Introduction

Leonardo Pisano, better know as Fibonacci, explained the development of natural growing phenomenon through his famous numerical sequence. He proved that this series was highly connected with the growing of dynamics structures, and the most important use is relationated with its ratios.

The objective of the present work is to demonstrate that the application of these rules, have an important probability of success in financial markets, and principally in FOREX.

We start with the premise that the human society is a dynamic system, and its behavior is represented in financial markets through prices.

That is the reason why we will try to prove that there is an important probability to predict the behavior of prices in Forex, joining Fibonacci numbers with Zig Zag Oscillator.

So, we will try to determine the objectives zones, or where the prices tend to go using Fibonacci. We will study the prices corrections against the major trend.

The Method: Fibonacci, and his legacy

In the beginning, we start using the most important correction ratios discovered by Fibonacci. These ratios came from the famous Sequence.

Many contributions were applied to mathematics science by Fibonacci, but the most relevant discover was denominated by the French mathematician, Edouard Lucas, as Fibonacci Sequence in the XIX Century.


The sequence. Properties and principal characteristics

This sequence is a rule that explain the development of natural growing phenomenon. Formed by adding the last two numbers to get the next one.

The formula is:


The Fibonacci Sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc.…

Fibonacci proved that this sequence could be found in the evolution of many natural phenomenons. He used as example the rabbit reproduction process. He wanted to know how many rabbits will be born in a year, knowing that:

a) A couple of rabbits could birth since the fist month, but the others couples just can do it since the second month.

b) Each labor brings two new rabbits as result.

If we suppose that any rabbit die, the process will be the like this:

1. In the first month there will be born two rabbits. So, we will have two couples.

2. During the second month, the initial couple will born another couple, and then will be three pair of rabbits.

3. In the third month, the initial couple, and the second one, will produce new couples. Then, there will be five couples.

Continuing with the present analysis, we could see in the next table the results of the rabbit’s couples forming the Fibonacci Sequence.


Despite all this, we find the major utility of the sequence in these fundamentals properties:

1. If we divide two consecutives numbers, 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, etc. We could find that the results tend to 0.618.

2. If we divided two non consecutive numbers from the sequence, ½, 1/3, 2/5, 3/8, 5/13, 8/21, etc. We could see that the result obtained tend to 0.382.

3. If we calculate the division between any numbers of the sequence to the next lower, 21/13, 13/8, 8/5... the results tend to 1.618, which is the opposite of 0.618.

4. If we calculate the division between any numbers of the sequence to the higher low non consecutive, 21/8, 13/5, 8/3... the results tend to 2.618, which is the opposite of 0.382.

E.g.; 144 / 233 = 0,618 144/89= 1.6179


The ratio 1.618, or the opposite 0,618 were denominated by the Old Greeks “Golden ratio” or “golden section”, and they are represented with the Greek letter phi, referenced by the greek author, Phidias. Chirstopher Carolan mentions in his book that Phidias was the author of the Athens statues in the Parthenon and The Zeus in Olympus. He considered very important the phi number in Art, and in nature.

This ratio, who’s opposite is the same number more the unit, characterize all the progressions of this kind, where ever it is the initial number.

The most important ratios are 0,618 and it’s opposite 1,618, but not the only ones. We can continue on the ratios derivation of the Sequence, just increasing or decreasing the distance between the Fibonacci numbers.

So, each number is relationated with the higher next trough the 0,382 ratio, and with the lower next with the opposite ratio, 2,618.

E.g.: 144/377=0,3819 144/55=2,618

In the same way, the division between a number and the third next, bring as a result, 0,236, and the proportion between a number and the third lower next is 4,236.

E.g.: 89/377=0.236 144/21=4,238


The same occurs with 0,618 and 1,618, these ratios are more exactly, when we use higher fibonacci numbers. The next table shows some examples:

Carolan emphasized that the Fibonacci ratios could be order as follows: 0,146, 0,236, 0,382, 0618, 1, 1,1618, 2,618, 4,236, and 6,854. Then we could find and additive sequence with the properties of the Fibonacci Sequence, because each number is the sum of the immediately two before, and moreover, each number is 1,618 times the number before.


