Алексей Боровков – Fibonacci in trading: the mathematics of market movement. A practical guide to using Correction, Expansion, and Fan Lines in the Forex Market (страница 1)
Алексей Боровков
Fibonacci in trading: the mathematics of market movement. A practical guide to using Correction, Expansion, and Fan Lines in the Forex Market
Part 1: Fundamentals. Numbers that Rule the World
Chapter 1: The History and Mathematics of Fibonacci Numbers
• A Brief History of Leonardo of Pisa
The name of Leonardo of Pisa (c. 1170 – c. 1250), better known by his nickname Fibonacci (son of Bonacci), is inextricably linked to the revolution that took place in European mathematics and commerce at the end of the Middle Ages.
1. The Son of a Merchant and a Traveler
Leonardo was born in the Italian trading republic of Pisa, the son of a wealthy merchant and government official, Guglielmo Bonacci. His father was appointed a customs official in the largest port of North Africa, Bejaia (modern-day Algeria). It was there that he took his young son to learn the intricacies of commerce.
In Bejaia, young Leonardo was immersed in a world that was vastly different from Europe. Here, Arab, Indian, Byzantine, and African trade routes converged. He quickly realized the tremendous advantage of the Arabic and Indian methods of calculation over the cumbersome Roman numeral system used by European merchants. Try quickly multiplying CXVII by XXIV in your head, and you will understand his frustration.
2. The Student of the East
Fibonacci did not limit himself to commercial arithmetic. He spent years traveling along the Mediterranean coast (Egypt, Syria, Sicily, Greece), studying the works of the great mathematicians of the Islamic world – Al-Khwarizmi, Abu Kamil, and others – from Arab and Byzantine teachers.
His genius was not just in studying, but in synthesizing. He was able to combine the knowledge of Greek geometry (Euclid), Indian arithmetic, and Arabic algebra into a unified, logical system.
3. "The Book of Abacus" (Liber Abaci) – a commercial code of the new era
Returning to Pisa around 1200, Fibonacci summarized his knowledge in the fundamental work "Liber Abaci" (The Book of Calculations), published in 1202. This was not a dry mathematical theory, but a practical guide for merchants, bankers, and scholars.
In the book, he:
· Introduced the Indo-Arabic numerals (0-9) and the decimal system to Europe. He clearly demonstrated how much more effective they are than Roman numerals for any calculations, from currency conversions to profit calculations.
· He analyzed in detail the tasks that are vital for trading: calculating interest rates, exchanging currencies, distributing profits in partnerships, and resolving commercial disputes.
· He presented the famous rabbit problem, which described the numerical sequence (1, 1, 2, 3, 5, 8, 13…) later named after him. For Fibonacci himself, it was just a curious example, not a central discovery.
The book caused a sensation. Leading Italian merchants and bankers quickly recognized the power of the new tool. "Liber Abaci" became a textbook for generations of European merchants, laying the mathematical foundation for the future flourishing of banking and stock trading during the Renaissance.
4. Legacy: From Trading Calculations to the Laws of Nature
Fibonacci was respected at the court of Emperor Frederick II, participated in mathematical tournaments, and wrote several other significant works on geometry and number theory.
Why is this story important for a trader?
Because it reminds us of the most important thing: Fibonacci's mathematics were born out of the practical needs of trading. His sequence and proportions were not just abstract theories; they became a tool for analyzing, predicting, and managing risks in a world where everything was subject to movement and exchange.
Ironically, eight centuries later, his name is associated not with the debit and credit of medieval merchants, but with the search for harmony and order in the seemingly chaotic movement of prices in financial markets. As before, Fibonacci numbers provide us with a coordinate system, a language for understanding the hidden logic of a market where the interests of millions of modern "traders" collide.
Moving on to mathematics:
This historical foundation brings us to the most important part—the numbers themselves. How did a simple recursive sequence born from the hypothetical reproduction of rabbits become the key to the "golden ratio" that defines the proportions of the Parthenon, the spiral of a seashell, and, as we will see, the behavior of the crowd in the Forex market? This will be discussed in the next section of the chapter.
• The Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
This seemingly simple chain of numbers is the mathematical heart of the method we are studying. Understanding its nature is the key to using all the following tools.
1. How it is constructed: A simple rule that generates complexity.
The sequence starts with 0 and 1. Each subsequent number is the sum of the two previous numbers.
· 0 + 1 = 1
· 1 + 1 = 2
· 1 + 2 = 3
· 2 + 3 = 5
· 3 + 5 = 8
· 5 + 8 = 13
· … and so on indefinitely.
Formally, this is written as: F(n) = F(n-1) + F(n-2), where F (0) =0, F (1) =1.
This is a recurrent sequence – its future is completely determined by its past. It is this idea of “dependence on the previous state” that resonates with the behavior of the market, where the current price is always a derivative of past price movements and the psychology of participants.
2. The main property: The desire for the "Golden ratio" (Phi, φ ≈ 1.618).
This is where the magic begins. If we start dividing each number in the sequence by the previous one, we will see an amazing pattern:
· 3/2 = 1.5
· 5/3 ≈ 1.666…
· 8/5 = 1.6
· 13/8 = 1.625
· 21/13 ≈ 1.615…
· 34/21 ≈ 1.619…
· 55/34 ≈ 1.6176…
· 89/55 ≈ 1.61818…
The further we move along the sequence, the closer the ratio of adjacent numbers approaches the irrational number ≈ 1.6180339887…, known as the Golden Ratio (Phi, φ).
The same is true for the inverse operation (dividing the previous by the subsequent), which approaches the number 0.618 (1 / 1.618).
3. Why is it important for us? The mathematics of mass psychology.
The golden ratio and its derivatives (0.236, 0.382, 0.5, 0.786, 1.618, etc.) are not just beautiful numbers. They are universal proportions of harmony, balance, and natural development, observed in the spiral structure of galaxies, the nautilus shell, the branching of trees, and the proportions of the human body.
The key takeaway for the trader: The market is a product of the collective psychology of thousands and millions of people. This psychology, as part of human nature, perceives Fibonacci-based proportions as levels of natural tension, saturation, and reversal on an unconscious level.
When the price makes a strong movement (impulse), market participants intuitively expect a correction. But how deep is it? Experience shows that the most likely points of correction completion often lie at levels derived from the golden ratio: 23.6%, 38.2%, 50.0%, 61.8%, and 78.6%.
· 38.2%: A shallow, "healthy" correction.
· 61.8% (1/φ): Deep, but still within the trend. The most important level.
· 50.0%: Although it is not a Fibonacci number, it is a key psychological balance level that the market often respects.
4. Derived key levels used in trading:
How do we get the other percentages that are important to us?
· 23.6%: This is the result of dividing a number in the sequence by a number two positions ahead (for example, 8/34 ≈ 0.235).
· 38.2%: Dividing a number by a number one position ahead (13/34 ≈ 0.382).
· 61.8%: Reverse The value of K is φ (1 / 1.618 ≈ 0.618).
· 78.6%: The square root of 0.618 (√0.618 ≈ 0.786).
· 161.8%: The Golden Ratio itself (1.618).
Conclusion of the section: