Posted In Technical Indicators 1 comment. So what is a fractal? It is a rough or fragmented shape that can be split into parts, each of which is at least similar to a reduced size copy of the original. The following video illustrates the beauty of fractals in 3D and is well worth watching in full screen:. Notice how no matter what scale you view the Koch Curve in it looks very similar?
This characteristic is called self similarity and defines a fractal shape. Can you see anything strange about the chart below?
Without being told would you have known that the left half of the chart above was 5 years of monthly bars and the right half was 15 days on 30 minute bars? Probably not, because price movements look similar no matter what time frame we are viewing them in, this is self similarity and why the financial markets are considered fractals.
The Wiener process is also a fractal and looks very much like stock price movements. It is a continuous time stochastic process that charts Brownian Motion and is used in the mathematical theory of finance as well as the Black-Scholes option pricing model.
It is clearly self similar and the average features of the function do not change while zooming in. Usually we think of things in whole, rather than partial dimensions and at first I found this a foreign concept, but try thinking of it this way: A shape like a stock chart is too big to be one dimensional but too thin to be two dimensional, so its Fractal Dimension results in a reading between 1 and 2.
It is easy to understand the number of dimensions in a line, square or cube. A line has 1 dimension — length, a square has 2 — length and width and a cube has 3 — length, width and depth. However if we use this very simplistic thought process to try and reveal the Dimension of a fractal like the Sierpinski triangle then we run into problems.
It is clear that we need a more intelligent approach for identifying the dimension of self similar shapes so lets look at it in another way: If we break a line segment into 4 self similar parts of the same length, a magnification factor of 4 is needed to reveal the original shape. If we break a line segment into 7 self similar parts, a magnification factor of 7 will yield the original shape… 20 parts, magnification of 20 etc.
If we break a square into four self similar sub squares then a magnification factor of 2 is needed to reveal the original shape. Using this thought process, lets look again at the dimension of the Sierpinski triangle.
How do we find the exponent in this case? Now we are going to need logarithms log.
Log reveals the power that a number needs to be raised to in order to produce a given result. Unless otherwise stated the base number is 10, therefore: The Sierpinski triangle is made up of 3 self similar pieces that require a magnification factor of 2 to reveal the original shape.
The Sierpinski triangle also contains 27 self similar pieces that require a magnification factor of 8 to reveal the original shape so:.
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So the Fractal Dimension is really a measure of how complicated a self similar shape is. A line is smaller and more basic than that a square, while the Sierpinski triangle sits somewhere between the two.
However all three have the same number of self similar parts; they can all be divided to infinity.
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He used this method as a component of his Fractal Adaptive Moving Average FRAMA and presented it again as a standalone indicator in the June edition of Technical Analysis of Stocks and Commodities — Fractal Dimension As A Market Mode Sensor. Covering a price curve with a series of small boxes however is far too cumbersome.
But because price samples are uniformly spaced each bar is 1 day, 1 week, 10 min etc Ehlers decided that the average slope of the curve could be used as an estimation of the box count. This is far less complicated than it sounds as the slope is found by simply taking the highest price over a period minus the lowest price during that period and dividing the result by the number of periods.
I have put together an Excel Spreadsheet that calculates the Fractal Dimension and made it available for FREE download. Find it at the following link near the bottom of the page under Downloads — Technical Indicators: The lower the Fractal Dimension the closer a stock chart is to a straight line and therefore the stronger the trend.
High readings on the other hand reveal a complex fractal; the shape of a range bound market. These two different market types require very different strategies in order to maximize profits and minimize losses. Ehlers uses this measure in the FRAMA to dynamically adjust the alpha of an exponential moving average so that it reacts quickly in a trending market and slowly when prices are congested.
Accurately being able to identify the strength of a trend has endless uses and therefore is worthy of much research. Woodshedder wrote thought provoking article On Fractals and Market Crashes. Here is an interesting open source platform called Fragmentarium that allows to create Fractals yourself like this:. Home About Derry Brown My Story Contact Testimonials. Topics ETF HQ Report Investing Education Stock Market for Beginners Technical Analysis Technical Indicators.