is through the pattern table presented in Figure 1 (Seo, 2017). Many applied researchers in time series and statistics will agree that patterns in the column 1, 2, 3 and 4, from first regularity to fourth regularity, are the mainly extracted features and patterns in their everyday research and operation. It is also agreeable that cyclic wave pattern can co-present with trend together. This concept is the main assumption behind the classic decomposition theory in the time series analysis. In the time series pattern table created by Gardner in 1987 (Figure 2) represents this concept clearly. The first row in the pattern table (Figure 1) shows the data in which no trend or weak trend exists. The second, third and fourth rows shows the co-existence of trend and waves.

Until now, many forecasting or industrial scientists use such concept to build forecasting models. Likewise, there are many applied software to create the forecasting or prediction model of this kind. Some example forecasting software with such modelling capability includes:
1. Stata (www.stata.com)
2. Eviews (www.eviews.com)
3. IBM SPSS (www.ibm.com/products/spss-statistics)
4. SAS (www.sas.com)
5. MatLab (www.mathworks.com)
6. And many others

 
Figure 1: Five Regularities and their sub price patterns with inclining trends. Each pattern can be referenced using their row and column number. For example, exponential trend pattern in the third row and first column can be referenced as Pattern (3, 1) in this table.

 
Figure 2: The original Gardner’s table to visualize the characteristics of different time series data (Gardner, 1987, p175). Gardner assumed the three components including randomness, trend and seasonality in this table.

Now the fifth column in Figure 1 presents the equilibrium fractal wave. This is extended part from the original Gardner’s table. When we list the equilibrium fractal wave in the fifth column, we can see that the pattern table (Figure 1) shows a systematic pattern. From left column to right column, we can see that the number of distinctive cycles in the data increases. For example, we can assume the pure trend does not have any periodic cycle. Therefore, number of the distinctive cycle is zero for pure trend series. Under the second and third columns, we can have one to several distinctive cycles depending on if the series follows daily, monthly, and yearly cycles. Under fourth column, we can have many more distinctive cycles outside daily, monthly and yearly cycles but the number of the cycles is finite. Fourier analysis or principal component method can be used to reveal the number of cycles for any series under column 4. From column 1 to column 4, you might be following this systemic pattern pretty well.
However, you might question why equilibrium fractal wave in column 5 possesses such infinite number of distinctive cycles. This is indeed the right question to ask. To understand this, you have to understand the fractal wave first.

A lot of research on fractal analysis was done by B. Mandelbrot (1924-2010). The Book: fractal geometry of nature (Kirkby, 1983) describes the nature of fractal geometries in scientific language. What is the difference between fractal wave and equilibrium fractal wave in this article? Fractal wave views a series as the subject of fractal analysis. Equilibrium Fractal wave views a series as the co-subject of fractal analysis and trend analysis. Hence, equilibrium fractal wave believes co-existence of trend and wave pattern in a single data series. The significance of equilibrium fractal wave is that we can model the trend and fractal wave in two separate steps or in one-step.
Indeed, scientists use the two-step process to model the data in column 2, 3 and 4 in economic and financial research. For example, price series under column 4 can be modelled with trend in the first step. Then the reminding data can be modelled using cycles in the second step. Likewise, for a data series under column 5, we can model a trend part first, then we can model a fractal wave patterns in separate steps. This explains the Proposition 1. This also imposes the fractal analysis under non-stationary condition when the trend component is strong in the data series. In this case, two-step modelling process might be advantageous. When the trend component is less dominating comparing to fractal wave component, the entire price series can be modelled using fractal analysis only. Proposition 1 states that the choice on the modelling process, either one-step or two-steps, is conditional upon the characteristics of the price series.

In the Book: fractal geometry of nature (Kirkby, 1983), the main characteristics of fractal wave is described as the repeating patterns in varying scales. To give you some idea of repeating patterns in varying scales, we can create a synthetic data like that using Weierstrass function. This function is famous for being continuous everywhere but non-differentiable nowhere among the math community. Of course, real world data will never look like this. However, this synthetic data describe what is repeating pattern in varying scale very well for our readers in Figure 3. You will see the same patterns everywhere in the data. Small pattern are combined to become the bigger pattern. The resulting bigger patterns look the same like small patterns. As the combing process continues, the size of the pattern can increase infinitely. This is referred to as repeating patterns in varying scale or varying size. This is the core assumption on any fractal analysis.

Now let us walk backwards from this combining process. Let us assume that we can extract those patterns in the same scale from rest and we can put them on the separate paper for each scale. When we separate those patterns in the smallest scale from rest, then the extracted series become the first cycle of our data. This extracted series with one cycle is not different from data or a series in column 2, 3 and 4. Likewise, we can separate the second smallest patterns from rest. This will become second cycle of our data. In this time, the frequency of second cycle will be less comparing to the first cycle because the period of second cycle is greater than first cycle. We can keep continue this separating process to create another cycles. Since we can combine to create the repeating patterns infinitely, we can separate the repeating pattern infinitely too. This describes the proposition 2, the infinite number of distinctive cycle.

 
Figure 3: Weierstrass function to give you a feel for the Fractal-Wave process. Note that this is synthetic Fractal-Wave process only and this function does not represent many of real world cases.

Now the Proposition 2 can lead to the Proposition 3 naturally. As you can see from Figure 4, from left to right columns, the number of distinctive cycle increases. Therefore, it is not so hard to say that equilibrium Fractal Wave is a superclass of all the periodic wave patterns we know in column 1, 2, 3 and 4. Figure 4 shows this concept clearly to our reader.
 
Figure 4: Visualizing number of distinctive cycle periods for the five regularities. Please note that this is only the conceptual demonstration and the number of cycles for second, third and fourth regularity can vary for different price series.

Finally, in many real world data, we do not possess the highly regular patterns as in a synthetic data like that using Weierstrass function in Figure 3. The highly regular repeating patterns are described as the stick self-similarity in In the Book: fractal geometry of nature (Kirkby, 1983). Instead of the strict self-similarity, the real world data will form loose self-similarity shown in Figure 5. For the financial price series, we can observe the repeating zigzag patterns made up from so many triangles. The triangles are only similar. However, each triangle in the data will be never identical to the other triangles. This is the typical example of loose self-similarity. This sort of loose self-similarity is much harder to model comparing to the strict self-similarity shown in a synthetic data like that using Weierstrass function (Figure 3).

 
Figure 5:  Loose self-similarity in the financial price series.

2. Empirical Research on Equilibrium Fractal Wave

As we have described, the concept of equilibrium fractal wave allow us to model the series as the co-subject between trend and fractal wave or as the single subject of fractal wave. The modelling choice will depend on the characteristics of data. Regardless of the modelling choice, Empirical research on equilibrium fractal wave must concern the fractal patterns in data series. Empirical research on equilibrium fractal wave in the price series data is relatively small because mainstream academic research is based on the algorithm utilizing the entire data sets like multiple regression techniques instead of detecting patterns.

One exception is the financial trading community. In the trading community, the repeating patterns or repeating geometry was used as early as 1930s. Some pioneers include R. Schabacker (1932), H.M. Gartley (1935) and R.N. Elliott (1938) in time order. In their books, the various repeating patterns were described for various US stock market data (Figure 6, 7 and 8). Until now, millions of traders are using these patterns in their practical applications for the profiting purpose in forex, future, and stock markets. Figure 6, 7 and 8 shows the commonly used repeating patterns by the financial trader. Having said that these repeating patterns in Figure 6, 7 and 8 were not modelled as the co-subject between trend and fractal wave. Instead, those repeating patterns are only modelled as the subject of fractal wave. Only exception is the trend filtered ZigZag indicator and excessive momentum indicator created recently (Seo, 2018). This is understandable consequence because the idea of equilibrium fractal wave and the two-step modelling process were only introduced in 2017. The modelling technique using trend and fractal wave patterns are only available recently. One very purpose of this article is to inform you that it is possible to model the financial price series as the co-subject between trend and fractal wave in two separate steps.

At the same time, another purpose to create equilibrium fractal wave was to connect the contemporary science to many repeating patterns used by the financial traders. Considering that millions of the financial traders now use the repeating patterns for their every day trading, this is a phenomenal level of activity by the society. Many traders are much happier to use the repeating patterns than the traditional math or technical indicators. Unfortunately, the connection between the repeating patterns and the contemporary science is very poor. It seems no literature is positioning those repeating patterns in the scalable scientific framework. Neither the financial trading community have much idea on what these repeating patterns are and why they are using these patterns. Simply speaking the communication between two communities is blocked. If R.N. Elliott (1938) had a chance to meet B. Mandelbrot (1924-2010), then things may have changed bit. However, they lived in two different time.

The pattern table in Figure 1 shows that repeating patterns are merely the extended concept from the conventional mathematical knowledge. We know that it is not so hard to put these five regularities together under the same table. Potential for academic and applied research in equilibrium fractal wave is huge. The main concern is that many techniques used for periodic wave pattern analysis may not work with equilibrium fractal wave because of the infinite number of the distinctive cycle in the data. To the best knowledge, Fourier analysis and many other similar techniques will not handle the infinite number of the distinctive cycle. Therefore, developing new analytical techniques remain as the main challenge for the empirical research in equilibrium fractal wave. In many cases, the algorithm or pattern recognition modelling the price series as the co-subject of trend and fractal wave will improve the prediction accuracy much more.

 
Figure 6: List of triangle and wedge patterns.

 
Figure 7: Ascending Triangle pattern found in USDCAD in H1 chart.

 
Figure 8: Repeating Gartley patterns in Hourly EURUSD Chart Hourly.

3. Further ideas in the Modified Quantum Physics

This section discusses the separate concern from this article. However, this is the last purpose for this article. As we know, Fourier analysis and the quantum mechanics have a strong connection. Fourier analysis can be used to decompose a typical quantum mechanical wave function. The wave and trend concept in Figure 1 closely resemble the wave and particle duality of the quantum physics. For example, the price series with periodic cyclic wave in column 2, 3 and 4 (Figure 1) can be modelled well using Fourier analysis too. Likewise, the trend and particle shares many common analysis techniques in the statistics, signal processing, and object-tracking field too.

As we can extend the classic wave pattern into equilibrium fractal wave pattern (Figure 1), we might be able to extend the quantum physics further to deal with the infinite number of distinctive cycles as in the concept of equilibrium fractal wave. It is often heard that many quantum physics or quantum mechanics based algorithms fail to bring the profits or good prediction in the financial trading. The reason might be that the contemporary quantum physics is not dealing with the infinite number of distinctive cycles present in the data. I am just guessing that the modified quantum mechanics might work much better in the financial trading than the contemporary quantum physics. As a bonus, the modified quantum physics can lead to the technological breakthrough in developing better medicine and better spaceship in the future. This is just some research ideas for those working in physics.
 

