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006_fibonacci_retracement

Fibonacci Retracement: Mathematical Origins and Trading Applications

Fibonacci Retracement: Mathematical Origins and Trading Applications

I. Historical and Mathematical Background

Fibonacci retracement levels derive from the Fibonacci sequence, discovered by Leonardo of Pisa in 1202. The sequence appears throughout nature and has found remarkable application in financial markets.

II. The Fibonacci Sequence

Definition 2.1 (Fibonacci Sequence)

The Fibonacci sequence {F_n} is defined by the recurrence relation: Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2} with initial conditions F_0 = 0, F_1 = 1.

First terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...

Theorem 2.1 (Binet's Formula)

The n-th Fibonacci number can be computed directly: Fn=ϕn−ψn5F_n = \frac{\phi^n - \psi^n}{\sqrt{5}} where φ = (1+√5)/2 ≈ 1.618 (golden ratio) and ψ = (1-√5)/2 ≈ -0.618

Proof: The characteristic equation of the recurrence is x² = x + 1. Roots: φ = (1+√5)/2 and ψ = (1-√5)/2. General solution: F_n = Aφ^n + Bψ^n. Applying initial conditions F_0 = 0, F_1 = 1 yields A = 1/√5, B = -1/√5. ∎

Theorem 2.2 (Golden Ratio Convergence)

lim⁡n→∞Fn+1Fn=ϕ≈1.61803...\lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \phi \approx 1.61803...

Proof: Let r_n = F_{n+1}/F_n. From the recurrence: rn=Fn+Fn−1Fn=1+1rn−1r_n = \frac{F_n + F_{n-1}}{F_n} = 1 + \frac{1}{r_{n-1}}

Taking limits: r = 1 + 1/r → r² - r - 1 = 0 → r = φ ∎

III. Fibonacci Ratios in Trading

Definition 3.1 (Key Fibonacci Ratios)

| Ratio | Origin | Approximation | |-------|--------|---------------| | 23.6% | 1 - φ^(-3) | F_n/F_{n+3} | | 38.2% | 1 - φ^(-1) = φ^(-2) | F_n/F_{n+2} | | 50.0% | 1/2 | Symmetry | | 61.8% | φ^(-1) | F_n/F_{n+1} | | 78.6% | √(0.618) | Square root | | 100% | 1 | Full retracement |

Theorem 3.1 (Ratio Relationships)

0.382×0.618≈0.2360.382 \times 0.618 \approx 0.236 0.6182≈0.3820.618^2 \approx 0.382 1−0.618=0.3821 - 0.618 = 0.382

IV. Retracement Level Construction

Definition 4.1 (Fibonacci Retracement)

Given a price swing from point A (low) to point B (high): RetracementLevel(r)=B−r(B−A)Retracement Level(r) = B - r(B - A)

For a downswing from A (high) to B (low): RetracementLevel(r)=B+r(A−B)Retracement Level(r) = B + r(A - B)

Example 4.1

Swing Low: 1.1000, Swing High: 1.1500, Range = 500 pips

  • 23.6% level: 1.1500 - 0.236 × 500 = 1.1382
  • 38.2% level: 1.1500 - 0.382 × 500 = 1.1309
  • 50.0% level: 1.1500 - 0.500 × 500 = 1.1250
  • 61.8% level: 1.1500 - 0.618 × 500 = 1.1191

V. Fibonacci Extensions

Definition 5.1 (Fibonacci Extension Levels)

Extensions project beyond the original move: ExtensionLevel(e)=B+e(B−A)Extension Level(e) = B + e(B - A)

Common extensions: 127.2%, 161.8%, 200%, 261.8%, 423.6%

Theorem 5.1 (Extension Relationships)

  • 127.2% = √(1.618)
  • 161.8% = φ
  • 261.8% = φ²
  • 423.6% = φ³

VI. Confluence Theory

Definition 6.1 (Fibonacci Confluence)

A confluence zone occurs when multiple Fibonacci levels align: Confluence={p:∃i,j,k:∣Fibi(Swing1)−Fibj(Swing2)∣<ϵ}Confluence = \{p : \exists i,j,k : |Fib_i(Swing_1) - Fib_j(Swing_2)| < \epsilon\}

Theorem 6.1 (Confluence Significance)

The probability of price respecting a level increases with the number of confluent factors. Empirically:

  • Single Fib level: ~40% respect rate
  • Double confluence: ~60% respect rate
  • Triple confluence: ~75% respect rate

VII. Statistical Validity

7.1 Academic Research

Studies show mixed results on Fibonacci effectiveness:

  • Osler (2000): Significant clustering at round numbers and Fibonacci levels
  • Batchelor & Furnham (2002): Self-fulfilling prophecy effect

7.2 Market Microstructure

Fibonacci levels work partly because:

  1. Many traders watch the same levels
  2. Orders cluster at these prices
  3. Creates supply/demand zones

VIII. Advanced Applications

8.1 Fibonacci Time Zones

Vertical lines at Fibonacci intervals: 1, 2, 3, 5, 8, 13, 21... bars from significant point.

8.2 Fibonacci Arcs

Curved lines incorporating both price and time: Arcr(t)=(t−t0)2+(r⋅(B−A)/S)2Arc_r(t) = \sqrt{(t-t_0)^2 + (r \cdot (B-A)/S)^2} where S is a scaling factor.

8.3 Fibonacci Fan

Lines from significant point through retracement levels.

IX. Exercises

Exercise 1: Prove that φ² = φ + 1.

Exercise 2: Calculate all Fibonacci retracement levels for EURUSD moving from 1.0800 to 1.1200.

Exercise 3: Show that 0.786 ≈ √0.618.

Exercise 4: Prove that consecutive Fibonacci ratios alternate above and below φ.

X. References

  • Fibonacci, L. (1202). Liber Abaci
  • Carney, S. (2010). Harmonic Trading
  • Pesavento, L. (1997). Fibonacci Ratios with Pattern Recognition
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