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2022 6/21

2021-11-10

ピボットプライス(P:Pivot Price)＝(H+L+C)÷3
H：前日高値、L：前日安値、C：前日終値D1＝H－P：高値とピボットの差
D2＝P－L：安値とピボットの差
D3＝H－L：高値と安値の差HBOP(High Break Out Price上方ブレークアウト)＝P+D2+D3＝2P-2L+H
R2（上値抵抗2：レジスタンス）＝P+D3＝P+H-L
R1（上値抵抗1：レジスタンス）＝P+D2＝2P-L
ピボット（P）＝（H＋L＋C）÷3
S1（下値支持1：サポート）＝P-D1＝2P-H
S2（下値支持2：サポート）＝P-D3＝P-H+L

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Fibonacci Retracement Levels

Cory Mitchell, CMT is フィボナッチ the founder of TradeThatSwing.com. He has been a professional day and swing trader since 2005. Cory is an expert on stock, forex and futures price action trading strategies.

Chip Stapleton is a Series 7 and Series 66 license holder, CFA Level 1 exam holder, and currently holds a Life, Accident, and Health License in Indiana. He has 8 years experience in finance, from financial planning and wealth management to corporate finance and FP&A.

Kirsten Rohrs Schmitt is フィボナッチ フィボナッチ an accomplished professional editor, writer, proofreader, and fact-checker. She has expertise in finance, investing, real estate, and world history. Throughout her career, she has written and edited content for numerous consumer magazines and websites, crafted resumes and social media content for business owners, and created collateral for academia and nonprofits. Kirsten is also the founder and director of Your Best Edit; find her on LinkedIn and Facebook.

What Are Fibonacci Retracement Levels?

Fibonacci retracement levels—stemming from the Fibonacci sequence—are horizontal lines that indicate where support and resistance are likely to occur.

Each level is associated with a percentage. The percentage is how much of a prior move the price has retraced. The Fibonacci retracement levels are 23.フィボナッチ 6%, 38.2%, 61.8%, and 78.6%. While not officially a Fibonacci ratio, 50% is also used.

The indicator is useful because it can be drawn between any two significant price points, such as a high and a フィボナッチ フィボナッチ low. The indicator will then create the levels between those two points.

Suppose the price of a stock rises $10 and then drops $2.36. In that case, it has retraced 23.6%, which is a Fibonacci number. Fibonacci フィボナッチ numbers are found throughout nature. Therefore, many traders believe that these numbers also have relevance in financial markets.

Fibonacci retracement levels were named after Italian mathemetician Leonardo Pisano Bigollo, who was famously known as Leonardo Fibonacci. However, Fibonacci did not フィボナッチ create the Fibonacci sequence. Fibonacci, instead, introduced these numbers to western Europe after learning about them from Indian merchants. Fibonacci retracement levels were formulated in Ancient India between 450 and 200 BCE.

Key Takeaways

Fibonacci retracement levels connect any フィボナッチ two points that the trader views as relevant, typically a high point and a low point.
The percentage levels provided are areas where the price could stall or reverse.
The most commonly used ratios include 23.6%, 38.2%, 50%, 61.8%, and 78.6%.
These levels should not be relied on exclusively, so it is dangerous to assume the price will reverse after hitting a specific Fibonacci level.
Fibonacci numbers and sequencing were フィボナッチ first used by Indian mathematicians centuries before Leonardo Fibonacci.

Numbers First Formulated in Ancient India

Despite its name, the Fibonacci sequence was not developed by its namesake. Instead, centuries before Leonardo Fibonacci shared it with western Europe, it was developed フィボナッチフィボナッチ and used by Indian mathematicians.

Most notably, Indian mathematician Acarya Virahanka is known to have developed Fibonacci numbers and the method of their sequencing around 600 AD. Following Virahanka's discovery, other subsequent generations of Indian mathematicians—Gopala, Hemacandra, and Narayana Pandita—referenced the numbers and method. Pandita expanded its use by drawing a correlation between the Fibonacci numbers and multinomial coefficients.

It is estimated that Fibonacci numbers existed in Indian society as early as 200 AD.

The Formula for フィボナッチ Fibonacci Retracement Levels

Fibonacci retracement levels do not have formulas. When these indicators are applied to a chart, the user chooses two points. Once those two points are chosen, the lines are drawn at percentages of that move.

Suppose the price rises from $10 to $15, and these two price levels are the points used to draw the retracement indicator. Then, the 23.6% level will be at $13.82 ($15 - ($5 x 0.236) = $13.82). The 50% level will be at $12.50 ($15 - ($5 x 0.5) = $12.50).