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Elliott, R. N., Douglas, D. C., Sherwood, M. W., Laidlaw, D. & Sweet, P. (1948) The wave principle (Note that first formal publication date of The Wave Principle was August 31 1938).

Elliott, R. N. (1982) Nature’s Law: The Secret of the Universe, Institute for Economic & Financial Research.

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Schabacker, R. (1932) Technical Analysis and Stock Market Profits, Harriman House Limited.

Golden Ratio and Financial Trading

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Golden Ratio and Financial Trading

www.algotrading-investment.com

20 Feb 2018
Written By Young Ho Seo
Finance Engineer and Quantitative Trader

Introduction to Golden Ratio 0.618 for financial trading
When I look back, even during my math class in the university, the golden ratio or Fibonacci number was not so popular topics. However, I still remember that a particular technique called a “Golden Section Search” was taught along with Newton’s method. Well, just like many of the students, I forgot about this technique after passing the exam. Many years later, the term “Golden ratio” keep coming back more and more during my research with financial market data (If you are not sure what the Golden ratio is, please check the appendix at the end of this article.) I realized that the importance of Golden ratio might be far more significant than what the financial trader think. Firstly, many geometry or shape found in nature including trees, leaves, flowers, etc, are often built upon the golden ratio and the derived ratios (i.e. Fibonacci ratios). Even there were some interesting research showing the relationship between the golden ratio and beauty. Now you can tell that the frequent occurrence of the Golden ratio is natural phenomenon. What do you think about the financial market? As you know, financial market is made by man. Would the Golden ratio play an important role in the financial market? If so, it would be quite surprising. The truth is yes. The Golden ratio 0.618 and other derived ratios (i.e. Fibonacci ratios) like 0.382 and 0.500 are considered as important. In fact, the belief about the Golden ratio was there for more than 85 years. I am referring to the work by Ralph Nelson Elliott in 1938. The use of Golden ratio for the financial market can go back even more. Whether you are user of the Golden ratio and the Fibonacci ratio for your financial trading (see appendix), you will be kept surprising reading this article until the end. We have built a scientific tool to reveal the precise structure of the financial market. The scientific tool can not only extract the useful information for your financial trading but also it can be used to make some interesting inference about the financial market. Now to start with, let us understand how to use the golden ratio and Fibonacci ratio for the financial trading first.
How to use the Golden Ratio and Fibonacci Ratio for financial trading
The most common way to apply the golden ratio and Fibonacci ratio is to use two price swing points in your chart. To identify the two swing points, you can simply use the peak trough analysis provided on our website. It is free of charge for use and for sharing (https://algotrading-investment.com). You can have a multiple options to identify the swing point in your chart. However, there are automated tools (the peak trough analysis) for the task, we will not discuss too much on how to detect the swings points manually.
 
Figure 1: Basics of Fibonacci ratio measurement (or Shape ratio measurement).
 
Figure 2: Basics of Fibonacci ratio measurement (or Shape ratio measurement).

Anyway, after you have identified the swing points, you can measure the ratio of two price swing points as shown in Figure 1 and 2. The ratio of price height of two swing points often expected to be close to the golden ratio or the Fibonacci ratio. We use this knowledge for our trading as shown in Figure 3 and Figure 4. In Figure 3 and Figure 4, we expect that the price will reverse at 38.2% (0.382) Fibonacci ratio. This analysis is called Fibonacci retracement analysis. This analysis is useful to check the corrective phase of the market. In the chart, we can easily spot where it reverse. Based on this idea, we can make our trading plan. This is the typical strategy used by millions of forex and stock market traders.

 
Figure 3: Fibonacci Retracement drawn over daily EURUSD candlestick chart for bearish setup.

 Figure 4: Fibonacci Retracement drawn over daily EURUSD candlestick chart for bullish setup.

Trading with Fibonacci retracement and expansion is relatively simple. Now there are advanced trading strategies using the Golden ratio and Fibonacci ratio too. We can introduce the two trading strategies in brief. One of them are Harmonic pattern trading. The other one is Elliott wave trading. In harmonic Pattern trading, we identify three or four successive swing points to identify the reversal trading opportunities. Just like Fibonacci retracement and expansion, the ratios measured between three or four successive swing points are expected to be the Golden ratio or Fibonacci ratio. In Elliott wave trading, we use around three to five swing points to identify the trading opportunities. The ratios of these swing points are the Golden ratio and Fibonacci ratios. In Elliott wave trading, the Golden ratio 0.618 and 1.618 are highly emphasized whereas in harmonic pattern trading, the other Fibonacci ratios are equally used to construct the harmonic patterns.
 
Figure 5: Butterfly pattern formed in EURUSD H4 timeframe.
 
Figure 6: Impulse Wave 12345 pattern formed in EURUSD D1 timeframe.
 
Figure 7: Corrective Wave ABC pattern formed in EURUSD D1 timeframe.

Revealing the Financial Market Structure using Equilibrium Fractal Wave Index
So far, we have introduced three trading strategies based on the Golden ratio and the Fibonacci ratios. These trading strategies are based on the assumption that there will be the frequent occurrence of the Golden ratio and Fibonacci ratios in the financial market. However, not necessarily we have much scientific evidence to support this assumption. I think these trading strategies can become more popular if there is more scientific evidence to support the trading logic and rational behind the Golden ratio and the Fibonacci ratios. To reveal the financial market structure precisely, we have made a scientific framework called Equilibrium fractal wave. To reveal the market structure, we need to understand what ratios the market is made up including both Fibonacci ratios and non-Fibonacci ratio. Using the framework of the Fibonacci ratio analysis can limit our understanding since we can only study Fibonacci ratios. Therefore, we use the generic term called “Equilibrium Fractal Wave” to describe the price geometry made up from the two price swing points (or three points) in your chart as shown in Figure 1 and Figure 2.
By definition, an equilibrium fractal wave is a simple triangle made up from two price swing points. It is precisely identical to the triangle introduced in Figure 1 and Figure 2. We refer to the ratio (Y2/Y1) as the shape ratio in equilibrium fractal wave. The shape ratio represents the shape of each equilibrium fractal wave and it is an identifier used to reveal the market structure. The shape ratio can include any ratios including Fibonacci ratios and non-Fibonacci ratios in our study.

 
Figure 8: One unit (or one cycle) of equilibrium fractal wave.
 To reveal the market structure, we use the quantity called Equilibrium fractal wave (EFW) index. The equation of the EFW index is shown below:
Equilibrium Fractal Wave (EFW) Index = number of the particular shape of equilibrium fractal wave (the shape ratio = Y2/Y1) / number of peaks and troughs in the price series.
The equation is straightforward to calculate in any charting package. The EFW index is a quantity describing how frequently we can detect the particular shape ratio (Y2/Y1) in the financial market. For example, if the Golden ratio 0.618 is really dominating in the financial market, we should have a highest EFW index among all ratios. Otherwise, our belief on the Golden ratio can be wrong or less optimal. It is the same for other Fibonacci ratios. If you were using the Fibonacci ratios 0.382 (38.2%), you should expect the EFW index of 0.382 to be higher. Otherwise, you were trading less optimal strategy for your investment. To reveal the market structure, we can create a distribution of EFW index from the ratio 0.1 to the ratio 3.0. We list the distribution of EFW index for EURUSD, GBPUSD and USDJPY in Figure 9, 10 and 11.
 
Figure 9: EFW Index Distribution for EURUSD Daily Timeframe from 2009 09 02 to 2018 02 20 (Label inside callout box, left: Ratio, right: EFW Index, vertical axis: EFW index, horizontal axis: ratio from 0.1 to 3.0).
 
Figure 10: EFW Index Distribution for GBPUSD Daily Timeframe from 2009 09 02 to 2018 02 20 (Label inside callout box, left: Ratio, right: EFW Index, vertical axis: EFW index, horizontal axis: ratio from 0.1 to 3.0).

 
Figure 11: EFW Index Distribution for USDJPY Daily Timeframe from 2010 05 30 to 2018 02 20 (Label inside callout box, left: Ratio, right: EFW Index, vertical axis: EFW index, horizontal axis: ratio from 0.1 to 3.0).

You can immediately recognize several important factors in this analysis. Firstly, each financial market has the different footprint of the EFW index distribution. This justifies their own unique behaviour of each financial instrument. Secondly, our belief on the Golden ratio and the Fibonacci ratios are less optimal rather than wrong. We can tell that the Golden ratio and the Fibonacci ratios stay in the top of the league table for three currency pairs. However, still some other ratios are ranked highest in the table. For example, the ratio 0.66, 0.50 and 0.75 stayed in the top of the table. It should be noted that for each financial instrument, there is a preferred ratio for your trading. If you were trading using the ratio 0.618 for GBPUSD, then it was far less optimal. You should have used the ratio 0.500 instead. In Figure 12, we have calculated the EFW index over the rolling window for GBPUSD daily timeframe. The rank of each ratio does not change often. We can tell that the market structure is stable over the time. Therefore, the revealed market structure in Figure 9, 10 and 11 might be at least semi-permanent characteristics of each financial instrument.
 
Figure 12: EFW index for GBPUSD D1 timeframe from 2007 01 04 to 2018 01 20.

What is your belief now and how you are going to trade?
This article revealed some important information for your trading, that no one have revealed before. We were trying to answer the question on the Golden ratio and the Fibonacci ratio, which were not answered last 100 years. In our analysis, we have revealed the market structure of the financial market using the scientific tool called the EFW index. If you were trading using the Golden ratio and the Fibonacci ratio, you might be shocked a bit. Now you know what to do to improve your trading. It is only the scientific analysis can help you to win in the financial market. Many traders including myself might be curious why the Golden ratio is less optimal or not optimal for some financial instruments. Well, honestly I do not have the right answer for it. I think that no one has the right answer but we can only guess. In nature, the golden ratio or other Fibonacci ratios are repeating in much higher precision than the financial market. The less precise nature in the financial market might be due to the higher noise in the financial market because of too many diverse players. Another possible explanation might be that the profitability of the Golden ratio and some Fibonacci ratios might be exhausted because too many of us were using them every day. Therefore, the EFW index distribution in Figure 9, 10 and 11 might be showing the distorted image of the financial market. Please feel free to write me on FinancialEngineerPro21@gmail.com if you have a better explanation about why the Golden ratio is less or not optimal for the financial market.

Appendix (Golden ratios and Fibonacci ratios)
The Fibonacci Ratio is used by millions of forex and stock market traders every day. It is a mega popular tool in the trading world. If you do not know what the Fibonacci ratio is, here is the simple explanation. Fibonacci ratio is the ratio between two adjacent Fibonacci numbers. To have a feel about the Fibonacci ratios, here is the 21 Fibonacci numbers derived from the relationship: Fn = Fn-1 + Fn-2.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, …………………
Once the Fibonacci numbers are reasonably large, you can just pick up any two adjacent Fibonacci numbers above to derive the ratio. For example, we will find that 4181/6765 = 0.618 and 1597/2584 = 0.618. Here 0.618 is called as the golden ratio. The golden ratio is one of the most important Fibonacci ratios. The rest of Fibonacci ratios are derived by using simple mathematical relationship like inverse or square root or etc. Table below shows the list of Fibonacci ratios you can derive from the Golden ratio 0.618.

