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On the world's financial markets and stock exchanges. Part 9

1. Fibonacci sequence

1.1 History of the discovery

In the 13th century, Thomas Aquinas formulated one of the basic principles of aesthetics - that objects with correct proportions are pleasing to the human senses. Pleasant - i.e. understandable or correct. 
Thomas Aquinas referred to a direct link between beauty and mathematics, which can often be "measured" and found in nature. Human instincts have a positive response to regular geometric shapes, both in the natural environment and in man-made objects, such as works of art. He was referring to the principle that Fibonacci had discovered. 
In early 1175, Leonardo Fibonacci of Pisa, Italy, published his famous "Liber Abacci" (Abacus or Book of Calculus; abacus means account), which introduced to Europe one of the greatest discoveries of all time, namely the decimal numeral system, which included the position of zero as the first digit in a number series of entries. This system, which included the familiar symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, became known as the Hindu-Arabic system and is now in common use. It would almost be an understatement to say that Leonardo Fibonacci was the greatest mathematician of the Middle Ages. In all, he wrote three significant mathematical works: The Book of Abacus, published in 1202, Practical Geometry, published in 1220, and The Book of Squares. Although he was the greatest mathematician of the Middle Ages, Fibonacci's only monuments are the statue opposite the Tower of Pisa across the Arno River and the two streets that bear his name, one in Pisa and the other in Florence. It seems strange that so few visitors to the 179-foot Falling Tower have ever heard of Fibonacci or seen his statue. Fibonacci was a contemporary of Bonannes, the architect of the Tower of Pisa, whose construction began in 1174. Both made contributions to world history, but one, whose contribution far surpasses the other, is almost unknown. 
In the Abacus Book, one of the problems posed gives rise to the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on to infinity, known today as the Fibonacci sequence. 
The formula inherent in this sequence is the sum of any numbers placed next to each other in the sequence gives the next number of the sequence, namely 1+1=2, 1+2=3, 2+3=5, 3+5=8 and so on to infinity. 
Fibonacci sequence formula: 
F 1=1, F 2=1, Fn +2= Fn + Fn +1,
where n = 1, 2, 3, 4... 

Fn = (((1+√5)/2)ⁿ - ((1-√5)/2)ⁿ)/√5 Binet formula 


1.2 Basic concepts and definitions

After the first few numbers in a sequence, the ratio of any number to the next oldest number is about 0.618 to 1, and to the next youngest number is about 1.618 to 1. The further along the sequence, the closer the ratio approaches the number φ "phi", which is an irrational number of 0.618034... The ratio between numbers one through one in the sequence is approximately 0.382, which is the inverse of 2.618 (1:2.618). 
Thus these ratios asymptotically (slower and slower) tend towards some constant ratio, but it is irrational, i.e. with an infinite sequence of digits in the fractional part. 
1:1 = 1.00 - which is less than φ by 0.618;
2:1 = 2.0 - which is greater than φ by 0.382;
3:2 = 1.5 - which is greater than φ by 0.1180;
5:3 = 1.6667 - which is greater than φ by 0.0486;
8:5 = 1.6 - which is less than φ by 0.0180.

Phi is the only number which, after adding it to 1, gives its own inversion: 0.618+1=1:0.618. This alliance of additive and multiplicative properties gives rise to the following sequence of equalities:
0,6182 = 1 - 0,618;
0,6183 = 0,618 - 0,6182;
0,6184 = 0,6182 - 0,6183;
0.6185 = 0.6183 - 0.6184, etc. 

or, alternatively:
1,6182 = 1 + 1,618;
1,6183 = 1,618 + 1,6182;
1,6184 = 1,6182 + 1,6183;
1.6185 = 1.6183 + 1.6184, etc. 

Some formulations from the interrelated properties of these four ratios can be represented as follows:
1,618 - 0,618 = 1;
1,618 * 0,618 = 1;
1 - 0,618 = 0,382;
0,618 * 0,618 = 0,382;
2,618 - 1,618 = 1;
2,618 * 0,382 = 1;
2,618 * 0,618 = 1,618;
1,618 * 1,618 = 2,618.

Besides 1 and 2, any Fibonacci number multiplied by 4 and added to some selected Fibonacci number gives another Fibonacci number:
3 * 4 = 12; + 1 = 13;
5 * 4 = 20; + 1 = 21;
8 * 4 = 32; + 2 = 34;
13 * 4 = 52; + 3 = 55;
21 * 4 = 84; + 5 = 89, etc.

