The Fibonacci Sequence and the Golden Ratio in Stock Market Theories
Description: This technical analysis article introduces the Fibonacci sequence and Golden Ratio in Elliott Wave Theory as a useful technique for trading Forex.
In the 1940’s, R.N. Elliott enhanced his initial Wave Theory tenets to include the ratios between Fibonacci numbers that have now become one of the more popular Forex indicators used by traders.
Elliott made this change because he noted that the mass psychology underlying the markets’ movements displayed a tendency to repeat over time, and that they did so in numbers of waves that were included in the Fibonacci sequence.
As a result of this observation, he postulated that this phenomenon could be related to an important sequence of numbers that the mathematician Fibonacci tied to the reproductive increase of a theoretical population of rabbits.
Computing the Fibonacci Sequence
In the book Liber Abaci, published in the 13th century, Italian mathematician Leonardo Pisano Bigollo (who was also known as Fibonacci) posed a question regarding the reproduction of rabbits that went roughly as follows:
“How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?”
Producing the answer to this question involves calculating what is now known as the Fibonacci sequence. Basically, to generate the Fibonacci sequence, you can perform the following steps:
(1) Start with a pair of immature rabbits, one male and one female.
(2) Then you wait one month before breeding them.
(3) Then you wait one month for them to have a pair of babies - one girl and one boy.
(4) Then you wait one month when the original pair have babies, but their babies do not.
(5) Then you wait one month when the original pair have babies, and so do their first set of babies, but not their second set.
(6) Repeat the process over and over and over again until you get tired.
(7) Assume that no rabbits die.
(8) Count how many pairs of rabbits there are at the end of each month.
Steps (1) and (2) generate the first two numbers in the sequence, which are 1 and 1.
Step (3) generates the next number which is 2 (one original pair and one pair of baby rabbits of the opposite sex).
Step (4) generates the next number which is 3 (one original pair, their first pair of babies, and a second pair of babies from the original pair).
Step (5) generates the next number which is 5 (one original pair, their first pair of babies, their second pair of babies, and a new pair of babies from the original pair and the first pair of babies.)
This process can then be repeated indefinitely.
Eventually, you might notice an important short cut, which is that you can simply compute the current number of rabbit pairs by adding together the previous two numbers. This generates the infinite Fibonacci sequence as follows:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
In which the sequence is created numerically by taking 1 and 1, and then adding them together to make 2, and then adding 1 and 2 together to make 3, and so on.
Fibonacci Sequence Ratios and the Golden Mean
Interestingly, as this famous numerical sequence progresses, a very unusual thing happens. The ratio of one number divided by the next number in the sequence approximates 0.618, which yields roughly 1.618 when inverted.
This is known as the so-called Golden Mean or Golden Ratio, and mathematicians have found the fact that these two numbers, which are reciprocals of each other, have the same three decimal places to be rather fascinating. Furthermore, numerous theories about the significance and power of this numerical phenomenon have been proposed.
Even more interesting is the observation that the ratio of one number to that seen two further in the sequence approaches 0.382, while three further approaches 0.236.
Here is a further mathematical look at the Fibonacci Sequence and the Golden Ratio:
Using Fibonacci Ratios in Elliott Wave Theory
With the addition of the ½ or 0.5 ratio, and also the (1-0.236) = 0.764 ratio used by some analysts, the popular technical analysis technique of computing Fibonacci retracements was ready to be included by Elliott as part of his Wave Theory:
First of all, Elliott noted that market movements or waves tended to occur in sets of Fibonacci numbers. Elliott also observed that major market moves would tend to be corrected by a degree that approximated a Fibonacci ratio.
Accordingly, by first multiplying the extent of the initial move by these ratios and then projecting them in the opposite direction from that move off of the final price of the move, he proposed that traders could compute a series of likely Fibonacci retracement targets.
Furthermore, Elliott noted that impulsive waves in an unfolding trend tend to relate to each other either on a 1:1 or 1:2 ratio, or by adding the Fibonacci ratios, to get 1:1.236, 1:1.382, 1:1.5, 1:1.764 and so on. These form the set of Fibonacci projection ratios.
By using these ratios calculated for an observed impulse, Fibonacci projections can then be computed off of the end of an intervening correction to determine the likely extent of a subsequent impulse. While this projection technique can be helpful in providing targets when one impulse is known, it can be even more accurate when two impulses have already occurred.
*Jennifer Gorton is the content manager of ForexIndicators.net. Her main task is to make sure all the articles on her site are very educational and useful while also planning out new sections for more advanced trading tool information. She has recently expanded her knowledge on the mathematical aspects of her indicators and is quite thrilled to share the information she's discovered.