Type Ratio Calculation
Primary 0.618 Fn-1/Fn of Fibonacci numbers
Primary 1.618 Fn/Fn-1 of Fibonacci numbers
Primary 0.786 
Primary 1.272 
Secondary 0.382 0.382=0.618*0.618
Secondary 2.618 2.618=1.618*1.618
Secondary 4.236 4.236=1.618*1.618*1.618
Secondary 6.854 6.854=1.618*1.618*1.618*1.618
Secondary 11.089 11.089=1.618*1.618*1.618*1.618*1.618
Secondary 0.500 0.500=1.000/2.000
Secondary 1.000 Unity
Secondary 2.000 Fibonacci Prime Number
Secondary 3.000 Fibonacci Prime Number
Secondary 5.000 Fibonacci Prime Number
Secondary 13.000 Fibonacci Prime Number
Secondary 1.414 
Secondary 1.732 
Secondary 2.236 
Secondary 3.610 
Secondary 3.142 3.142 = Pi = circumference /diameter of the circle

Table 1: Fibonacci ratios and corresponding calculations to derive each ratio.

Winning Financial Trading with Equilibrium Fractal Wave

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Table of Contents

About the book 2
1. Introduction to Equilibrium Fractal Wave for Financial Trading 4
2. Five Characteristics of Equilibrium Fractal Wave 6
3. Hurst Exponent and Equilibrium Fractal Wave Index for Financial Trading 24
4. Shape Ratio Trading and Equilibrium Fractal Wave Channel 41
4.1 Introduction to EFW Index for trading 41
4.2 Trading with the shape ratio of equilibrium fractal wave 46
4.3 Introduction to Equilibrium Fractal Wave (EFW) Channel 51
4.4 Practical trading with Equilibrium Fractal Wave (EFW) Channel 59
5. Appendix 67

1. Introduction to Equilibrium Fractal Wave for Financial Trading
The concept of Equilibrium fractal wave was first introduced in the Book: Financial Trading with Five Regularities of Nature: Scientific Guide to Price Action and Pattern Trading (2017). In the book, I have categorized the three distinctive market behaviours (regularities) for financial trading (see appendix). The second behaviour was further split into the three sub categories. Therefore, the five distinctive market behaviours were introduced for financial trading in total (see appendix). Most of trading strategies can be categorized under these five categories except the correlation (i.e. fundamentals). Having said that, the correlation is still the main cause behind these five distinctive market behaviours. Therefore, we are still studying the effect of correlation while we are studying these five distinctive market behaviours. If the five categories sound too much, forget about it. I always like things simple and stupid. Just remember the three categories for your trading. The three categories include equilibrium (first), equilibrium wave (second), and equilibrium fractal wave (third). In brief, equilibrium is equivalent to the trend. Equilibrium wave is equivalent to the market cycle with some definable cycle period. Equilibrium fractal wave is equivalent to the infinitely repeating price patterns in the financial market.
Many people might feel curious to see equilibrium wave and equilibrium fractal wave instead just wave and fractal wave. They or you might question why “equilibrium“ in front of “wave” and “fractal wave”? In fact, this is due to my personal understanding on particle-wave duality from Quantum Physics. Equilibrium is in fact a term to represent the particle or the behaviour of the particle in the financial market. Many scientist believe that they can not apply the quantum physics directly to the financial market. Indeed, what I believe is that we can use the Quantum physics but we just need a modified version of Quantum physics to better model the financial market due to the strong presence of Equilibrium fractal wave. Only discuss this to show you how the five categories (i.e. the five regularities) are inter-related to other branches of science. The focus in this article is to explain equilibrium fractal wave as simple as possible without any mathematical equation if possible.
Anyway, the way we capture each of these three market behaviours into our profit is very different because their distinctive characteristics. There are two cases where many traders make a serious mistake for their financial trading. Firstly, many traders often believe that the second and third categories are strongly similar in terms of how to capture them technically, because both have the term “wave” inside. Secondly, many others also believe that they should only capture the first category of the market behaviour ignoring the second and third category. First case is due to the lack of technical knowledge. If you are trying to define the cycles where the cycle period is not definable, then your model will start to break down. We cannot torture data to see what we want. If you can define the periodic cycles, yes, it is great opportunity for our trading. Go ahead with it. However, if the equilibrium fractal wave is present strongly, then due to its infinite scalability, modelling the market with periodic cycle become very difficult task. The second case is due to the overly simplified belief on the financial market. Just imagine that financial market is the transformation of infinite number of internal and external variables into the two dimensional space between price and time. Therefore, the price represents the complex crowd behaviour. In highly liquid and competitive financial market, an overly simplified assumption can offer you the immature entry and exit only for your trading.
To avoid the above two cases, it is helpful to understand the three distinctive behaviours of the financial market in details. Indeed the main book: Financial Trading with Five Regularities of Nature: Scientific Guide to Price Action and Pattern Trading (2017) will provide the good introduction over all three distinctive behaviours of the financial market. In this introductory book, we will only cover the basics of the Equilibrium fractal wave for your trading.

2. Five Characteristics of Equilibrium Fractal Wave

The basic building block of the fractal geometry in the financial market looks like the triangle for up market as shown in Figure 2-1. For down market, you can just flip the triangle vertically. One triangle is made when the price makes two price movements. For example, either peak-trough-peak or trough-peak-trough in the price series will make one triangle as shown in Figure 2-2. Since these triangles are propagating to reach the market equilibrium price, we can call these triangles as the equilibrium fractal waves. By definition, the single equilibrium fractal wave is equivalent to a simple triangle made up from two price movements. Since equilibrium fractal wave is a fractal geometry, we only concern its shape regardless of its size. Equilibrium fractal wave can have many different shapes. Since equilibrium fractal wave is made up from two price moves, the one possible way to describe the shape of equilibrium fractal wave is by relating these two price moves. One can take the ratio of current price move to previous price move (Y2/Y1) to describe the shape of the equilibrium fractal wave typically.

The shape ratio of equilibrium fractal wave = current move in price units (Y2)/ previous move in price units (Y1).

Using the shape ratio, we can differentiate a specific shape of equilibrium fractal wave from the other shapes. For example, Figure 2-3 shows two identical equilibrium fractal waves in their shape. Their shape can be considered as identical as their shape ratio is identical. Likewise, if the shape ratios of two equilibrium fractal waves are different, then two equilibrium fractal waves can be considered as being non-identical in their shape (Figure 2-4).
 
Figure 2-1: Structure of one equilibrium fractal wave. It is made up from two price movements (i.e. two swings).

 

Figure 2-2: one unit cycle of an equilibrium fractal wave in the candlestick chart.
 
Figure 2-3: An example of two identical equilibrium fractal waves in their shape.

 
Figure 2-4: An example of non-identical equilibrium fractal waves in their shape.

To make use of equilibrium fractal wave for your trading, you have to understand the characteristics of equilibrium fractal wave. In this book, we outline the five most important characteristics for your trading. When you trade with equilibrium fractal wave or other EFW derived patterns, you will find out that the trading strategies are based on one or few of these characteristics.
 
The first characteristic of equilibrium fractal wave is the repeatability. While the price is moving to its equilibrium price level, we observe the zigzag path of the price movement. After extensive price rise, the price must fall to realize the overvaluation of the price. Likewise, after extensive price fall, the price must rise to realize the undervaluation of the price. This price mechanism builds the complex zigzag path of the price movement in the financial market. During the zigzag path, the price shows the four possible triangle shapes as shown in Figure 2-5. These four equilibrium fractal waves are the mirrored image of each other. Therefore, they are the fractal. The complex price path in the financial market is in fact the combination of these four equilibrium fractal waves in alternation.

 
Figure 2-5: Equilibrium fractal wave in the Zig Zag price path where RFU = Rise Fall UP pattern, RFD = Rise Fall Down pattern, FRD = Fall Rise Down pattern and FRU = Fall Rise Up pattern.
The second characteristic of equilibrium fractal wave is that equilibrium fractal wave can be extended to form another bigger equilibrium fractal wave as shown in Figure 2-6. During the important data release or market news release, the financial market can experience a high volatility or shock. When the market experiences the high volatility or shock, the last leg of equilibrium fractal wave can extends to adapt the shock or volatility introduced in the market. Even after the extension, the equilibrium fractal wave still maintains its fractal geometry, the triangle. Hence, the fractal nature of financial market is unbreakable. This price extension often determines the reversal or breakout movement around the important support and resistance levels.
 
Figure 2-6: Illustration of price transformation (extension) from path 1 to path 2 to meet new equilibrium price due to an abrupt introduction of new equilibrium source in the financial market.

Third characteristic of equilibrium fractal wave is that they can overlap on each other. For example, when the equilibrium fractal wave is propagating, we can observe the jagged patterns repeatedly as shown in Figure 2-7. To untrained eyes, this complex pattern might look like random patterns. They are not random pattern. Later part of the 1st training, we will show you how even untrained individual can readily identify equilibrium fractal waves in your chart with the peak trough analysis.

 
Figure 2-7: Pictorial representation of jagged equilibrium fractal wave with a linear trend.

The fourth characteristic of equilibrium fractal wave is the infinite scales. The infinite scales mean that you will see the similar patterns repeatedly in the price series while their sizes are keep changing. The repeating pattern can come in any size from small to large. For example, if we stack the varying size of equilibrium fractal waves with the particular shape ratio, then literarily we can stack the infinite number of triangle as shown in Figure 2-8. This implies the infinite number of cycle periods in the Price Pattern Table (see appendix). This is exactly why “Equilibrium Fractal-Wave” process is much harder to be handled by traditional technical indicators or mathematical models because they were not designed to deal with the infinite scaling problem mostly.

 
Figure 2-8: Infinite number of stacked triangles.

The fifth characteristic of equilibrium fractal wave is the loose self-similarity (heterogeneity). In nature, it is easy to find the strict self-similarity. However, we can only expect the loose self-similarity in the financial market due to the highly heterogeneous players, participating in the market. Even though all the equilibrium fractal waves will have the triangular form, their shape ratio will be different to each other. For example, if we display the shape ratios of all the series of equilibrium fractal waves in the chart, then we will expect the different shape ratios to its adjacent one (Figure 2-9). This does not mean that we will never have the similar shape ratios in history. In fact, we can get lots of them repeating in the history. For example, we get to see the shape ratio of 0.618 all the time in the financial market. However, we are just saying that the same shape ratios will not come in the successive manner. This heterogeneous characteristic also implies that the financial market have the shapes more frequently occurring than the other shapes. For example, last hundreds years, traders had a solid belief in using the Fibonacci ratios like 0.618, 0.382 and 1.618 for their trading. Some traders used these ratios for Elliott wave analysis or some traders used these ratios for the Fibonacci retracement measurement. Likewise, each financial instrument has different shapes dominating than the rest. Hence, each financial instrument shows more idiosyncratic behaviour of their own.

