Fibonacci, Schmibonacci
The Fibonacci sequence of numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...) was discovered by the Italian mathematician Leonardo de Pisa in the 13th century, while considering the increasing size of a theoretical population of rabbits through many generations. The first two numbers in the sequence are 0 and 1 and each successive number in the sequence is the sum of the previous two numbers.The Fibonacci ratio between any two successive numbers in the Fibonacci sequence is known as phi. The ratio between a number in the sequence and the next higher number converges to approximately 0.618 (also known as the Golden Mean, Section or Ratio). The ratio between a number in the sequence and the next lower number converges to approximately 1.618 (the inverse of the Golden Mean).
These numbers and ratios have been found in nature, for example, in the number and arrangement of seeds and petals on flowers, and in the logarithmic spirals found in shells. There are good reasons for this, such as the optimum packing of seeds, optimum presentations of petals to light, etc.
In the financial markets, some technical analysts believe that fibonacci ratios exist, chiefly in the form of "retracements" of prior moves. For example, if a market made a high at 3000 followed by a low at 2000 and then a subsequent high at 2618, we would have a fibonacci retracement of 618, or 61.8% of the move from 3000 to 2000. These levels are considered to be important places of support and resistance in the markets, points where the market is said to be more likely to change direction.
Of course, besides .618, people have gone on to claim that a variety of fibonacci retracements levels exist in the markets, such as .382 (.618 squared), .236 (.618 cubed), .500 (1/2), 1.000 (1/1), and 2.000 (2/1). As a side note, Gann, a well-known trader, thought retracements of one-half (50%) very important, as well as quarters (25%, 50%, 75%), and thirds (33%, 66%).
In my opinion, these retracements do not occur in the financial markets any more frequently than would be expected by chance. Unlike many complex, esoteric theories in the financial markets, it is a relatively straighforward task to define and measure retracements, and thus test the hypothesis that these levels are indeed important. Highs and lows can be defined (in hindsight) by testing to see if a given point in the center of a range is the max or min of the range.
Using this method of choosing highs and lows, pick an x-day high, followed by a y-day low, followed by a z-day high. Divide the amount of the y to z retracement by the initial move from x to y to get the retracement percentage. [Note: you could also look at low-high-low sequences.]
Do this for every x-y-z sequence that occurs in the data. Once you've done this, look at the chart of a frequency histogram of the retracements you've calculated. This is just a graph of the number of occurrences of retracements at each particular level. The outline of the histogram will probably be fairly "noisy", and you'll need a LOT of data to get anything approaching a smooth curve. If you do have enough data and the proper bin sizes, you will see the bulk of the data falls along the lines of a normal bell shaped curve. This suggests that the distribution of retracements is something approaching a random, gaussian distribution.
However, if the technical analysts are correct, you should see "bumps" on the histogram curve at specific points. In other words, we would expect to see significantly more retracements occurring around the levels of .618, 1.618, or other "important" levels than at the surrounding levels. I've tried this, but in my experience, I have not seen any levels that looked more important than any others. True, I haven't tried every possible x, y, and z value, but after a few dozen tries, including ALL of the levels touted by practitioners, one tends to get the basic idea.
I tried this with 20 years of daily data, and frankly, more data would be nice, as you want a LOT of data in each bin. If I had more interest in pursuing the topic, for a given x-y-z combination I would run 10 to 20 years worth for all the major commodities and combine the results. Set the "bins" for the histogram so that you have enough fineness of detail to see what's going on, and center the important values you are looking for (i.e. .618, etc.) within a bin.
Why is there so much talk about fibonacci retracements? On a conceptual level, I believe that like many other superstitions, it is born out of a response to fear and uncertainty. People latch onto ideas that may appear sound without really examining them. On a more practical level, I believe people use a lot of retracement levels, and they end up interpreting any given retracement as belonging to whatever "important" level is closest. For example, a 55.9% retracement falls exactly in the middle of the often cited 50% and 61.8% retracements. By all rights, this should be no mans land, an absolute non-event, yet fib traders will inevitably label any successful rebound at this level to be a hit for either the 50% or 61.8% levels. In other words, there is no level that will not be labeled as a success by believers. Throw in enough numbers and enough leeway and you can attribute the results to just about anything.