Applications of Fractals
In economy, perhaps the most important thing is to be able to predict more or less accurately what will happen to the market after some time. Until very recently, the dominant theory that was used for this was the so-called Portfolio Theory. According to it, the probability of various changes of the market can be shown using the standard bell curve:
Assuming this theory is accurate, we can conclude that very small changes happen most often, while very big changes happen extremely rarely. However, this is not true in practice. While on the bell curve, one can observe the probability of rapid changes to approach zero, they can, be seen almost every month at the real market. Recently, in about 20 years after discovering fractals, Benoit Mandelbrot introduced a new fractal theory that can be used much more efficiently than the Portfolio Theory to analyze the market. Consider taking a year of market activity and graphing the price for every month. You will get a broken line with some rises and falls. Now, if you take one of the months and graph it in a more detailed way with every week shown, you will get a very similar line with some rises and falls. If you make it more and more detailed by showing every day, every hour, and even every minute or second... you will still get the same, only smaller, rises and falls. There is your Brownian self-similarity! Mandelbrot came up with a method of creating fractals that fit the above description. He based in on simple generator iteration and created base-motif fractals that could model the market. In the February 1999 issue of Scientific American, he published some of his fractal "forgeries" next to real market lines, showing how remarkably similar they were. In his method, you start with a shape, called the generator. The generator must be composed of three line segments, in order to contain both a rise and a fall in price:
You then take this exact picture and substitute every line segment with it:
Continuing to substitute (see iteration) you get something looking like this:
Now, compare it to a Portfolio Theory model:
Which one seems more appeasing to you?
Chaos : Applying Expert Techniques to Maximize Your Profits (Wiley Finance Editions)
This book is entirely devoted the economical applications of fractals and chaos. Although not very hard to read in general, this book requires some knowledge of both fractals and economy.