 
Figure 2-9:  Equilibrium fractal waves with different shape ratios.

To give you some idea of equilibrium fractal wave, let us have some real world example using currency pairs. Regardless of how long the market goes on, the market can be described with few cycles of equilibrium fractal waves due to the fractal nature of the financial market. For example, the financial prices series with 20 years of history can be described using two unit cycles of equilibrium fractal wave (Figure 2-10). Likewise, the price series with 2 weeks historical data can be described using two unit cycles of equilibrium fractal wave too (Figure 2-11). The main difference is that there are more jagged patterns inside the financial price series for 20 years comparing to the two weeks data.

 
Figure 2-10: EURUSD twenty years’ historical data from 1992 to 2016.

 
Figure 2-11: EURUSD two weeks historical Data from 2015 August 28 to 2015 September 16.

Each equilibrium fractal wave can be combined to form the patterns that are more complex. Several popular tradable patterns can be derived by combining several equilibrium fractal waves. For example, Harmonic patterns are typically made up from three equilibrium fractal waves. Impulse Wave 12345 pattern in Elliott Wave Theory is made up from four equilibrium fractal waves. Corrective Wave ABC pattern in Elliott Wave Theory is made up from two equilibrium fractal waves. Like the case of Elliott Wave patterns and Harmonic patterns, some derived patterns can have some definite number for equilibrium fractal wave for the defined patterns. However, there are some derived patterns does not have the definite number of equilibrium fractal wave. For example, rising wedge, falling wedge and triangle patterns does not require the definite number of equilibrium fractal wave. Rising wedge, falling wedge and triangle patterns are envelops connecting highs and lows of each equilibrium fractal wave.

EFW Derived patterns Number of equilibrium fractal waves Number of points
ABCD pattern 2 4
Butterfly pattern 3 5
Bat pattern 3 5
Gartley pattern  3 5
Impulse Wave 12345 4 6
Corrective wave ABC 2 4
Falling wedge pattern Not defined Not defined
Rising wedge pattern Not defined Not defined
Symmetric triangle Not defined Not defined
Ascending triangle Not defined Not defined
Descending triangle Not defined Not defined

Table 2-1: List of derived patterns for trader from equilibrium fractal waves

The properties of these derived patterns remain identical to the equilibrium fractal wave because the derived patterns are also fractals by nature. Therefore, the derived patterns are repeating in different scales. For example, the size of butterfly pattern detected in EURUSD today will be different to the butterfly pattern detected 1 month ago. In addition, the size of butterfly pattern detected in EURUSD will be different to the butterfly pattern detected in GBPUSD. The detected patterns can have slightly different shape too. It is also possible to have nested patterns inside larger patterns. For example, we can have a small bullish butterfly pattern inside the greater bullish butterfly pattern. Likewise, we can have a nested bullish Impulse Wave 12345 pattern inside greater bullish Impulse Wave 12345 pattern. Another important point about these derived patterns is that they will serve for the price to propagate in the direction of the market equilibrium. The formation of the repeating patterns will typically guide the price to the end of the equilibrium price. Some derived patterns like Harmonic Patterns can pick up the trend reversal. Some patterns like impulse wave 1234 can help you to predict trend continuation. Therefore, these derived patterns provide good clue about trading direction for us. Presence of these derived patterns can represent the existence of fifth regularity, equilibrium Fractal-Wave process in the financial price series.
 
Figure 2-12: Butterfly pattern formed in EURUSD H4 timeframe.
 
Figure 2-13: Impulse Wave 12345 pattern formed in EURUSD D1 timeframe.

 
Figure 2-14: Corrective Wave ABC pattern formed in EURUSD D1 timeframe.
 
Figure 2-15: Rising Wedge pattern A (left) and another Rising Wedge pattern B (right) formed in EURUSD H4 timeframe.
3. Hurst Exponent and Equilibrium Fractal Wave Index for Financial Trading
The term fractal was used for the first time by Benoit Mandelbrot (20 November 1924 – 14 October 2010). This is how he defined fractals: “Fractals are objects, whether mathematical, created by nature or by man, that are called irregular, rough, porous or fragmented and which possess these properties at any scale. That is to say they have the same shape, whether seen from close or from far.” This is a general description of the fractals from the father of fractals. At the most plain language, the fractal is the repeating geometry. For example, in Figure 3-1, a triangle is keep repeating to form larger triangles. How big or small we zoom out or zoom in, we can only see the identical triangle everywhere. When the pattern or structure is composed of regular shape as shown in Figure 3-1, we call such a pattern as the strict self-similarity.
 
Figure 3-1: Example fractal geometry with strict self-similarity.

Fractal geometry can be found in nature including trees, leaves, mountain edges, coastline, etc. The financial market has also strong fractal nature in it. Since the price of financial instruments is drawn in time and price space, the fractal in the financial market comes in waveform over the time. However, we are not talking about the typical cyclic wave as in the sine or cosine wave, which can be defined with a definite cycle period. In the financial market, we are talking about the repeating geometry or patterns over the time without definite cycle period. Another important fractal characteristic of the financial market is a loose self-similarity in contrast to the strict self-similarity in Figure 3-1. Loose self-similarity means that the financial market is composed of slightly different variation of the regular shape (Figure 3-2). Therefore, to understand the financial market, we need some tools to visualize its structure. If we understand the fractal nature of the financial market, we can definitely improve our trading performance. Many investment banks and fund management firms do spend considerable amount of efforts and time to reveal the fractal properties of the financial market. They use such a knowledge for their trading and investment decision. From the next chapter, we introduce few important scientific tools to reveal the market structure and behaviour of the financial market for your trading.

 
Figure 3-2: Loose self-similarity of the financial market.

Financial market is one of the most interesting topics in science. The fractal nature of the financial market was studied more than decades in both academic and industrial research. Many investment banks and fund management firms spend a considerable amount of time and efforts to reveal the fractal properties of the financial market so they can use such a knowledge for their trading and investment decision. Since fractal geometry in the financial market is complex, we need scientific tools to study the structure and the behaviour of the financial market. If we understand the structure and the behaviour of the financial market, we can create better trading strategies for sure. In this article, we will help you to understand two important fractal based scientific tools including Hurst Exponent and Equilibrium Fractal wave index. We explain these two tools in a simple language for the example of financial trading.
The name “Hurst exponent” or “Hurst coefficient” was derived from Harold Edwin Hurst (1880-1978), the British hydrologist. Among the scientists, Hurst exponent is typically used to measure the predictability of time series. In fact, Hurst exponent is theoretically tied to the Fractal dimension index coined by Mandelbrot in 1975. Therefore, when we explain Hurst exponent, we can not avoid to mention about the Fractal Dimension index. The relationship between Hurst exponent and Fractal dimension index is like this:
Fractal dimension index (D) = 2 – Hurst exponent (H).
Even if we had a definite mathematical relationship between these two, we should interpret them independently.  For example, Fractal dimension index can range from one to two. This value corresponding to the typical geometric dimension we know. For example, everyone knows that one dimension indicates a straight line whereas the two dimension indicates an area. Three dimension is a volume. Of course, for some big science fiction fans, four dimension might be an interesting topic. Now we know that the fractal dimension index can range from 1 to 2. What does 1.5 dimension means? Fractal dimension index 1.5 is simply the filling capacity of the geometric pattern. If the geometric patterns are highly wiggly and then can fill more space than a straight line, the geometric patterns will have higher fractal dimension index. If the geometric pattern is simple, then the pattern will have lower fractal dimension index close to one (i.e. straight line). For the financial market, the fractal dimension index can range somewhere between 1.36 and 1.52. You can imagine how complex they are. It is important to note that the fractal dimension index is not a unique descriptor of shape. Therefore, the number does not tell how the shape of the fractal geometry.
Hurst exponent can range from 0.0 and 1.0. Unlike the fractal dimension index, Hurst exponent tell us the predictability of the financial market. For example, if the Hurst exponent is close to 0.5, this indicates the financial market is random. If the Hurst exponent is close to 0.0 or 1.0, then it indicates that the financial market is highly predictable. The best-known approach using the Hurst exponent for the financial trading is to classify the financial market data into momentum (i.e. trending) and mean reversion (i.e. ranging) characteristics. For example, if Hurst exponent of the financial market is greater than 0.5, then we can assume that the financial market have a tendency for trending. If Hurst exponent is less than 0.5, we can assume that the financial market have a tendency for ranging. Hurst exponent is generally calculated over the entire data. It is used as a metric to describe the characteristic of the financial market. However, there are some traders using Hurst exponent like a technical indicator by calculating them for short period. When you calculate Hurst exponent over short period, you might run the risk of incorrect range analysis (Figure 3-3). For example, it is well known that with small data set, the estimated standard deviation can be far off from the true standard deviation of the population. However, at the same time, if you are using overly long period to calculate Hurst exponent, you will get the lagging signals (Figure 3-4). If you are using Hurst exponent for reasonably long calculating period, then Hurst exponent will not alternate between trending (> 0.5)and ranging region (<0.5) but the value will stay only one side (Figure 3-5). In Figure 3-5, Hurst exponent stayed over 0.57 always when we have the calculating period 3000 for EURUSD H1 timeframe. It is also important to note that Hurst exponent does not tell you the direction of the market.

 
Figure 3-3: Hurst Exponent indicator with period 30 on EURUSD H1 timeframe. The green dotted line is at 0.5.

 
Figure 3-4: Hurst Exponent indicator with period 100 on EURUSD H1 timeframe. The green dotted line is at 0.5.

 
Figure 3-5: Hurst Exponent indicator with period 3000 on EURUSD H1 timeframe. Hurst exponent value is always greater than 0.57.

The Equilibrium fractal wave index was first introduced in the Book: Financial trading with Five Regularities of Nature: Scientific Guide to Price Action and Pattern Trading (2017). If Hurst exponent was created to extract insight for the overall data of the financial market, the Equilibrium fractal wave index was created to extract insight for the fractal geometry in the loose self-similarity system like the financial market. In the Equilibrium fractal wave index, the building block of the fractal geometry is assumed as the simple triangular waveform called equilibrium fractal wave. Remember that in the strict self-similarity system, the fractal geometry is composed of infinite number of regular shape as in Koch Curve and Sierpinski Triangle as shown in Figure 3-1. In the loose self-similarity structure, the fractal geometry is composed of infinite number of slightly different version of the regular shape. Likewise, many different variation of the triangular shape shown in Figure 3-6 can become the equilibrium fractal wave in the financial market. The variation of shape in the equilibrium fractal wave can be expressed as the Shape ratio of latest price move to previous price move at the two swing points (i.e. the shape ratio = Y2/Y1). Figure 3-7 and 3-8 show the example of identical shape and non-identical shape of equilibrium fractal wave. Since the financial market is the complex system with loose self-similarity, the financial market is composed of infinite number of some identical and some non-identical shape of equilibrium fractal waves as shown in Figure 3-9. The Equilibrium fractal wave index simply tells you how often the identical shape of equilibrium fractal wave is repeating in the financial market. To help you understand further, the mathematical equation for the Equilibrium Fractal Wave index is shown below:

Equilibrium fractal wave index = number of the particular shape of equilibrium fractal wave / number of peaks and troughs in the price series.