The square of any Fibonacci number is equal to the number in front of it multiplied by the number after + or - 1:
5 * 5 = (3 * 8) + 1;
8 * 8 = (5 * 13) - 1;
13 * 13 = (8 * 21) + 1.

The plus and minus alternate constantly. This is another manifestation of an integral part of Elliott wave theory - the rule of alternation.

1.3 Psychological rationale for understanding the Golden Ratio

1.618 (or 0.618) is known as the Golden Ratio or Golden Ratio - widely used in geometry, mathematics and architecture in ancient Greece. Its harmony is pleasing to the eye and is an important phenomenon in music, art, architecture and biology (Luca Pacioli - the divine proportion)
Any segment can be divided so that the ratio between its smaller and larger parts equals the ratio between the larger part and the whole segment (Figure 3-3). This ratio is always 0.618. 

Figure 2.

AB = 1
1/φ = φ/(1- φ)
Φ * φ + φ = 0
Φ = (1 +√5)/2 =1,618034...

Man subconsciously seeks divine proportion - it is needed to satisfy his need for comfort (see Thomas Aquinas one of the basic principles of aesthetics - the feelings of man are pleasant to objects having correct proportions).
Pleasant means intelligible.
See chart of the market.

1.4 Practical application

The golden ratio is ubiquitous in nature. Indeed, the human body is an embodiment of the Golden Ratio in everything from external dimensions to facial structure. Man is divided in the belt by the Fibonacci ratio. The average value is approximately 0.618. This ratio remains true separately for men and separately for women, a perfect sign of creation "in the likeness". Is everything in the development of mankind also a creation "in the likeness"?
The absurd Fibonacci rabbits pop up in the most unexpected places. These numbers are undoubtedly part of a mystical natural harmony, which is pleasant to touch, pleasant to look at and even pleasant to sound at. Music, for example, is based on the 8-note octave. On the piano, this is represented by 8 white keys and 5 black keys - a total of 13. It is no coincidence that the musical harmony that seems to bring the ear the greatest pleasure is the major sixth sound. The note E (E) sounds like a ratio of 0.625 to the note C (C). Only 0.006966 more than the exact Golden Ratio, the ratios of the major six-voice cause a pleasant oscillation in the cochlea of the inner ear, an organ which is just shaped like a logarithmic spiral.
The continuous finding of Fibonacci numbers and the golden spiral in nature precisely explains why the proportion 0.618034 to 1 is so attractive in art. Man sees a representation of life in art which is based on the golden ratio.

Nature uses the Golden Ratio in its most intimate building blocks and in its most advanced patterns, from such small forms as atomic structures, brain microcapillaries and DNA molecules to such huge ones as planetary orbits and galaxies. It concerns phenomena as diverse as the arrangement of quasicrystals, planetary distances and orbital periods, the reflection of light rays off glass, the brain and nervous system, musical arrangements and the structure of plants and animals. Science is quick to prove, there really is a basic law of proportion in nature. By the way, you hold an object with two of your five extensions (two arms, two legs and the head), which have three articulated parts (shoulder, forearm and hand), five extensions at the ends (fingers) with three articulated parts (phalanges) - the wave sequence 5-3-5-3.
Pyramids in Mexico:
1 tier 16 steps;
2nd tier 42 steps;
3rd tier 68 steps;
Why such a number of steps at different levels?
16 * 1.618 = 26; 16 + 26 = 42 steps;
26 * 1, 618 = 42; 42 + 26 = 68 steps.
The earliest adherents were the architects of the Egyptian pyramids near the city of Giza, who encoded phi knowledge into their designs about 5000 years ago. The Egyptian constructors deliberately implemented the Golden Ratio into the Great Pyramid, giving its facade an inclined height of 1.618 times half its base so that the vertical height of the pyramid at the same time was the root of the square of the length of half the base multiplied by 1.618.
The Egyptians of the time of the pharaohs, considered phi not as a number, but as a symbol of the creative function or reproduction in an infinite sequence. To them it symbolised "the fire of life, the male seed, the logos - to which the Gospel of St John refers". Logos, a Greek word, was variously defined by Heraclitus and later pagan, Jewish and Christian philosophers to denote the rational order of the universe, the inherent law of nature, the life-giving force hidden in action, the creative power of the universe to govern and satiate the world.
As you read this difficult description, bear in mind that those people could not see clearly all that they felt. They had no charts and the Law of Waves to make a clear model of the development of nature, and they did what they could to describe those organisational principles shaping the natural world that they discerned. If those ancient philosophers were right that a universal creative force governs and pervades the universe, then why should it not govern and saturate the human world? If forms throughout the universe, including the human body, brain and DNA reflect phi forms, can human activity reflect it in the same way? If phi is the life force in the universe, can it be the impetus behind the development of human productive activity? If phi is a creative function, can it drive human creative activity? If human development is based on production and reproduction in an 'infinite sequence', is it not reasonable that such a process has a spiralling phi form and that this form is discernible in the movement of the aggregate assessment of its productive capacity, i.e. the financial markets? Just as the dedicated Egyptians studied the hidden truths of construction and growth in the universe behind visible randomness and chaos (something that was finally rediscovered by modern "chaos theory" in the 1980s), so too can financial markets, in our view, be properly interpreted by considering its essence rather than what it appears to be upon superficial examination. Financial markets are not a random shapeless mess reacting to current events, but a remarkably accurate record of the rigorous structure of human development.
How do we use φ? In what units do we measure? Points, currency - no, in %!
An important mathematical point that confirms the philosophical point is that it is a fractional, irrational number and it will never be a whole. A perfect proportion can be aspired to, but it does not exist in the world, nor does a perfect price. Everything created by nature or God is created according to that proportion, from quanta to the universe. The market is part of our lives and we walk in those proportions - it is not chaos!

