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 2)
Thus, the Fibonacci sequence is not just a numerical puzzle. It is a mathematical bridge that connects the laws of natural growth with the collective psychology of the market. The price, like a living organism, tends to follow these natural proportions in its movement. The trader's task is not to guess, but to use objective Fibonacci levels to identify areas where the market's response is most likely. In the following chapters, we will learn how to apply these levels to a chart and create comprehensive trading strategies.
• "The Golden Ratio" (Phi, φ ≈ 1.618) and its reciprocal (0.618)
If the Fibonacci sequence is a musical scale, then the number Phi (φ) is its main, fundamental note that creates harmony. It is the key that transforms a series of numbers into a system for analyzing proportions.
1. What is the Golden Ratio?
It is an irrational number (≈ 1.6180339887…), which arises as a solution to a simple but elegant problem: to divide a segment into two parts in such a way that the ratio of the whole segment (C) to the larger part (A) is equal to the ratio of the larger part (A) to the smaller part (B).
C / A = A / B ≈ 1.618
This ratio is φ (Phi). In other words, if you take a line segment, the golden ratio point will divide it approximately in a 62% / 38% ratio.
2. Connection to the Fibonacci sequence
As we have already seen, the ratio of adjacent Fibonacci numbers converges to φ. But the connection goes deeper:
· φ can be derived from the properties of the sequence: φ = (1 + √5) / 2.
· φ has unique mathematical properties.
For example:
· φ + 1 = φ² (2.618… = 1.618…²)
· 1 / φ = φ – 1 ≈ 0.618 – this is its reciprocal, a number that is no less important in trading.
· φ can be obtained by continuing the Fibonacci sequence in the opposite direction (…-8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3…) – the ratio remains the same.
3. The inverse value: 0.618 – the main correction level
The number 0.618 (61.8%) is 1 / φ. In trading, this is not just a mathematical transformation. It is the most significant and powerful Fibonacci correction level.
· Psychological meaning: If the price movement retraces 61.8% of the previous impulse, it is perceived by the market as a deep but natural correction that does not disrupt the trend structure. This is the point of maximum balance, after which it is highly likely that the initial movement will continue. Many traders are waiting for a reaction at this level.
4. Why do these proportions "work" in the market?
· Harmony and expectation: φ and 0.618 are universal proportions of aesthetic harmony and growth that are inherent in nature. Human perception (including the perception of charts and proportions) subconsciously seeks these harmonious relationships. When the price approaches these proportions, there is a collective expectation of a reversal or reaction.
· Self-fulfilling prophecy: Since millions of traders around the world are watching the same levels, they are concentrating their orders (stop-losses, take-profits, pending orders) around them. This concentration of liquidity causes the market to react to these zones.
· Decision-making level: For large players (institutional investors, banks), the 61.8% and 38.2% levels are often logical decision-making zones for entering a trend at a "good price" after a pullback.
5. Derived levels: the whole Fibonacci family
From φ and 0.618, other key levels occur that we use:
· 0.382 (38.2%): This is 1 – 0.618. The level of a shallow, "first" correction.
· 0.236 (23.6%): This is 0.618 * 0.382 or 0.618². A weak level, often used to confirm the strength of a trend.
· 0.786 (78.6%): This is the square root of 0.618 (√0.618). A deep and important level, an alternative to 61.8%.
· 1.272 and 1.618: These are extensions (√φ and φ), targets for continuing the movement.
A practical conclusion for a trader:
Remember this pair as the foundation of the foundations:
· 1.618 (φ) is the target (extension, expansion of momentum).
· 0.618 is the key reversal point (correction, depth of retracement).
They do not guarantee that the price will reverse exactly at these points, but they indicate areas where the probability of a strong market reaction is much higher than in any random location on the chart. Your task is to learn how to find these zones and confirm signals in them with other tools.
Next, we will look at how these abstract numbers turn into specific tools on the chart – correction and expansion levels.
• Derived key ratios: 0.236, 0.382, 0.5, 0.618, 0.786, 1.0, 1.272, 1.618
Now that we know the ancestors – φ (1.618) and 0.618 – it's time to get acquainted with the whole "family". These derived ratios are the very lines that we will put on the chart. Each of them has its own mathematical genesis and unique trading interpretation.
They can be divided into three groups: correction levels, base level, and expansion levels.
I. Correction Levels (Depth of Rejection)
These levels answer the question: "How deep can the retracement be against the main trend?". They are measured as a percentage of the previous movement.
· 0.236 (23.6%) – Weak level.
· Origin: 0.618³ or the ratio of a Fibonacci number to a number three positions ahead (for example, 8/34 ≈ 0.235).
· Trading meaning: A very shallow correction. It often serves as a temporary pause in a powerful trend. A rebound from this level indicates the exceptional strength of the current movement. Its breakout is the first signal that the correction may deepen.
· 0.382 (38.2%) – Moderate correction.
· Origin: 1 – 0.618 or 0.618².
· Trading meaning: The most common and “healthy” depth of a pullback. It is believed that the trend retains a good chance of continuation after a rebound from this level. This is the first serious zone for finding a trend entry point.
· 0.500 (50.0%) – Balance level (NOT a Fibonacci number, but a key one!).
· Origin: Simply half (1/2). Although it is not formally a Fibonacci level, the market respects it as the deepest psychological level of symmetry and balance between bulls and bears.
· Trading meaning: It often acts as strong support/resistance. Many professional traders (such as followers of Elliott Wave Theory) consider it mandatory to use alongside the "classic" Fibonacci levels.
· 0.618 (61.8%) – Golden correction (Main level).
· Origin: 1 / φ (the inverse of the golden ratio).
· Trading meaning: The most important and significant level. A deep but still "legitimate" correction within the trend. It is considered the last boundary, and if the price holds it, the trend is alive. This is where the maximum number of orders and traders' attention is concentrated. There is a high probability of a strong price reaction.
· 0.786 (78.6%) – Deep correction.
· Origin: √0.618.
· Trading meaning: Alternative deep level. Often is the border between deep correction and the beginning of a trend reversal. Especially important in some modern trading techniques. If the price breaks through this level, the probability of a trend change increases dramatically.
II. Basic Level
· 1.000 (100.0%) – Level of complete correction.
· Origin: Complete return of the price to the beginning of the impulse.
· Trading meaning: In fact, it says that there is no trend. The movement has been completely "compensated." However, in the context of extensions (as we will see later), this level becomes a key target for the next impulse (extension).
III. Expansion Levels (Movement Targets)
These levels answer the question: "Where can the trend extend after the correction is complete?" They are measured in percentages of the original impulse's length.
· 1.272 (127.2%) – The first expansion target.
· Origin: √φ (square root of 1.618).
· Trading meaning: A frequent and conservative target for taking part of the profit. A move towards this level indicates a resumption of trend strength.
· 1.618 (161.8%) – The Golden Expansion (Classical Target).
· Origin: The Golden Ratio itself, φ.