 
Figure 3-6: Structure of one equilibrium fractal wave. It is made up from two price movements (i.e. two swings).

 
Figure 3-7: An example of two identical equilibrium fractal waves in their shape.

 
Figure 3-8: An example of non-identical equilibrium fractal waves in their shape.

 
Figure 3-9: Financial market with loose self-similarity. The shape ratio (Y2/Y1) corresponds to each equilibrium fractal wave.

So how to use Equilibrium fractal wave index for financial trading? If the Hurst exponent tells you the predictability of the financial market, then the Equilibrium fractal wave index can reveal the internal structure of the financial market. For example, Table 3-1 shows the internal structure of EURUSD for around 12 years of history data. We can tell how the six different variation of equilibrium fractal waves exist in EURUSD in different proportion. Some variation of equilibrium fractal wave appears more frequently than the other shape ratios. For example, the shape ratio 0.618 (i.e. the golden ratio) and 0.850 appears more frequently than the other shape ratios in EURUSD. The higher the Equilibrium fractal wave index means that the shape ratio indicates reliable trading opportunity whereas the lower the Equilibrium fractal wave means that they are not so significant to trade. With Equilibrium fractal wave index, you can also cross compare the internal structure of different financial instruments. Table 3-2 shows how GBPUSD is composed of these six variation of equilibrium fractal waves. You can tell the composition is not similar to the case of EURUSD (Table 3-1). This simply tells you that each financial instrument have their own behaviour. In addition, with Equilibrium fractal wave index, we can cross-compare the composition for multiple of financial instruments. For example, in Table 3-3, we cross compared the composition of the shape ratio 0.618 for 10 different currency pairs. You can tell that the shape ratio of 0.618 take up the higher proportion in some currency pairs whereas it is not so significant in other currency pairs. For example, the appearance of shape ratio in USDJPY is roughly 25% more than the appearance of the shape ratio in AUDNZD (Table 3-3). This indicates that you will be better off to trade with USDJPY than AUDNZD if your trading strategy involves using the golden ratio 0.618.
Shape Ratio Start End Number of Equilibrium Fractal Wave Number of Peaks and troughs EFW Index
0.618 2006 09 20 2018 01 20 108 321 33.6%
0.382 2006 09 20 2018 01 20 99 321 30.8%
0.500 2006 09 20 2018 01 20 102 321 31.8%
0.300 2006 09 20 2018 01 20 65 321 20.2%
0.450 2006 09 20 2018 01 20 101 321 31.5%
0.850 2006 09 20 2018 01 20 138 321 43.0%
Sum     613 321 190.97%
Average     102.17 321 31.83%
Stdev     23.28 0.00 N/A

Table 3-1: Internal structure of EURUSD D1 timeframe from 2006 09 20 to 2018 01 20 with six different shape ratios.
Shape Ratio Start End Number of Equilibrium Fractal Wave Number of Peaks and troughs EFW Index
0.618 2007 01 04 2018 01 20 116 339 34.2%
0.382 2007 01 04 2018 01 20 95 339 28.0%
0.500 2007 01 04 2018 01 20 124 339 36.6%
0.300 2007 01 04 2018 01 20 62 339 18.3%
0.450 2007 01 04 2018 01 20 114 339 33.6%
0.850 2007 01 04 2018 01 20 147 339 43.4%
Sum     658 321 194.10%
Average     109.67 321 32.35%
Stdev     28.79 0.00 N/A

Table 3-2: Internal structure of GBPUSD D1 timeframe from 2007 01 04 to 2018 01 20 with six different shape ratios.

Instrument Start End Number of Equilibrium Fractal Wave Number of Peaks and troughs EFW Index 0.618
EURUSD 2006 09 20 2018 01 20 108 321 33.6%
GBPUSD 2007 01 04 2018 01 20 116 339 34.2%
USDJPY 2008 04 01 2018 01 20 134 326 41.1%
AUDUSD 2008 03 08 2018 01 20 117 333 35.1%
USDCAD 2008 02 19 2018 01 20 120 328 36.6%
NZDUSD 2007 08 15 2018 01 20 122 330 37.0%
EURGBP 2008 05 01 2018 01 20 130 342 38.0%
AUDNZD 2007 08 03 2018 01 20 107 325 32.9%
AUDCAD 2006 08 26 2018 01 20 137 342 40.1%
AUDJPY 2007 04 17 2018 01 20 121 315 38.4%
Average     121.20 330.10 36.7%
Stdev     9.56 8.54 2.60%

Table 3-3: Counting number of equilibrium fractal wave with the shape ratio 0.618 on D1 timeframe for over 3000 candle bars.

Just like Hurst exponent, you can turn the Equilibrium fractal wave index into the technical indicators too. In this case, you can monitor the EFW index over time to check the dominating shape ratio for the financial instrument. Just like the case of Hurst exponent, if you are using too small calculating period, you have the risk of under or over estimating the index values. Therefore, it is important to use the reasonably long calculation period to avoid the risk of under or over estimating the index values.

 
Figure 3-10: EFW index for EURUSD D1 timeframe from 2006 09 20 to 2018 01 20.

There are many different ways of using Hurst exponent and Equilibrium fractal wave index for the practical trading. In this section, we share one practical tips. In general, Hurst exponent value far away from 0.5 is preferred for your trading because they are more predictable. Based on this knowledge, you can select your best timeframe to trade. For example, in Table 3-4, we can tell that M30 and H4 timeframe is easiest to trade among the six timeframes for EURUSD because they are more predictable than the other timeframes.
  M5 M15 M30 H1 H4 D1
Hurst Exponent 0.553 0.539 0.588 0.58 0.594 0.532

Table 3-4: Hurst exponent for different timeframe for EURUSD.

Likewise, if you are going to trade using the Golden ratio, you can use the Equilibrium fractal wave index to select the best timeframe. For example, in Table 3-5, we can tell that M30 and H1 have more significant EFW index for the shape ratio 0.618. Therefore, it is easier to trade with M30 and H1 using the Golden ratio.
  M5 M15 M30 H1 H4 D1
EFW Index for 0.618 0.284 0.272 0.308 0.300 0.267 0.290

Table 3-5: Equilibrium fractal wave index of the shape ratio 0.618 for different timeframe for EURUSD.

Both Hurst exponent and Equilibrium fractal wave index can be used to select the financial instrument to trade. At the same time, you can use both Hurst exponent and Equilibrium fractal wave index to fine-tune your trading strategy.

4. Shape Ratio Trading and Equilibrium Fractal Wave Channel

4.1 Introduction to EFW Index for trading

By definition, an equilibrium fractal wave is a triangle made up from two price movements in opposite direction. When the price is moving towards the equilibrium price, the equilibrium fractal waves propagate. In the financial market, various shapes of equilibrium fractal wave exist. They are often mixed and jagged to form more complex price patterns. The shape of each equilibrium wave can be described by their shape ratio. This shape ratio can be used to identify the shape of an individual equilibrium fractal wave in the complex price patterns. As you can tell from the equation, the shape ratio of equilibrium fractal wave is independent from their size.
The shape ratio of equilibrium fractal wave = current move in price units (Y2)/ previous move in price units (Y1).

 
Figure 4-1: One unit cycle of Equilibrium Fractal Wave is a triangle made up from two price movements.

 
Figure 4-2: one unit cycle of an equilibrium fractal wave in the candlestick chart.
Two important shape classes for equilibrium fractal wave include Fibonacci based ratios and non-Fibonacci based ratios. Trader can trade both ratios if they wish. However, traders are required to have a knowledge on which shape of equilibrium fractal wave is more suitable for your trading. To find out the suitable EFW shape, you can simply use the “Equilibrium Fractal Wave (EFW) Index” to do a simple exploratory analysis. The EFW index can be calculated using following equation.

Equilibrium fractal wave index = number of the particular shape of equilibrium fractal wave / number of peaks and troughs in the price series.

The very best part of equilibrium fractal wave trading is that it combines both the exploratory analysis and trading in one practice. In the exploratory analysis, you will build your trading logic. In the trading, you will use the logic to build the best outcome for your trading. In the exploratory analysis, you will use the EFW index exclusively. With the EFW index, you can answer the following questions:
• What particular shape of equilibrium fractal wave exists in the price series?
• Which particular shape of equilibrium fractal wave is dominating in the price series?
• How frequently have they occurred in the past?
• Which financial instruments like currency pairs and stocks prices are easier to trade than rest of the market?
• Is the fifth regularity the most dominating characteristics of this financial market?
For example, Figure 4-3 shows the EFW indices for EURUSD daily timeframe for the three ratios including 0.618, 0.500 and 0.382. We have shown the three EFW indices over the time. From the chart, it is possible to figure out that 0.618 is the most dominating ratio for EURUSD followed by the ratio 0.500. Would this tendency hold the same for GBPUSD too? Let us check the Figure 4-4 for this. You can tell that the ratio 0.500 is more frequently occurring than the ratio 0.618. For GBPUSD and EURUSD, the ratio 0.382 is the least occurring shape of the equilibrium fractal wave. By inspecting the EFW indices, we can tell that EURUSD and GBPUSD have a strong presence of equilibrium fractal wave. For this reason, we can use any trading analysis and strategies designed for the fifth regularity. To calculate the EFW index, we typically recommend using as much data as you can. For example, in Figure 4-3 and Figure 4-4, we have used more than 3000 bars (i.e. around 10 years long history) to calculate each EFW index. You might be able to use more data if you wish.
 
Figure 4-3: EFW index for EURUSD D1 timeframe from 2006 09 20 to 2018 01 20.
 
Figure 4-4: EFW index for GBPUSD D1 timeframe from 2007 01 04 to 2018 01 20.

4.2 Trading with the shape ratio of equilibrium fractal wave

The simplest way to trade with a single equilibrium fractal wave is to trade with their shape ratio. The shape ratio is an identifier of the shape of individual equilibrium fractal wave in the financial market. Hence, each equilibrium fractal wave has one corresponding shape ratio. The way the shape ratio trading works is very similar to the Fibonacci retracement trading. Fibonacci retracement trading is a popular trading technique. In the Fibonacci retracement trading, we predict the potential reversal area by projecting 38.2%, 50% or 61.8% the retracement. Anyone understanding this simple Fibonacci retracement trading can readily understand the trading operation with the shape ratio too because they are similar in term of operation. However trading with shape ratio has several distinctive advantages against the Fibonacci retracement trading. Trader must thoroughly understand the difference between shape ratio trading and Fibonacci retracement trading to yield the better performance.