 

Corrections sometimes roll back by the Fibonacci percentage of the previous wave. As shown in Figure 4-1. 4-1, a sharp correction tends to roll back 61.8% or 50% of the previous wave, Lateral corrections more often tend to roll back 38.2% of the previous impulse wave, as shown in Figure 4-2. 4-2. 
The most important thing in the market is : move-correction-move-corridor. 
And where to buy? Where will the price reach? There are many levels: 38.2%, 50.0%, 61.8%, rarely 26.3%, and 76.4%. 
If the market corrects from 38.2% to 61.8%, further moves are likely to continue. You can measure the size of the correction or its targets by Fibonacci levels. 


There are many trading systems based on Fibonacci levels described by Fisher, Demark, Murphy, Tom Joseph, Williams, and Dinapolli.
Let's analyze on real examples: 
Fibonacci + candlestick configuration is a simple trading strategy. It can be aggressive or conservative. Upon completion of any movement, we use the Fibonacci levels to determine the size of the expected retracement and open a trade position. Once the correction is completed at levels of 38.2%, 50% or 61.8%, we can determine the target area for a further move and reopen the trade position. Fibonacci levels should be set aside in the direction of price movement!
Aside from the lines, there are also: Gann fan, ellipses and Fibonacci arcs.

 
2.Elliott Wave Theory

The law of waves is Ralph Nelson Elliott's discovery of how the behaviour of society or of a crowd develops and changes in the form of recognisable patterns. Using stock market data as his main research tool, Elliott identified thirteen movement patterns or "waves", which are repeated over and over again in the flow of market prices. He gave them names, definitions, illustrated them and described the regularities of their appearance and development. Now this law is named after him - the Elliott Wave Law. 

2.1 History of discovery

Ralph Nelson Elliott was an engineer. After a serious illness in the early 1930s, he began analysing stock prices, especially the Dow Jones index. After a series of successful predictions he published a series of articles in Financial World Magazine in 1939. They were the first to present his viewpoint that Dow Jones index movements follow certain rhythms. According to Elliott, all these movements follow the same law as the tide - the tide is followed by an ebb, an action (action) by a counteraction (reaction). This pattern is time-independent because the structure of the market, taken as a whole, remains unchanged. 
Elliott wrote: "The law of nature includes the most important element in consideration - rhythmicity. The law of nature is not a system, not a method of playing the market, but a phenomenon characteristic of the course of any human activity. Its application in forecasting is revolutionary!" 
Elliott based his discoveries on the Law of Nature. Elliott's law of waves suggests that the same law that creates living things and galaxies is inherent in the mood and activity of man in the masses. The stock market is a creature of man and therefore reflects his character.
What is psychology? It is the reaction of people or groups to certain events. What do we see in the market? Psychology! From the way we behave, the prediction of the situation is derived. 

What drives any moment of activity? Motive, desire. Secret desires or explicit desires have one foundation - instincts! The basic human instincts:
self-preservation 
herd (the crowd) 
food 
sexual 
The herd instinct contradicts all 3. Here comes the main difficulty of trading - the struggle with the self! And the herd is primitive and easy to predict using Elliott waves.