Image by Sabrina Jiang Â© Investopedia 2021

How to Calculate Fibonacci Retracement Levels

As discussed above, there is nothing to calculate when it comes to フィボナッチフィボナッチ Fibonacci retracement levels. They are simply percentages of whatever price range is chosen.

However, the origin of the Fibonacci numbers is fascinating. They are based on something called the Golden Ratio. Start a sequence of numbers with zero and one. Then, keep adding the prior two numbers to get a number string like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987. with the string continuing indefinitely.

The Fibonacci retracement levels are all derived from this number string. After the sequence gets going, dividing one number by the next number yields 0.618, or 61.8%. Divide a number by the second number to its right, and the result is 0.382 or 38.2%. All the ratios, except for 50% (since it is not an official Fibonacci number), are based on some mathematical calculation involving this number string.

The Golden Ratio, known as the divine proportion, can be found in various spaces, from geometry to human DNA.

Interestingly, the Golden Ratio of 0.618 or 1.618 is found in sunflowers, galaxy formations, shells, historical artifacts, and architecture.

What Do Fibonacci Retracement Levels Tell フィボナッチ You?

Fibonacci retracements can be used to place entry orders, determine stop-loss levels, or set price targets. For example, a trader may see a stock moving higher. After a move up, it retraces to the 61.8% level. Then, it starts to go up again. Since the bounce occurred at a Fibonacci level during an uptrend, the trader decides to buy. The trader might set a stop loss at the 61.8% level, as a return below that level could indicate that the rally has failed.

Fibonacci levels also arise in other ways within technical analysis. For example, they are prevalent in Gartley patterns and Elliott Wave theory. After a significant price movement up or down, these forms フィボナッチ of technical analysis find that reversals tend to occur close to certain Fibonacci levels.

Market trends are more accurately identified when other analysis tools are used with the Fibonacci approach.

Fibonacci retracement levels are static, unlike moving averages. The フィボナッチ static nature of the price levels allows for quick and easy identification. That helps traders and investors to anticipate and react prudently when the price levels are tested. These levels are inflection points where some type of price action フィボナッチフィボナッチ is expected, either a reversal or a break.

Fibonacci Retracements vs. Fibonacci Extensions

While Fibonacci retracements apply percentages to a pullback, Fibonacci extensions apply percentages to a move in the trending direction. For example, a stock goes from $5 to $10, and then back to $7.50. The move from $10 to $7.50 is a retracement. If the price starts rallying again and goes to $16, that is an extension.

Limitations of Using Fibonacci Retracement Levels

While the retracement フィボナッチ levels indicate where the price might find support or resistance, there are no assurances the price will actually stop there. This is why other confirmation signals are often used, such as the price starting to bounce off the level.フィボナッチ

The other argument against Fibonacci retracement levels is that there are so many of them that the price is likely to reverse near one of them quite often. The problem is that traders struggle to know which one will フィボナッチ be useful at any particular time. When it doesn't work out, it can always be claimed that the trader should have been looking at another Fibonacci retracement level instead.

Why are Fibonacci Retracements Important?

In technical analysis, Fibonacci retracement levels indicate key areas where a stock may reverse or stall. Common ratios include 23.6%, 38.2%, and 50%, among others. Usually, these will occur between a high and low point for a security, designed to predict the フィボナッチフィボナッチフィボナッチ future direction of its price movement.

What Are the Fibonacci Ratios?

The Fibonacci ratios are derived from the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on. Here, each フィボナッチ number is equal to the sum of the two preceding numbers. Fibonacci ratios are informed by mathematical relationships found in this formula. As a result, they produce the following ratios 23.6%, 38.2%, 50% 61.8%, 78.6%, 100%, 161.8%, 261.8%, and 423.6%. Although 50% is not a pure Fibonacci ratio, it is still used as a support and resistance indicator.

How Do You Apply Fibonacci Retracement Levels in a Chart?

As one of フィボナッチ the most common technical trading strategies, a trader could use a Fibonacci retracement level to indicate where he would enter a trade. For instance, if the trader notices that after significant momentum, a stock has declined 38.2%. As フィボナッチ the stock begins to face an upward trend, he decides to enter the trade. Because the stock reached a Fibonacci level, it is deemed a good time to buy, with the trader speculating that the stock will then retrace, フィボナッチ or recover its recent losses.

How Do You Draw a Fibonacci Retracement?