 
Figure 4-5: Fibonacci retracement trading example with ratio 0.618 on EURUSD daily timeframe.

Firstly, in the shape ratio trading, we do not limit our trading opportunity to Fibonacci ratios only. In the Fibonacci retracement trading, traders assume that the Fibonacci ratios like 0.382, 0.500 or 0.618 or some other Fibonacci ratios are only ratios they can trade. In the shape ratio trading, this assumption is not valid any more. Trader can trade with any shape ratios including the Fibonacci ratios and non-Fibonacci ratios. Since the EFW index tells us exactly which shape ratio is dominating in the particular financial market, it is possible we can pick up the shape ratio based on the EFW index. For example, trader can even trade the shape ratio 0.850 or 0.450 if the EFW index indicates the strong presence of the shape ratio 0.850 or 0.450 in the financial market. Of course, the ratio 0.850 and 0.450 are not the Fibonacci ratios. As we have shown in the previous chapter, it is possible to have the higher EFW index with non-Fibonacci ratios. For example, in EURUSD daily timeframe, the ratio 0.850 had much stronger presence than the golden ratio 0.618.

 
Figure 4-6: Shape ratio trading example with ratio 0.850 on EURUSD daily timeframe.

Secondly, in the shape ratio trading, we believe that some ratios will perform better than the other ratios. At the same time, we also believe that the same ratio can perform differently for other financial instrument. This is related to the loose self-similarity (heterogeneity) characteristic of equilibrium fractal waves. For this reason, we do not blindly apply any ratios for our trading even they are Fibonacci ratios or even golden ratios. We can get the guidance for choosing the ratios from the EFW index too. By applying the EFW index, we can get the good ideas on which shape ratios we should avoid and which ratios we should use for the particular financial market.

Thirdly, in the Fibonacci retracement trading, trader assumes that price will reverse at the projected level. In the shape ratio trading, we do not assume that the price will reverse at the projected level, but we are open to both reversal and breakout (expansion) trading. It is related to the extension (transformation) characteristic of equilibrium fractal waves. We have already covered that the last leg of equilibrium fractal wave can be extended to form the bigger equilibrium fractal wave. This extension can happen when new equilibrium source arrived to market including any economic data release or any significant market news release. The extension will never be able to break the fractal nature of the financial market because the extension creates merely another bigger equilibrium fractal wave (i.e. another bigger triangle). For this reason, in the shape ratio trading, we prefer to bet on the size of equilibrium fractal wave rather than assuming the reversal. How to trade is nearly identical to the support and resistance trading. We will take buy or sell action when the price enter the buy and sell trigger level around the projected level.
 
Figure 4-7: Illustration of price transformation (extension) from path 1 to path 2 to meet new equilibrium price due to an abrupt introduction of new equilibrium source in the financial market.
 
Figure 4-8: Shape ratio trading with breakout example on EURUSD H4 timeframe.

4.3 Introduction to Equilibrium Fractal Wave (EFW) Channel

Unlike many other EFW derived patterns including harmonic patterns and Elliott wave patterns, equilibrium fractal wave is relatively easy to use for our trading. In spite of its simplicity, equilibrium fractal wave can provide an extremely useful insight for our trading.  One of the very important usage of equilibrium fractal wave is a channelling technique. The Equilibrium fractal wave channel can be constructed in two steps. In first step, you need to connect the first and third points to draw the base line. Once base line is drawn in your chart, offset the baseline to the middle point of the equilibrium fractal wave to draw the extended line. Since the base line and extended line is parallel to each other, these two lines form a single channel as shown in Figure 4-9.

 
Figure 4-9: Drawing Equilibrium fractal wave channel.

In the previous chapter, we have spotted that channels are merely a pair of support and resistance lines aligned in parallel. In general, there is various way of drawing channels for your trading. Sometimes, you can draw the channel by connecting several peaks and troughs in your chart. The main difference between the typical channels and EFW channel is that EFW channel is drawn using only three points of a triangle whereas the typical channels are drawn with more than three points.

When you want to control the angle of channel, equilibrium fractal wave provide the most efficient way of controlling the angles. For example, sometimes you might prefer to trade with horizontal channel only. Sometimes, you might prefer to trade with a channel with stiff angle. With equilibrium fractal wave, the angle of channel is simply controlled by the shape ratio. The shape ratio close to 1.000 provides near the horizontal channel or a channel with a near flat angle (Figure 4-10). On the other hands, the shape ratio close to 0.000 provides a channel with a stiff angle (Figure 4-11). The shape ratio around 0.500 provides a channel with a moderate angle (Figure 4-12). Especially when you want to build a mechanical rule for your trading, this property of EFW channel becomes useful.
 
 Figure 4-10: Equilibrium fractal wave channel with the shape ratio around 1.000.

 
Figure 4-11: Equilibrium fractal wave channel with the shape ratio around 0.100.
 
Figure 4-12: Equilibrium fractal wave channel with the shape ratio around 0.500.

EFW Channel can be used for many different purposes for our trading. Trader can use channel for the reversal trading. At the same time, trader can use channel for the breakout trading. Trader can use channel for market prediction. For example, an experienced trader can predict the short-term or long-term market direction with a channel or with several channels. Typically, you can detect the four-market states with EFW Channel. Firstly, you can detect the turning point when the market changes from bullish to bearish (Figure 4-13). Likewise, you can detect the turning point when the market changes from bearish to bullish too (Figure 4-14).  At the same time, you can measure the momentum of the current market. For example, when the price moves over the upwards EFW Channel, it indicates the strong bullish momentum in the market (Figure 4-15). Likewise, when the price moves below the downwards EFW Channel, it indicates the strong bearish momentum in the market (Figure 4-16). This logic is very similar to the way Gann’s angle (or Fan) works.

 
Figure 4-13: Detecting the bearish turning point with EFW channel.

 
Figure 4-14: Detecting the bullish turning point with EFW channel.

 
Figure 4-15: Measuring the strong bullish momentum with EFW channel.
 
Figure 4-16: Measuring the strong bearish momentum with EFW channel.

4.4 Practical trading with Equilibrium Fractal Wave (EFW) Channel

Trading with the EFW channel is almost identical to the support and resistance trading. The main trading principle is that we are betting on the potential size of the equilibrium fractal wave. If the equilibrium fractal wave does not extend, the price will make the reversal movement. If the equilibrium fractal wave extends due to any surprise in the market, then the price will likely to show the breakout movement. To catch either reversal or breakout move, we can apply the threshold approach again from the concept of support and resistance trading in the previous chapter as shown in Figure 4-17 and Figure 4-18. Figure 4-17 shows the trading setup for the bearish turning point. Figure 4-18 shows the trading setup for the strong bullish momentum with the upwards EFW Channel. Trader can use the proportional approach to execute buy and sell. Since we are dealing with angle, it is much easier to use the proportional approach. To calculate the trigger level for buy and sell, we can use the same formula as before:
Y Buy = Proportion (%) x Y Height     and
Y Sell = Proportion (%) x Y Height, where Y Height = the height of the channel and Proportion is fraction of the height of the channel expressed in percentage.

Some proportions you can use include 20% and 30% for your trigger level. You can even use greater proportion like 50% if you wish. The upper and lower channel lines can be used as the minimum stop loss level. To avoid the tight stop loss, you should always have the greater stop loss size than the minimum stop loss level. You can set the take profit according to your preferred rewards/risk level. With the EFW channel, it is possible to achieve Reward/Risk ratio greater than 3. We also show some trading examples in Figure 4-19, 4-20, 4-21 and 4-22.

 
Figure 4-17: EFW Upwards Channel trading setup for the bearish turning point.

 
Figure 4-18: EFW Upwards Channel trading setup for strong bullish momentum.

 
Figure 4-19: EFW Upwards channel sell trading setup on EURUSD D1 timeframe.

 
Figure 4-20: EFW Upwards channel buy trading setup on EURUSD H4 timeframe.

 
Figure 4-21: EFW Downwards channel buy trading setup on EURUSD D1 timeframe.
 
Figure 4-22: EFW Downwards channel sell trading setup on EURUSD D1 timeframe.
5. Appendix

 
Figure 5-1: Five Regularities and their sub price patterns with inclining trends. Each pattern can be referenced using their row and column number. For example, exponential trend pattern in the third row and first column can be referenced as Pattern (3, 1) in this table.

 
Figure 5-2: Five Regularities and their sub price patterns with declining trend. Each price pattern can be referenced using their row and column number. For example, exponential trend pattern in the third row and first column can be referenced as Pattern (3, 1) in this table.

 
Figure 5-3: Visualizing number of cycle periods for the five regularities. Please note that this is only the conceptual demonstration and the number of cycles for second, third and fourth regularity can vary for different price series.
 
Figure 5-4: Five Regularities and their sub price patterns.
 
Figure 5-5: Trading strategies, indicators and charting techniques to deal with the fifth regularity.

 

Understanding Hurst Exponent and Equilibrium Fractal Wave Index for Financial Trading

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Understanding Hurst Exponent and Equilibrium Fractal Wave Index for Financial Trading
www.algotrading-investment.com

11 Feb 2018

Written By Young Ho Seo
Finance Engineer and Quantitative Trader
Overview
Financial market is one of the most interesting topics in science. The fractal nature of the financial market was studied more than decades in both academic and industrial research. Many investment banks and fund management firms spend a considerable amount of time and efforts to reveal the fractal properties of the financial market so they can use such a knowledge for their trading and investment decision. Since fractal geometry in the financial market is complex, we need scientific tools to study the structure and the behaviour of the financial market. If we understand the structure and the behaviour of the financial market, we can create better trading strategies for sure. In this article, we will help you to understand two important fractal based scientific tools including Hurst Exponent and Equilibrium Fractal wave index. We explain these two tools in a simple language for the example of financial trading.

1. Fractal Nature in the Financial Market
The term fractal was used for the first time by Benoit Mandelbrot (20 November 1924 – 14 October 2010). This is how he defined fractals: “Fractals are objects, whether mathematical, created by nature or by man, that are called irregular, rough, porous or fragmented and which possess these properties at any scale. That is to say they have the same shape, whether seen from close or from far.” This is a general description of the fractals from the father of fractals. At the most plain language, the fractal is the repeating geometry. For example, in Figure 1-1, a triangle is keep repeating to form larger triangles. How big or small we zoom out or zoom in, we can only see the identical triangle everywhere. When the pattern or structure is composed of regular shape as shown in Figure 1-1, we call such a pattern as the strict self-similarity.

 
Figure 1-1: Example fractal geometry with strict self-similarity.