2.2 The types of waves

Wave 3 - the most powerful, driving wave; Wave 5 - the wave of "latecomers" (remember the Dow Theory). The classic five-wave pattern is rare and fractal, i.e., each wave of a larger time period consists of a completed cycle of waves of a smaller time period. 
There are two types of waves: 
waves that cause price changes, i.e. driving waves - 1, 2, 3, 4, 5; 
Waves that counter-trend, i.e. corrective waves - A, B, C.
Why 5 and 3? Should the underlying structure necessarily consist of five and three waves? Think about it and you will realize that this is the necessary minimum to ensure a progressive movement with both forward and backward elements at the same time, and therefore the most effective way of such movement. One wave does not contain a pullback. The minimum set to form a rollback is three waves. Three waves in both directions will not provide a forward movement. In order to advance in one direction regardless of the duration of the rollback, the movement in the main direction must consist of at least five waves just to exceed the rollback from three waves and still contain those rollback waves (the well-known philosophical concept of unity and struggle of opposites). Although, to ensure this, there could be more waves than in our case, but the most rational figure of guaranteed forward movement is 5-3, and nature usually follows the most rational way - the Fibonacci numbers and the optimal way of development of anything. After the end of the wave cycle shown in the figure above, a second similar cycle of five upward waves begins, followed by three downward waves. A third phase of movement, also consisting of five upward waves, then develops. This third phase ends the five-wave movement with one wave level higher than the level of the waves it consists of. The result is shown in Figure 1-3 up to the peak indicated by (5).
 
At the peak of wave (5), a downward movement of an appropriately higher wave level begins, again consisting of three waves. These three waves of the next wave level "correct" the upward movement of five waves of the same level. The result is another complete cycle, but of a higher wave level, as shown in Figures 1-3. Let us now note that within the corrective pattern shown as wave [2] in Figures 1-3, waves (a) and (c), directed downward, consist of five sub-waves: 1, 2, 3, 4 и 5. Similarly, wave (b), which is directed upwards, consists of three sub-waves: a, b and c. This structure reveals a very important point: driving waves are not always directed upwards and corrective waves are not always directed downwards. The style of a wave is determined not only by its absolute direction, but mainly by its relative direction. With the exception of four digressions, which will be discussed later, waves are divided into a driving style wave (five waves) when travelling in the same direction as the wave one wave level above and of which they are a part, and into corrective style waves (three waves or one of their varieties) when travelling in the opposite direction. Waves (a) and (c) developing in the same direction as wave [2] are moving waves. Wave (b) is corrective because it corrects wave (a) and is developing in the opposite direction from wave [2]. In summary, the main essence of the Law of Waves is that on any wave level, a movement in the same direction as the movement of a wave one wave higher, is developed by five waves, while a pullback against the movement of a wave higher, is developed by three waves. Driving + Corrective = Cycle.
Highest level 1 + 1 = 2;
lower level 5 + 3 = 8;
next level down 21 + 13 = 34;
next level down 89 + 55 = 144. 

2.3 Practical application

Driving waves are composed of five waves of defined characteristics and always develop in the same direction as the driving wave one wave level above. They are straightforward and relatively easy to recognise and explain:
in moving waves, wave 2 never rolls back more than 100% of the size of wave 1 
wave 4 never retreats by more than 100% of the size of wave 3 
wave 3 always travels farther than the end of wave 1
Wave 3 is often the longest and never the shortest of the three active sub-waves ( 1, 3, and 5) of the driving wave.
The purpose of moving waves is to move forward, and these rules of wave construction guarantee that it will be so. 
There are two types of moving waves:
impulses; 
diagonal triangles (diagonal triangles).
The most common driving wave is an impulse.
Many wave impulses contain what Elliott called wave lengthening. Wave extensions are stretched pulses with an extended wave structure. A huge number of pulses do contain elongation in one and only one of its three active waves.
 

Truncation 
Elliott used the word "failure" to describe a situation in which the fifth wave does not exceed the top of the third wave. We prefer the less ambiguous term "truncation" or "truncated fifth". We can verify that this is indeed a wave truncation by verifying that the assumed fifth wave contains the necessary five sub-waves, as shown in Figures 1-11 and 1-12. Truncation usually occurs after an extremely strong third wave. 
Bull/Bear Market Truncation 

A diagonal (sloping) triangle is a driving pattern, although not yet an impulse, as it has one or two corrective traits. Diagonal triangles substitute for impulses at certain points in the wave structure. As in an impulse, no counteracting sub-wave (composite wave level smaller) rolls back more than the size of the previous active sub-wave here and the third sub-wave is never the shortest. However, diagonal triangles are the only five-wave structures developing in the direction of the main movement in which wave 4 almost always enters (overlaps) the price territory of wave 1. On rare occasions a diagonal triangle may end in a truncation, although in our experience the size of such truncations is minimal. 