Fibonacci’s applications

The verification of the Fibonacci Sequence in many real phenomenon’s, makes lots of people decide to study the relationship between these mathematics of nature and the behavior of the financial markets.

May be this is the most curious and captivating, so it has be proved that the Fibonacci Sequence appears in nature, forming physical structures and defining the process of chance of this dynamics structures. Many authors have mentioned this concept in theirs books.

At the end of XIX Century, a botanist calls A. H. Church, had discovered, studying the sunflower the presence of Fibonacci. His seeds are arranged around the middle in 89 curves, turning 55 of these in one direction and the others 34 to the opposite direction.

From here, the botanists have found Fibonacci numbers in different pieces of Nature. For example, the margarita forms spiral model similar of the sunflower in the middle of his flower. There are also many varieties of flowers where the numbers of petals are Fibo numbers.

A mathematician from Arizona University, Alan Newell, and his pupil Patrick Shipman have been studying recently the cactus to determinate the reason why this numeric pattern is universal. These researchers analyzed the form of this plant, his skin size, and another biomechanics that surge in his growth. When their introduced all of the data in the computer, found by surprise, that the more stable configuration followed the forms based of the Fibonacci Sequence.

We can find other applications, for example the spiral that many trees developed in their branches; the number of little branches in a big one, and the follow of the same vertical is a Fibonacci number, if we use to calculated one of the to branches.

The Fibonacci numbers appear also, in the human body. The men have five appendixes (two arms, two legs, and a head); each arm and each leg are divide in three parts, ending each of them in five appendixes (five fingers), divide each of them in three little phalanx, except two of them which have just two. In the same way, the head has three outstanding characteristics (two ears and a nose), and three inlaid characteristics (two eyes and a mouth). In the end, the human body has five physical senses: sight, ear, sense of smell, taste, and touch.


”The human body presents the golden number or phi”. Leonardo Da Vinci, in his famous picture, the Vitrubio man, illustrated the Luca Pacioli book “Divine Proportion” edited in 1509. This book, describes witch must be the proportion of artistic creations. He proposes that a human figure where each part of the body must respect a specific proportion to be harmonic. This perfect man, for Pacioli, is based on the following mathematical calculation: the high of the men, (side of the square) divided the distance between the navel and the extreme of the extended hand (circle radio), represents the divine number.


Also, many animals body have a trunk and five appendixes, (head and four legs); birds have 5 projections too: a head two legs, two wings.

Fibonacci is also present in music: see for example the piano. The division of the keyboard in scales of eight white keys and five black ones; the black keys are distributed all along the keyboard in groups of three and two. A complete keyboard has eleven scales, and could have one more key, meaning 89.
The chords that allow us to identify any tone are formed by the first, third, fifth and eight note of the scale.

Since professors Church and Hambridge, the interest on Fibonacci numbers by many researchers end in the creation in 1963 of the Fibonacci Society in California, formed by mathematics, witch main objective is exchange ideas and stimulate research on Fibonacci’s relationship with nature.

It has been proved that the Fibonacci sequence is highly connected with the progressive development of dynamical structures, an as society is a dynamical system, human history could be running according to this Nature Law, based on the proportion 3-5 or 0,618; if we add to this concept, the idea that financial markets are the reflect of mass behavior, we can conclude that Fibonacci sequence could be applicable to those markets.

In the next session based on these concepts, we will develop a model to prove the proposed objective on this work.

Application in the Objective Market

Definition of the sample


Once chosen the objective market, the study was focused in 4 (four) currency pairs, in the Forex International Market.

In order to make it as objective as possible, the study was based in those pairs with higher volume trade in the Forex market, because they accumulate the 85% of the daily transactions.