Fractal geometry can be found in nature including trees, leaves, mountain edges, coastline, etc. The financial market has also strong fractal nature in it. Since the price of financial instruments is drawn in time and price space, the fractal in the financial market comes in waveform over the time. However, we are not talking about the typical cyclic wave as in the sine or cosine wave, which can be defined with a definite cycle period. In the financial market, we are talking about the repeating geometry or patterns over the time without definite cycle period. Another important fractal characteristic of the financial market is a loose self-similarity in contrast to the strict self-similarity in Figure 1-1. Loose self-similarity means that the financial market is composed of slightly different variation of the regular shape (Figure 1-2). Therefore, to understand the financial market, we need some tools to visualize its structure. If we understand the fractal nature of the financial market, we can definitely improve our trading performance. Many investment banks and fund management firms do spend considerable amount of efforts and time to reveal the fractal properties of the financial market. They use such a knowledge for their trading and investment decision. From the next chapter, we introduce few important scientific tools to reveal the market structure and behaviour of the financial market for your trading.

 
Figure 1-2: Loose self-similarity of the financial market.

2. Hurst Exponent for Financial Trading
The name “Hurst exponent” or “Hurst coefficient” was derived from Harold Edwin Hurst (1880-1978), the British hydrologist. Among the scientists, Hurst exponent is typically used to measure the predictability of time series. In fact, Hurst exponent is theoretically tied to the Fractal dimension index coined by Mandelbrot in 1975. Therefore, when we explain Hurst exponent, we can not avoid to mention about the Fractal Dimension index. The relationship between Hurst exponent and Fractal dimension index is like this:
Fractal dimension index (D) = 2 – Hurst exponent (H).
Even if we had a definite mathematical relationship between these two, we should interpret them independently.  For example, Fractal dimension index can range from one to two. This value corresponding to the typical geometric dimension we know. For example, everyone knows that one dimension indicates a straight line whereas the two dimension indicates an area. Three dimension is a volume. Of course, for some big science fiction fans, four dimension might be an interesting topic. Now we know that the fractal dimension index can range from 1 to 2. What does 1.5 dimension means? Fractal dimension index 1.5 is simply the filling capacity of the geometric pattern. If the geometric patterns are highly wiggly and then can fill more space than a straight line, the geometric patterns will have higher fractal dimension index. If the geometric pattern is simple, then the pattern will have lower fractal dimension index close to one (i.e. straight line). For the financial market, the fractal dimension index can range somewhere between 1.36 and 1.52. You can imagine how complex they are. It is important to note that the fractal dimension index is not a unique descriptor of shape. Therefore, the number does not tell how the shape of the fractal geometry.
Hurst exponent can range from 0.0 and 1.0. Unlike the fractal dimension index, Hurst exponent tell us the predictability of the financial market. For example, if the Hurst exponent is close to 0.5, this indicates the financial market is random. If the Hurst exponent is close to 0.0 or 1.0, then it indicates that the financial market is highly predictable. The best-known approach using the Hurst exponent for the financial trading is to classify the financial market data into momentum (i.e. trending) and mean reversion (i.e. ranging) characteristics. For example, if Hurst exponent of the financial market is greater than 0.5, then we can assume that the financial market have a tendency for trending. If Hurst exponent is less than 0.5, we can assume that the financial market have a tendency for ranging. Hurst exponent is generally calculated over the entire data. It is used as a metric to describe the characteristic of the financial market. However, there are some traders using Hurst exponent like a technical indicator by calculating them for short period. When you calculate Hurst exponent over short period, you might run the risk of incorrect range analysis (Figure 2-1). For example, it is well known that with small data set, the estimated standard deviation can be far off from the true standard deviation of the population. However, at the same time, if you are using overly long period to calculate Hurst exponent, you will get the lagging signals (Figure 2-2). If you are using Hurst exponent for reasonably long calculating period, then Hurst exponent will not alternate between trending (> 0.5)and ranging region (<0.5) but the value will stay only one side (Figure 2-3). In Figure 3, Hurst exponent stayed over 0.57 always when we have the calculating period 3000 for EURUSD H1 timeframe. It is also important to note that Hurst exponent does not tell you the direction of the market.

 
Figure 2-1: Hurst Exponent indicator with period 30 on EURUSD H1 timeframe. The green dotted line is at 0.5.

 
Figure 2-2: Hurst Exponent indicator with period 100 on EURUSD H1 timeframe. The green dotted line is at 0.5.

 
Figure 2-3: Hurst Exponent indicator with period 3000 on EURUSD H1 timeframe. Hurst exponent value is always greater than 0.57.
3. Equilibrium Fractal Wave Index for Financial Trading

The Equilibrium fractal wave index was first introduced in the Book: Financial trading with Five Regularities of Nature: Scientific Guide to Price Action and Pattern Trading (2017). If Hurst exponent was created to extract insight for the overall data of the financial market, the Equilibrium fractal wave index was created to extract insight for the fractal geometry in the loose self-similarity system like the financial market. In the Equilibrium fractal wave index, the building block of the fractal geometry is assumed as the simple triangular waveform called equilibrium fractal wave. Remember that in the strict self-similarity system, the fractal geometry is composed of infinite number of regular shape as in Koch Curve and Sierpinski Triangle (Figure 1-1). In the loose self-similarity structure, the fractal geometry is composed of infinite number of slightly different version of the regular shape. Likewise, many different variation of the triangular shape shown in in Figure 3-1 can become the equilibrium fractal wave in the financial market. The variation of shape in the equilibrium fractal wave can be expressed as the Shape ratio of latest price move to previous price move at the two swing points (i.e. the shape ratio = Y2/Y1). Figure 3-2 and 3-3 show the example of identical shape and non-identical shape of equilibrium fractal wave. Since the financial market is the complex system with loose self-similarity, the financial market is composed of infinite number of some identical and some non-identical shape of equilibrium fractal waves as shown in Figure 3-4. The Equilibrium fractal wave index simply tells you how often the identical shape of equilibrium fractal wave is repeating in the financial market. To help you understand further, the mathematical equation for the Equilibrium Fractal Wave index is shown below:

Equilibrium fractal wave index = number of the particular shape of equilibrium fractal wave / number of peaks and troughs in the price series.

 
Figure 3-1: Structure of one equilibrium fractal wave. It is made up from two price movements (i.e. two swings).

 
Figure 3-2: An example of two identical equilibrium fractal waves in their shape.

 
Figure 3-3: An example of non-identical equilibrium fractal waves in their shape.

 
Figure 3-4: Financial market with loose self-similarity. The shape ratio (Y2/Y1) corresponds to each equilibrium fractal wave.
So how to use Equilibrium fractal wave index for financial trading? If the Hurst exponent tells you the predictability of the financial market, then the Equilibrium fractal wave index can reveal the internal structure of the financial market. For example, Table 3-1 shows the international structure of EURUSD for around 12 years of history data. We can tell how the six different variation of equilibrium fractal waves exist in EURUSD in different proportion. Some variation of equilibrium fractal wave appears more frequently than the other shape ratios. For example, the shape ratio 0.618 (i.e. the golden ratio) and 0.850 appears more frequently than the other shape ratios in EURUSD. The higher the Equilibrium fractal wave index means that the shape ratio indicates reliable trading opportunity whereas the lower the Equilibrium fractal wave means that they are not so significant to trade. With Equilibrium fractal wave index, you can also cross compare the internal structure of different financial instruments. Table 3-2 shows how GBPUSD is composed of these six variation of equilibrium fractal waves. You can tell the composition is not similar to the case of EURUSD (Table 3-1). This simply tells you that each financial instrument have their own behaviour. In addition, with Equilibrium fractal wave index, we can cross-compare the composition for multiple of financial instruments. For example, in Table 3-3, we cross compared the composition of the shape ratio 0.618 for 10 different currency pairs. You can tell that the shape ratio of 0.618 take up the higher proportion in some currency pairs whereas it is not so significant in other currency pairs. For example, the appearance of shape ratio in USDJPY is roughly 25% more than the appearance of the shape ratio in AUDNZD (Table 3-3). This indicates that you will be better off to trade with USDJPY than AUDNZD if your trading strategy involves using the golden ratio 0.618.
Shape Ratio Start End Number of Equilibrium Fractal Wave Number of Peaks and troughs EFW Index
0.618 2006 09 20 2018 01 20 108 321 33.6%
0.382 2006 09 20 2018 01 20 99 321 30.8%
0.500 2006 09 20 2018 01 20 102 321 31.8%
0.300 2006 09 20 2018 01 20 65 321 20.2%
0.450 2006 09 20 2018 01 20 101 321 31.5%
0.850 2006 09 20 2018 01 20 138 321 43.0%
Sum     613 321 190.97%
Average     102.17 321 31.83%
Stdev     23.28 0.00 N/A

Table 3-1: Internal structure of EURUSD D1 timeframe from 2006 09 20 to 2018 01 20 with six different shape ratios.
Shape Ratio Start End Number of Equilibrium Fractal Wave Number of Peaks and troughs EFW Index
0.618 2007 01 04 2018 01 20 116 339 34.2%
0.382 2007 01 04 2018 01 20 95 339 28.0%
0.500 2007 01 04 2018 01 20 124 339 36.6%
0.300 2007 01 04 2018 01 20 62 339 18.3%
0.450 2007 01 04 2018 01 20 114 339 33.6%
0.850 2007 01 04 2018 01 20 147 339 43.4%
Sum     658 321 194.10%
Average     109.67 321 32.35%
Stdev     28.79 0.00 N/A

Table 3-2: Internal structure of GBPUSD D1 timeframe from 2007 01 04 to 2018 01 20 with six different shape ratios.

Instrument Start End Number of Equilibrium Fractal Wave Number of Peaks and troughs EFW Index 0.618
EURUSD 2006 09 20 2018 01 20 108 321 33.6%
GBPUSD 2007 01 04 2018 01 20 116 339 34.2%
USDJPY 2008 04 01 2018 01 20 134 326 41.1%
AUDUSD 2008 03 08 2018 01 20 117 333 35.1%
USDCAD 2008 02 19 2018 01 20 120 328 36.6%
NZDUSD 2007 08 15 2018 01 20 122 330 37.0%
EURGBP 2008 05 01 2018 01 20 130 342 38.0%
AUDNZD 2007 08 03 2018 01 20 107 325 32.9%
AUDCAD 2006 08 26 2018 01 20 137 342 40.1%
AUDJPY 2007 04 17 2018 01 20 121 315 38.4%
Average     121.20 330.10 36.7%
Stdev     9.56 8.54 2.60%

Table 3-3: Counting number of equilibrium fractal wave with the shape ratio 0.618 on D1 timeframe for over 3000 candle bars.
Just like Hurst exponent, you can turn the Equilibrium fractal wave index into the technical indicators too. In this case, you can monitor the EFW index over time to check the dominating shape ratio for the financial instrument. Just like the case of Hurst exponent, if you are using too small calculating period, you have the risk of under or over estimating the index values. Therefore, it is important to use the reasonably long calculation period to avoid the risk of under or over estimating the index values.