Ending Diagonal 
An Ending Triangle is a particular type of wave that develops mainly in place of the fifth wave when the previous movement (wave 3) has gone "too far and too fast", as Elliott put it. A very small percentage of terminal triangles appear in place of wave C in structures A-B-C. In all cases, they are found in the final components of the model one wave level higher, showing the exhaustion of the strength of the movement in this senior wave level. Final triangles take the form of a wedge between two converging lines and each of their composite sub-waves, including waves 1, 3, and 5, is subdivided into a 'triplet', which in other cases is a corrective wave feature. The final triangle is illustrated in Figures 1-15 and 1-16 and is shown in its typical position in the impulse waves of the upper wave level. 

An upward diagonal triangle heralds a bearish mood and is usually followed by a sharp fall in prices, at least to the level where the triangle began. A diagonal descending triangle is also a bullish omen, which usually triggers an upward correction in prices. A fifth wave lengthening, a truncated fifth wave, and an ending diagonal triangle all encapsulate the same fact: an impending spectacular change in direction. At some pivot points, two such phenomena have occurred together at different wave levels, multiplying the strength of the subsequent move in the opposite direction. 
Market prices move against the direction of the senior wave level only with seeming effort. It may appear that resistance from the senior wave level is preventing the pullback from developing into a full-blown driving structure. This struggle between the two opposing wave levels makes corrective waves less recognizable than driving waves, which always develop with relative ease in the direction of the movement of the senior wave level. As a result of this conflict between level movements, corrective waves are slightly more diverse than driving waves. Moreover, they sometimes increase or decrease in complexity and develop so that a structure that is formally a subwave of the same wave level may appear to belong to a different wave level because of its complexity or time duration. For all these reasons, it is at times difficult to fit corrective waves into a recognizable pattern until they are over and behind us. Since the ending of corrective waves is less predictable than for driving waves, the analyst should be more cautious in his analysis when the market is in a fluctuating, corrective mood than when prices are moving steadily in one direction. The single most important rule, which can be formulated from the study of various corrective patterns, is that pullbacks are never "fives". Only the driving waves are "fives". For this reason, an initial five-wave movement against a higher wave level is never the end of a correction, only part of it. 
Corrections occur in two ways:
Sharp pullbacks bend steeply against the direction of movement of the senior wave level. 
Lateral corrections, though always performing a final pullback from the previous wave, usually contain movement towards its starting point or even beyond, forming the appearance of a sideways movement. 
Individual corrective patterns fall into four main categories: 

Zigzags (5-3-5): 
single;,
double;
triple. 

Flats (3-3-5):
regular;
expanded;
running. 

Triangles (3-3-3-3):
Four types, three of which are convergent:
ascending;
descending;
symmetrical;
and one divergent type:
reverse symmetrical. 

The single zigzag in a bull market is a simple three-wave descending pattern marked A-B-C. The sequence of sub-waves is 5-3-5 and the top of wave B is noticeably lower than the beginning of wave A, as shown in Figure 1-22 and 1-23.
 
Figure 1-22 Figure 1-23 

In a bear market, the correction in the form of a zigzag develops in the opposite direction as shown in Figure 1-24 and 1-25. For this reason, a zigzag in a bear market is referred to as an inverted zigzag. 


Sometimes zigzags are formed twice or at most three times in a sequence, especially when the first zigzag has not reached the standard target. In these cases, each zigzag is separated by an intermediate "triplet", forming what is called a double zigzag or triple zigzag. These structures are similar to impulse wave lengthening, but less common. 

As it turns out, triangles reflect the balance of forces that cause sideways movement, which is usually associated with a decrease in volume and the amplitude of price fluctuation has become entrenched in the Russian language as "volatility". Triangles contain five overlapping waves, which are subdivided into 3-3-3-3 models and labelled a-bc-d-e. The contour of any triangle is formed by a pairwise connection of the endpoints of waves a and c, b and d. Wave e may not touch or cross the a-c line and, in fact, our experience tells us that this is more likely to happen than not. 

There are two kinds of triangles: 
convergent;
divergent. 

Within the convergent varieties, there are three types:
symmetrical,
ascending
descending, as shown in Figure 1-42. 

The rarer divergent triangle has no variation. It always forms as shown in Figure 1-42, which is why Elliott called it an "inverted symmetric" triangle

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