Fibonacci retracements are trend lines drawn between two significant points, usually between absolute lows and absolute highs, plotted on a chart. Intersecting horizontal lines are placed at フィボナッチフィボナッチ the Fibonacci levels.

The Bottom Line

Fibonacci retracements are useful tools that help traders identify support and resistance levels. With the information gathered, they can place orders, identify stop-loss levels, and set price targets. Although useful, traders often use other フィボナッチ indicators to make more accurate assessments of trends and make better trading decisions.

Fibonacci in Nature

Master Fibonacci

The Fibonacci sequence of numbers forms the best whole number approximations to the Golden Proportion, which, some say, is most aesthetically beautiful to humans. “Empirical フィボナッチ investigations of the aesthetic properties of the Golden Section date back to the very origins of scientific psychology itself, the first studies being conducted by Fechner in the 1860s” (Green 937). Debate remains as to whether or not humans naturally prefer Golden Ratio (1.61803…) proportions in the organization and structural symmetry of art, music or nature, and some even deny that the Golden Ratio is as ubiquitous in nature as others proclaim. Nevertheless, mathematical principles do appear to govern the development of many patterns and structures in nature, and as time passes, more and more scientific research finds evidence that the Fibonacci numbers and the Golden Ratio are prevalent in natural objects, from the microscopic structure proportions in the bodies of living beings on Earth to the relationships of gravitational forces and distances between bodies in the universe (Akhtaruzzaman and Shafie).

As it relates to the development and structure of a plant, it is not uncommon to find representations of the Fibonacci numbers or the Golden Ratio. Structural symmetry is one of the simplest ways an organism will demonstrate this fascinating phenomenon (Livio 115). For example, pentagonal symmetry (five parts around a central axis, フィボナッチ 72° apart) is quite common in the natural world, particularly among the more “primitive” phyla, such as the water net (Hydrodictyaceae Hydrodictyon), a green algae (“Live”). Higher in the plant kingdom, many flowers exhibit Fibonacci-number petal symmetry, including fruit フィボナッチ blossoms, water lilies, brier-roses and all the genus rosa, honeysuckle, carnations, geraniums, primroses, marsh-mallows, campanula, and passionflowers. Besides symmetrical number and arrangement of parts or petals, plants often illustrate the Fibonacci sequence in their seed sections or in the フィボナッチ spirals that are formed as new parts and branches grow.

Flowers, Fruits, and Vegetables

Spanish poet Salvador Rueda (1857-1933) eloquently said, “las flores son matematicas bellas, compass, armonia callada, ritmo mudo,” (flowers are a beautiful mathematics, compass, silent harmony, mute rhythm) (Spooner 38).

Many flowers display figures adorned with numbers of petals that are in the Fibonacci sequence:

1 petal: White Calla Lily 2 petals: Euphorbia 3 petals: Lily, Iris, Euphorbia 5 petals: Buttercup, wild Rose, Larkspur, Columbine (Aquilegia), Hibiscus フィボナッチ 8 petals: Delphiniums, Bloodroot 13 petals: Ragwort, Corn Marigold, Cineraria, Black Eyed Susan 21 petals: Aster, Shasta Daisy, Chicory 34 petals: Plantain, Pyrethrum, Daisy 55, 89 petals: Michaelmas Daisies, the Asteraceae family (Sinha; Akhtaruzzaman and Shafie)

One of the フィボナッチフィボナッチ largest families of the vascular plants, compositae, contains nearly 2000 genera and over 32,000 species (“Plant List”) of flowering plants. Compositae (or Asteraceae) is commonly referred to as the aster, daisy, composite, or sunflower family. Family members are distributed worldwide and have a recognizable, “unique disc-shaped inflorescence, composed of numerous pentamerous florets packed on an involucrate head, surrounded by ray florets (petals) on the outside.” The numbers of ray-florets and disc-florets vary from one plant to another, フィボナッチフィボナッチ but they all are all “beautiful phyllotactic configurations” due to the arrangement of seeds in the seed head.

Daisy with Equiangular Spiral

The head of a composite displays definite equiangular spirals running counter-clockwise and clockwise. These bi-directional spirals intersect each other, such as: フィボナッチ 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, … The numerators or the denominators of this series are recognizable as the Fibonacci sequence. The petal counts of Field Daisies are usually thirteen, twenty-one or thirty-four and, in the close-packed arrangement of フィボナッチ tiny florets in the core of a daisy blossom, we can see the equiangular spiral phenomenon clearly as twenty-one counterclockwise spirals swirl in delicate, picturesque motion with thirty-four clockwise spirals. In any daisy, the floral tango of logarithmic spirals フィボナッチ generally consists of successive terms of the Fibonacci sequence (Britton; Livio 112).