 
Figure 3-5: EFW index for EURUSD D1 timeframe from 2006 09 20 to 2018 01 20.

4. Practical trading tips with Hurst exponent and Equilibrium fractal wave index
There are many different ways of using Hurst exponent and Equilibrium fractal wave index for the practical trading. In this section, we share one practical tips. In general, Hurst exponent value far away from 0.5 is preferred for your trading because they are more predictable. Based on this knowledge, you can select your best timeframe to trade. For example, in Table 4-1, we can tell that M30 and H4 timeframe is easiest to trade among the six timeframes for EURUSD because they are more predictable than the other timeframes.
  M5 M15 M30 H1 H4 D1
Hurst Exponent 0.553 0.539 0.588 0.58 0.594 0.532

Table 4-1: Hurst exponent for different timeframe for EURUSD.

Likewise, if you are going to trade using the Golden ratio, you can use the Equilibrium fractal wave index to select the best timeframe. For example, in Table 4-2, we can tell that M30 and H1 have more significant EFW index for the shape ratio 0.618. Therefore, it is easier to trade with M30 and H1 using the Golden ratio.
  M5 M15 M30 H1 H4 D1
EFW Index for 0.618 0.284 0.272 0.308 0.300 0.267 0.290

Table 4-2: Equilibrium fractal wave index of the shape ratio 0.618 for different timeframe for EURUSD.

Both Hurst exponent and Equilibrium fractal wave index can be used to select the financial instrument to trade. At the same time, you can use both Hurst exponent and Equilibrium fractal wave index to fine-tune your trading strategy.

5. Conclusion
In this article, we have briefly covered the loose self-similarity of the financial market. Hurst exponent can be used to measure the predictability of the financial market. At the same time, Hurst exponent can be used to classify the financial market as either trending or ranging market. With Equilibrium fractal wave index, we can reveal the internal structure of the financial market. With Equilibrium fractal wave index, we can cross compare the internal structure for the different financial instruments. Both Hurst exponent and Equilibrium fractal wave index can be used to select the best timeframe and the financial instrument for your trading. At the same time, you can use these two tools to fine-tuning your trading strategy.

Best Fibonacci Ratio and Shape Ratio for Winning Technical Analysis with 100 Years of Belief

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Best Fibonacci Ratio and Shape Ratio for Winning Technical Analysis with 100 Years of Belief
www.algotrading-investment.com

10 Feb 2018

Written By Young Ho Seo
Finance Engineer and Quantitative Trader

The Fibonacci Ratio is used by millions of forex and stock market traders every day. It is a mega popular tool in the trading world. If you do not know what the Fibonacci ratio is, here is the simple explanation. Fibonacci ratio is the ratio between two adjacent Fibonacci numbers. To have a feel about the Fibonacci ratios, here is the 21 Fibonacci numbers derived from the relationship: Fn = Fn-1 + Fn-2.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, …………………
Once the Fibonacci numbers are reasonably large, you can just pick up any two adjacent Fibonacci numbers above to derive the ratio. For example, we will find that 4181/6765 = 0.618 and 1597/2584 = 0.618. Here 0.618 is called as the golden ratio. The golden ratio is one of the most important Fibonacci ratios. The rest of Fibonacci ratios are derived by using simple mathematical relationship like inverse or square root or etc. Table below shows the list of Fibonacci ratios you can derive from the Golden ratio 0.618.

Type Ratio Calculation
Primary 0.618 Fn-1/Fn of Fibonacci numbers
Primary 1.618 Fn/Fn-1 of Fibonacci numbers
Primary 0.786 0.786=√0.618
Primary 1.272 1.272=√1.618
Secondary 0.382 0.382=0.618*0.618
Secondary 2.618 2.618=1.618*1.618
Secondary 4.236 4.236=1.618*1.618*1.618
Secondary 6.854 6.854=1.618*1.618*1.618*1.618
Secondary 11.089 11.089=1.618*1.618*1.618*1.618*1.618
Secondary 0.500 0.500=1.000/2.000
Secondary 1.000 Unity
Secondary 2.000 Fibonacci Prime Number
Secondary 3.000 Fibonacci Prime Number
Secondary 5.000 Fibonacci Prime Number
Secondary 13.000 Fibonacci Prime Number
Secondary 1.414 1.414=√2.000
Secondary 1.732 1.732=√3.000
Secondary 2.236 2.236=√5.000
Secondary 3.610 3.610=√13.000
Secondary 3.142 3.142 = Pi = circumference /diameter of the circle

Figure 1: Fibonacci ratios and corresponding calculations to derive each ratio.
Then how do we use the Fibonacci ratio for our trading? Well, the common approach is to take the two price movements at each swing point and then just divide the latest price move  Y2 by the previous price move Y1 (The ratio = Y2/Y1) as shown in Figure 2. As shown in Figure 3 and Figure 4, many swing traders uses this Fibonacci retracement to pick up the potential reversal point for their trading. You can also use it for breakout trading too (i.e. Fibonacci expansion). Up to this point, I guess everyone is happy. Now the real question is “Why do the Fibonacci ratios work or not work for our trading?”. Does anyone have an answer to this question? You will find that the Fibonacci ratios are without doubt the popular topics in many major trading websites including www.investopedia.com or www.stockcharts.com. Even after reading dozens of articles about Fibonacci ratio, it is not easy to spot any rational behind the method. The best I can find is the reference to the Elliott Wave developed by Ralph Nelson Elliott in 1938. Then it became popular among trader. Being popular might be good rational. However, can we actually prove it scientifically? Have you asked these two questions:
Question 1: Are all the Fibonacci ratios equally effective for our trading?
Question 2: Can we use some other ratios rather than the Fibonacci ratios for our trading?
Whether you have asked these two questions or not, we will try to answer to these two question in this article because it would be helpful for our trading.
 
Figure 2: Basics of Fibonacci ratio measurement (or Shape ratio measurement).

 
Figure 3: Fibonacci Retracement drawn over daily EURUSD candlestick chart for bearish setup.

 Figure 4: Fibonacci Retracement drawn over daily EURUSD candlestick chart for bullish setup.

Now, let us use the term Shape ratio to describe the ratio (Y2/Y1) in Figure 2 because we can have the ratios other than the Fibonacci ratios like 0.222 or 0.888, etc. The Shape ratio (Y2/Y1 in Figure 2) can be any ratios including both non-Fibonacci ratios and Fibonacci ratios. To measure the usefulness of each shape ratio, we can actually devise one simple index using the following equation:
Index = number of a particular Shape ratio (Y2/Y1) / number of swing highs and swing lows in the price series.
We are counting number of a particular shape ratio in regards to the potential swing highs and swing lows in the price series. For example, if we have 50 times 0.618 ratio among 200 swing highs and swing lows in EURUSD daily timeframe, then the index will be 0.25 (or 25%). If the shape ratio is not significant, then we will have a poor index value. If the index is small, then it means that the ratio will not provide us a good trading opportunity. If the shape ratio is significant then we will have a strong index value. This means that the shape ratio will provide us good trading opportunities.
Have you noticed that the index equation above is in fact quite similar to something? Yes, the index with above formula is identical to the “Equilibrium Fractal Wave (EFW) index” as described in the book “Financial Trading with Five Regularities of Nature: Scientific guide to Price Action and Pattern Trading”. The original equation looks like this in the book:

Equilibrium Fractal Wave (EFW) Index = number of the particular shape of equilibrium fractal wave (the shape ratio = Y2/Y1) / number of peaks and troughs in the price series.

However, the name does not really matter. Two equations are the same. The valid theory or concept can be valid from many different angles. Anyway, what is important is that the EFW index will describe the significance of each shape ratio for your trading whether they are Fibonacci ratio or non-Fibonacci ratios.

Now, let us test the EFW index to answer the two questions using EURUSD Daily timeframe. In this testing, I am using around 10 Years history data for EURUSD in daily timeframe around 2200 candle bars. For this task, we will be selecting three non-Fibonacci ratios and three Fibonacci ratios for comparison. Therefore, the shape ratios in our testing include the typical Fibonacci ratios like 0.382, 0.500 and 0.618. The non-Fibonacci ratios include 0.250, 0.570 and 0.680. Please note that these ratios 0.250, 0.570 and 0.680 are not Fibonacci ratios. Considering that the strong belief on the Fibonacci ratio was held around 100 years in the trading world, our result is very interesting. In general, the Golden ratio 0.618 has the EFW index of 0.274 (or 27.4%). This means that we have nearly 2.7 trading opportunity with the Golden ratio for every 10 peaks and troughs in our chart. Golden ratio looks significant as well as the other two Fibonacci ratios. The Shape ratio 0.382 is least significant among the three Fibonacci ratios only yielding the EFW index 0.261 (or 26.1%). Now let us have a look at the non-Fibonacci ratios. The shape ratio 0.250 has only the EFW index 0.135 (13.5%). This is insignificant. However, the shape ratio 0.570 and 0.680 respectively scored the EFW index 0.278 and 0.291. In fact, both the shape ratio 0.570 and 0.680 have higher the EFW index than the Golden ratio 0.618. This is an interesting observation. This means that we can make slightly better edge using the Shape ratio 0.680 than the Golden ratio 0.618. Now we can answer to the two question above.

Question 1: Are all the Fibonacci ratios equally effective for our trading?
Answer 1: No, different Fibonacci ratio will perform differently for our trading. In general, using the Fibonacci ratio is not a bad choice. However, some Fibonacci ratio can perform better than the other Fibonacci ratio.

Question 2: Can we use some other ratios rather than the Fibonacci ratios for our trading?
Answer 2: Yes, you can search better opportunity with other ratios comparing to the Fibonacci ratios.

From my experience, the results are very specific to the financial instruments. It means that you cannot assume that every financial instrument will behave the same. Therefore, if possible, you should conduct the similar experiments for the financial instruments you want to trade. When you have the results, you can fine tuning your trading strategy with the EFW index. Of course, you can gain the better profit. In the world of trading, the scientific mind set can help you to win. If you want to learn the Price Action and Pattern Trading in the scientific way, we do really recommend reading the book “Financial Trading with Five Regularities of Nature: Scientific guide to Price Action and Pattern Trading”. The book will provide new breakthrough in the trading science. You will get to learn one unified trading framework and practical trading guide never told in other books.
Shape Ratio Type EFW Index Number of identical Shape ratio Total Peaks and Troughs
0.382 Fibonacci ratio 0.261 60 230
0.500 Fibonacci ratio 0.274 63 230
0.618 Fibonacci ratio 0.274 63 230
0.250 Non Fibonacci ratio 0.135 31 230
0.570 Non Fibonacci ratio 0.278 64 230
0.680 Non Fibonacci ratio 0.291 67 230
 Sum 1.513 348 
 Average 0.252 58 230
 Standard deviation 0.058 13.42 0

Figure 5: EFW Index for six shape ratios including three Fibonacci ratios and three non-Fibonacci ratios.