The Fractal Dimension

Fractal stock market analysis began with two books by the creator of the new mathematics of fractal geometry, the late, great Benoit Mandelbrot (1924-2010). His original field of applied research was in econometrics and financial models, applying ideas of scaling and self-similarity to arrays of data generated by financial analyses. His two finance books are Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (1997) and The Misbehavior of Markets: A Fractal View of Financial Turbulence (2007) with Richard L. Hudson.

I discovered this work more than 10 years ago. All common technical analyses – moving averages, point & figure, candlesticks, etc. – are well-known and programmed into hundreds of quantitative models at the big hedge funds. I don’t want to try to compete with hundreds of PhD’s using supercomputers or millions of traders who can pull up an indicator on stockcharts.com. But almost no one is using fractals – or if they are, they never talk about it.

Classical geometry is based on the study of smooth, continuous curves and shapes. Fractals are irregular and complex patterns that represent the way the world really is. Mandelbrot created fractal geometry to deal with these real-world patterns. There’s a hard way and an easy way to describe what a fractal is. Mandelbrot defined them this way: “A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension.”

That’s the easy way – kidding! The easy way is this: “A fractal pattern is made up of smaller patterns identical to the large one that are in turn made up of even smaller identical patterns.” Fractal patterns are self-similar in all time-frames. The smaller components have the same basic shape and pattern as the larger components. Fractals are common in nature, in trees, snowflakes, rivers, and shells.

So how does this apply to financial markets? Fractal market analysis applies chaos theory to understand the order hidden within seemingly random chaos and determine the probability of future events.

Financial markets are fractal. An unlabeled price chart looks the same whether it is a monthly, weekly, daily, hourly, or even a five minute chart of the trading action. They all look the same. Markets are fractal.

If a market trended perfectly over a time period – every price change exactly equal – it would have a fractal dimension of 0. If a market was completely chaotic over that same time period, with no relationship from one price to the next, the fractal dimension would be 100.

In practice, trends last until they stop in the 20-30 area, and then the trend consolidates in a chaotic way until the fractal dimension gets back over 55. Consolidation sometimes continues until the fractal dimension is over 65, but it almost never gets over 75.

It’s very helpful to know whether a market is trending or consolidating, e.g., whether a drop is the beginning of a new downtrend or just a consolidation of the prior uptrend. So the main advantage of knowing the fractal pattern is identifying the balance point where the potential energy may resolve in one direction or another.

Usually, trends resume in the same direction they were going before the consolidation began. If the fractal dimension gets fully consolidated over 55 and then drops back through 55, signaling a new trend, it’s really important to be able to tell early that the direction has resumed or has reversed.

Most analysts look for direct linear relationships in the stock market – Input: sales, profit margins, earnings, growth rates, price action, liquidity; Output: stock prices or stock price changes. But the stock market actually is a chaotic, non-linear system. We get an advantage by understanding the fractal dimension that might last for the next 10 to 20 years.