The seeds are packed this way on the seed head presumably to “reduce the size of the florets to the optimum [size] necessary for quick production of an adequate number of single-seeded fruits” (Majumder and Chakravarti). The distribution of the ray- florets on the heads in Fibonacci number structure is indicative of “perfect growth,” according to Majumder and Chakravarti.

Passion Flower with 3 Sepals and 5 Outer Green Petals

Research also indicates that “individual flowers emerge at a uniform speed at fixed intervals of time along a logarithmic spiral, with an initial angle at = 137.5° (Mathai and Davis, 1974)” (Majumder and Chakravarti).Passion flower, also known as Passiflora Incarnata, is a perfect example of フィボナッチ a flower regally displaying Fibonacci Numbers, for three sepals protect the bud at the outermost layer, while five outer green petals are followed by an inner layer of five more, paler, green petals. With an array of purple and フィボナッチ white stamens, there are 5 greenish T-shaped stamens in the center and three deep brown carpels at the uppermost layer (Akhtaruzzaman and Shafie).

Insufficient data and “careless methodological practices” cause many scientists to doubt or outright refute the notion フィボナッチフィボナッチ フィボナッチ that Fibonacci numbers or the Golden Ratio are an absolute “law of nature” (Green 937). Jonathan Swinton and Erinma Ochu aimed to remedy the lack of scientific evidence by studying and recording the occurrence of Fibonacci structure in the spirals (parastichies) of 657 sunflower (Helianthus annuus) seed heads at the MSI Turing’s Sunflower Consortium.

Sunflower Fibonacci

The sunflower has 55 clockwise spirals overlaid on either 34 or 89 counterclockwise spirals, a phi proportion (Phi Φ =1.618 …)(Wright). The most reliable data subset of 768 clockwise or anticlockwise parastichy numbers revealed a clear indication of a dominance of Fibonacci structure: 565 were Fibonacci numbers and 67 had a predefined type of Fibonacci structure. They also found “more complex フィボナッチ Fibonacci structures not previously reported in sunflowers” and seed heads without Fibonacci structure (nearly 20%). Some seed heads without Fibonacci structure nevertheless had a tendency for counts to cluster near the Fibonacci number; in those, “parastichy numbers equal to フィボナッチ one less than a Fibonacci number were present significantly more often than those one more than a Fibonacci number.” The research also revealed the “existence of quasi-regular heads, in which no parastichy number could be definitively assigned” (Swinton and Ochu).フィボナッチ

Apple Fibonacci

The bumps and hexagonal scales (bracts) on the surface of pineapples form three distinct spirals in increasing steepness, creating a recognizable pattern of Fibonacci numbers (usually 5, 8, and 13) and the Romanesco Broccoli (looks and tastes like a cross between broccoli and cauliflower) has a shape almost like a pentagon with florets organized in spirals in both directions around the center point, where the florets are smallest (Posamentier and Lehmann; Knott). Other fruits have Fibonacci numbers in their seeds’ sectional arrangements. Three sections are easy to see in the cut cross-sections of the Banana, Cantaloupe, Cucumber, Kiwano fruit (African cucumber), and Watermelon. Star Fruit, Okra, and Apple seeds are arranged in a pentagram shape of five sections (Akhtaruzzaman and Shafie).

Spirals, Branches, and Leaves

Artichoke Fibonacci

According to Scotta and Marketos, the Fibonacci spiral is “fundamental to organic life.” They admit that it is “not always clear why these numbers appear,” but it appears that they “reflect minimization or optimization principles of some sort, namely the notion that nature is efficient yet ‘lazy,’ making the most of available resources” (Scotta and Marketos). New growth may simply form spirals so that the new leaves, petals, and branches フィボナッチ will not block older leaves, etc. from sunlight or air, or so that the maximum amount of rain or dew will get directed down to the roots (Akhtaruzzaman and Shafie). Others suggest the logarithmic spiral may be a “natural フィボナッチフィボナッチ outcome of the supply of genetic material in the form of pulses at constant intervals of time and obeying the law of fluid flow” (Majumder and Chakravarti).

In the world of nature, things grow by adding some unit, even フィボナッチ if the unit is as small as a molecule. Michael Wright says phi is “an ideal rate of growth for things which grow by adding some quantity,” such as the nautilus and sunflower (Wright). In addition to pineapples, the nautilus, and the sunflower, spirals are found in pinecones, ginger plants, artichokes, and other plants, including numerous cacti (Britton; Livio 110-111).

