## Golden Ratio and Financial Trading

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Golden Ratio and Financial Trading

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

Introduction to Golden Ratio 0.618 for financial trading
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 (http://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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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

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

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.