• Pair EUR (Euro)/USD (United States Dollars)

Since it’s apparition in December 1999, the Eur, soon replace the German Mark, and becomes the second currency in the world, getting day by day more importance. The strength of EUR is based on the power of the European Economic Community, no matter how many political factors may affect it.


• Pair GBP (Great Britain Pound)/USD

It was the reference currency till Second War, and most of the transaction involving it. Took place in London, the biggest international market regardless his small volume during American market sessions.


• Pair USD/JPY (Japanese Yen)

This is the third currency trade in the world, making market liquid 24 hours a day. Notice that oriental economy moves according to Japan, and so, Yen is very sensitive to oriental agricultural production, technological factors, salaries and NIKKEI.


• Pair USD/CHF (Helvetic Confederation Franc)

This is the other European currency not included in Eur or G-7, but at the same time, it seems favour related to politic uncertainly of the European Community. Practically, we can say that Swiss Franc moves almost the same way that EUR in relation with the USD.


Sample: scope

This work was developed based in the following time frames, because they represent a prominent quantity of subjacent quotation time, and allows reducing “noise”, in short time:

• Daily sessions: 24 hours of transactions or quotations. We use it to deeply analyze the trend in Medium Term (weeks) and Long Term (months).

• 4 (Four) hours sessions, that gives us more detail of temporality, due to in a 24 hours day trading there are moments with higher transaction volume, like the opening or close of the biggest world financial centers (Tokyo, London, Frankfurt and New York).

Anyway, we invite the readers to extend this analysis to sessions with more or less duration, where you can find similar results.


Field work

Once introduced Leonardo Pisano and his invaluable contribution to science, we will stop at his more important ratios, specifically in the target zones created because of them.

Based on what we can see in financial markets, there are retracements or backward movements in a certain percentage. According to Fibonacci, in a strong tendency, a minimum retracement generally address in its first impulse to the zone of 23.6% of the rally; and in case this zone is broken, the quotation usually goes to the zone near the 38.2%, then to the 50 % zone, and in a weaker tendency, the maximum retracement could reach the 61.8%; but if this point is broken, the quotation will continue to a point not consider by Leonardo Pisano, but very important to remark, because of the results given in our diary work, the 76,4 % to finally reach the 100 %.

Once the quotation runs over the 100 % retracement, and confirms that point, we can suppose that the dominant tendency has changed, and price will look for other objectives, that according to Fibonacci, will be at first place the 161.8%, then the 261.8% and finally the 423.6%.

In Table 2, you can see Fibonacci ratios, coming from the division of each number of the numerical sequence he developed by the one before it.

Ratios of Fibonacci

Now we propose combine the price Fibonacci retracements with the Zigzag Oscillator, in a major defined trend, to corroborate the accomplishing of the target quotations.

The popularity and use of Zigzag oscillator are based in three main characteristics: is a good “noise” filtering; it represents the main trend clearly, and is a simple indicator for the market price final interpretation.

However, this oscillator has as main disadvantage his natural dynamic: the last line of its draw marking trend could be tricky and needs confirmation.

It works simply presenting the major movement by connecting picks (high prices) and depressions (low prices) with straight lines.

The inclination parameter of the slope in the specific quotation specifies the percentage that this price has to move to draw a new line or Zigzag line.

Its formula is:

ZZO= 100 * (CL-BASE)/BASE

Where base is the initial price (maximum or minimum) or the Zigzag leg CL or Last Closing of the before session

This oscillator filters the changes in the subjacent chart, smaller than the quantity specified in the inclination parameter of the scope. It only shows significant changes. The minimum price movements are fixed as percentages, and could be based in close price, or in maximum/minimum ranges.

For example, the Zigzag established in a 10% respect to the OHCL (Open-High-Close-Last) candles, will draw a line that will only change direction if the changes between maximum and minimum exceed the 10%. This means any smaller variation will be ignored.

Then, after we defined the system used to calculate the bullish or bearish rallies objectives, through Zigzag Oscillator, we start the empiric confirmation of the information for each pair under study, main subject of the next session.