The American accountant R.N. Elliott published in 1939 a number of articles in which he explains his theory about the behaviour of the stock markets. He supposed that the continuous fluctuations in the market are caused by the states of mind of the mass, which are wavered between pessimism en optimism.
figure 1
figure 2
figure 3
In figure 1 we see an elementary undulation. First the price rises fast and then relapses. During a longer period of time we recognize a pattern like (stylistically) shown in the second figure. We see three ascending and two descending intervals on the left side of the apex and 2 descending intervals and 1 ascending interval on the right side of the apex. After intensive study over a longer period of time about the behaviour of the stock market of Wall Street, Elliott found out that in every exchange diagram this characteristic pattern existing of 5 intervals left from the top and 3 right from the top is repeated contiguously and on each scale. Let's take a closer look at this phenomenon.
If we project figure 1 and 2 on each other we see that the top and the edges coincide. You can regard figure 2 as a refinement of figure 1 caused by measuring over a longer period of time. (Note that this is a stylistic picture of the reality; the form can vary, and there may be 20 peaks, but the pattern of 5 intervals left and 3 right from the top is always there.) Figure 3 is a stock market graph over a still longer period of time. We see that this graph is a refinement of figure 2. We may continue this process over an unlimited period of time. Let us put the number of intervals in a table:
figure
Right
Left
Together
figure 1
1
1
2
figure 2
3
5
8
figure 3
13
21
34
We recognize the sequence of Fibonacci.
Let's find an explanation for that.
Note that there are two kinds of intervals. The first kind (A)
will be divided into 3 parts and the other (B) into 5 parts.
We split up the previous table:
figure
Right (A)
Right (B)
Left (A)
Left (B)
figure 1
1
0
0
1
figure 2
1
2
2
3
figure 3
5
8
8
13
The numbers of the first table follow immediately from this table.
In the kth row of this table we suppose that the numbers are F3k-4, F3k-3, F3k-3 and F3k-2.
This is correct for k=1, because according to the recursion F0 = 0 and F-1 = 1.
Let N be a positive integer. Suppose that the supposition was already proven for k<=N. We want to show that the supposition is also true for k=N+1.
Division of the intervals into 3 and 5 pieces results in the N+1-th row with:
Right (A): F3N-4 + 2F3N-3 = F3N-2 + F3N-3 = F3N-1;
Right (B): 2F3N-4 + 3F3N-3 = 2F3N-2 + F3N-3 = 2F3N-1 + F3N-2 = F3N;
Left (A): F3N-3 + 2F3N-2 = F3N-1 + F3N-2 = F3N;
Left (B): 2F3N-3 + 3F3N-2 = 2F3N-1 + F3N-2 = 2F3N + F3N-1 = F3N+1.
This shows, that if a row in this table consists of Fibonacci numbers, then the next row also contains Fibonacci numbers. Thus our supposition is correct for every row.