In 1868, Wilhelm Hofmeister suggested that new cells destined to develop into leaves, petals, etc. (primordia) “always form in フィボナッチ フィボナッチ the least crowded spot” on the meristem (growing tip of a plant). Each successive primordium of a continuously growing plant “forms at one point along the meristem and then moves radially outward at a rate proportional to the stem’s growth” (Seewald). The second primordium grows as far as possible from the first, and the third grows at a distance farthest from both the first and the second primordia (Seewald). In the 1830s, scientist brothers found that the rotation フィボナッチ tends to be an angle made with a fraction of two successive Fibonacci Numbers, such as 1/2, 1/3, 2/5, 3/8 (Akhtaruzzaman and Shafie). “As the number of primordia increases, the divergence angle eventually converges to a constant value” of 137.5° thereby creating Golden Angle Fibonacci spirals (Seewald).

Aloe Vera Fibonacci

The fact that branches and leaves of plants follow certain mathematical growth patterns was first noted in antiquity by Theophrastus (ca. 372 B.C. – ca. 287 B.C.) but フィボナッチ the phenomenon was first called phyllotaxis (“leaf arrangement” in Greek) in 1754 by the Swiss naturalist Charles Bonnet (1720-1793) (Livio 109-110). Patterns with the other fractions are also observed, though uncommonly (Okabe). Professor Emeritus H. S. M. Coxeter at the University of Toronto, in his Introduction to Geometry admits that some plants exhibit phyllotaxis numbers that “do not belong to the sequence of f’s [Fibonacci numbers] but to the sequence of g’s [Lucas numbers] or even to the フィボナッチフィボナッチフィボナッチ still more anomalous sequences 3,1,4,5,9,… or 5,2,7,9,16,….” He concludes we must face the fact that phyllotaxis is really not a universal law but only a fascinatingly prevalent tendency favored フィボナッチフィボナッチ by nature (Coxeter).

The Golden Ratio

The Sneezewort is a simple plant that exhibits the Fibonacci sequence. New shoots commonly spring from the main stem at an axil. Horizontal lines drawn through the axils highlight obvious stages of development in the plant. The フィボナッチフィボナッチフィボナッチ pattern of development mirrors the growth of the rabbits in Fibonacci’s classic problem; that is, the number of branches at any stage of development is a Fibonacci number. “Furthermore, the number of leaves in any stage will also be フィボナッチ a Fibonacci number” (Britton).

Palms are ideal specimens of the plants that display spiral phyllotaxis because their large leaves are prominently arranged (and therefore easily observed) on the trunk. Palm leaves are arranged in Fibonacci sequence spiral formation, overlap least and provide an “angular deflection between consecutive leaves that, together, comprise a photosynthetic surface optimally accessible to illumination” (Davis; Majumder and Chakravarti).

The initial leaves are often 180° apart. As the stem matures, it thickens, and the spiral pitch between leaves decreases. “The result of this process is that angular divergence of new leaves gradually approximates the golden angle. This gives rise to an approximate logarithmic spiral of touching leaves” (Green). On the oak tree, for example, フィボナッチ the branch rotation is a Fibonacci fraction, 2/5, which means that five branches spiral two times around the trunk to complete one pattern. Other trees with the Fibonacci leaf arrangement are the elm tree (1/2), the beech (1/3), the willow (3/8) and the almond tree (5/13) (Livio 113-115).

Okabe refers to Fibonacci phyllotaxis as evidence of natural selection, which eliminates plants whose growth patterns do not turn following the Golden Angle 2πα0 = 137.5◦ He says those フィボナッチ フィボナッチ which follow this process are “favored in nature” because the Golden Angle is structurally “the most stable” because it undergoes the least (though inevitable) phyllotactic structural changes (stepwise transitions between phyllotactic fractions) during early stages of the growing process to a mature plant (Okabe).

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ピボットプライス(P:Pivot Price)＝(H+L+C)÷3
H：前日高値、L：前日安値、C：前日終値
D1＝H－P：高値とピボットの差
D2＝P－L：安値とピボットの差
D3＝H－L：高値と安値の差
HBOP(High Break Out Price上方ブレークアウト)＝P+D2+D3＝2P-2L+H
R2（上値抵抗2：レジスタンス）＝P+D3＝P+H-L
R1（上値抵抗1：レジスタンス）＝P+D2＝2P-L
ピボット（P）＝（H＋L＋C）÷3
S1（下値支持1：サポート）＝P-D1＝2P-H
S2（下値支持2：サポート）＝P-D3＝P-H+L