The Fibonacci Series and Trading

The Fibonacci Series forms the essential mathematical basis of Elliott Wave theory.

The existence of the Fibonacci Series was first announced to the world by the twelfth century Italian mathematician, Leonardo Bigollo Fibonacci of Pisa. Whilst taking a sabbatical in the Middle East (the birth place of contemporary mathematics), it was in Egypt when studying the Great Pyramid of Gizeh that he uncovered a unique and astounding mathematical sequence of numbers, the importance of which was not fully appreciated until recent developments in the fields of non-linear mathematics – the mathematics of the improbable.

Indeed, to his credit, Fibonacci is the person who first adopted the decimal system and introduced Arabic arithmetic to the Western world in his treatise the Liber Abaci. Before this Roman numerals were still being used. The term Fibonacci series or Fibonacci sequence was in fact not used until the 19th century when it was first used by the French mathematician and number-theory specialist, Edouard Lucas.

The Fibonacci Series

The Fibonacci Series itself is quite a simple concept, based entirely on whole numbers. Starting with the number 1, the series unfolds as follows:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …

The series is unending and embodies the following properties:

1. The sum of any two consecutive numbers in sequence forms the next number of the series. E.g. 1+2=3; 2+3=5; 3+5=8; 5+8=13; 8+13=21; 13+21=34 … and so on.
   
   
2. Dividing any number into the second number higher in sequence, yields a quotient of 2 with a remainder equal to the number found immediately before the divisor in the sequence. E.g. dividing 34 by 13, gives a quotient of 2 with a remainder of 8.
   
   
3. The ratio of any number to the next higher number in the series is always 1 to 1.618 (to 4 significant figures). E.g. dividing 610 by 377 yields 1.618037; dividing 987 by 610 yields 1.618033.
   
   
4. The ratio of any number to the number immediately below it in the series is always 0.618 to 1.
   
   
5. The ratio of each number to the second number below it in the series is always 1 to 2.618.
   
   
6. The following formula relates each number of the series to its immediate predecessor: n-1 = ½(v5 - 1)n = 0.618n
   
   
7. The following formula relates each number of the series to its immediate successor: n+1 = ½(v5 + 1)n = 1.618n
   
   

The Fibonacci Series in Nature and the Arts

In Nature’s Law, published in 1946, and subtitled The Secret of the Universe, Elliott makes reference to the Great Pyramid of Gizeh as exemplifying the mathematical relationships of the Fibonacci Series. Indeed, the height of the pyramid is exactly 5,813 inches (ref. the consecutive Fibonacci numbers 5, 8 and 13) – coincidence or not? – and the ratio of the base of the pyrimad to its height is exactly 61.8 to 100. But it doesn’t end there – the so-called Golden Ratio (i.e. 0.618:1) can be used to explain some of the most complex geometry of the pyramid which otherwise would remain totally obscure.

Further, the ratio can also be found in such diverse areas as the shape of the Parthenon in Athens, the paintings of Leonardo Da Vinci and other great artists, the music of Bach and the rhythmic pulse of the poetry if Keats. In other words, throughout the fine arts and architecture. This is in addition to a multitude of natural phenomena.

In Nature’s Law, Elliott cites the following examples:

• Our personal anatomy follows the Summation Series, in accordance with the five plus three pattern which characterises a complete stock market cycle. From the torso there are five extremities: the head, two arms and two legs. Each extremity can be broken down to the “cycle of the next smaller degree”, of three sections. The legs and arms can be broken downinto the next smaller degree also, the arms terminating in five fingers and the legs in five toes. These fingers can be further extrapolated into three sub-sections, the 5-3 pattern remaining throughout. (One may find an anomaly with regard to the big toe. However, in the case of monkeys, apes and other similar forms, the fingers and the toes are similar in anatomical structure. The big toe of a monkey breaks down into three sub-sections as do the other minor extremities.)
  
  
• In music we have the octive of 8 diatonic notes. We have the chromatic scale with 13 notes, while the staff has five lines. There are three basic elements in music: melody, harmony and rhythm. Natural harmonies follow the Golden Ratio.
  
  
• There are three primary colors. The blending of these three primaries produces all other colors.
  
  
• The Washington Monument in Washington DC has a capstone with a base measuring 34 feet square with a height of 55 feet. The ratio is 0.618. The base of the shaft of the monument measures 55 feet square, while the rim of the shaft measures 34 feet. The foundation has 8 steps. There are 8 windows. The monument has three basic components: the base, the shaft and the capstone, the latter being in the shape of a pyramid.
  

However, of greater interest to the student of financial market movements – and it is emphasised that such movements are the result of emotional rather than rational responses to news events – is the manner in which the Fibonacci Series can be found in the emotional cyclical behaviour patterns.

Elliott’s recognition of the true forces behind share price behaviour gives additional credibility to his Wave Theory, and he felt that the typical investor’s general lack of success in the stock market made it all the more important, stating:

“Despite the attention given the stock market, success, both in the accuracy of prediction and the bounties attendant thereto, has necessarily been haphazard because those who have attempted to deal with the market’s movements have failed to recognise the extent to which the market is a psychological phenomenon. They have not grasped the fact that there is regularity underlying fluctuations of the market, or, stated otherwise, that price movements in shares are subject to rhythms, or an ordered sequence. Thus market predictions, as those who have had any experience in the subject well know, have lacked certainty or value of any but an accidental kind.”

“But the market has its law, just as is true of other things throughout the universe. Where there is no law, there could be no centre about which prices could revolve and therefore, no market. Instead, there would be a daily series of disorganised, confused price fluctuations without reason or order anywhere apparent. A close study of the market, however, as will be subsequently disclosed, proves that this is not the case. Rhythm, or regular, measured, and harmonious movement, is to be discerned. This law behind the market can be discovered only when the market is viewed in its proper light, and then is analysed from this approach. Simply put, the stock market is a creation of man and therefore reflects human idiosyncrasy.”

Thus the Elliott Wave Theory, with its foundation in the Fibonacci Series, can be seen to represent man’s response to external stimuli, showing itself in price fluctuations of the stock market and other financial markets, where mass behaviour patterns react to the forces of supply and demand. Once the waves are comprehended, the theory can be applied to any market movement, since the same principle which affects the price of stocks, also affects those of Government Securities, Metals, Grains, Coffee, Cotton (i.e. commodities), international currencies (the Forex markets), and other mass markets.

Forward to: The Fibonacci Series Applied