From the December 2013 issue of Futures Magazine • Subscribe!

Using fractals in forex

Here, we will illustrate the profitability of trading a currency position using strategies based on the Fractal Market Hypothesis as discussed by Edgar Peters and Benoit Mandelbrot. We’ll look at the fractal from the slant of a time series analysis provided by Mandelbrot in 1963.

Mandelbrot found that cotton prices (1900-1963) were not normally distributed and instead showed clusters around the mean with a greater frequency of extreme variations (the tails) than that found in a normal distribution. This type of distribution is known as leptokurtic: A distribution that displays a positive value of excess kurtosis or sharpness of the peak of the graph of distribution. In other words, it has a higher peak than a normal curve and “fat tails” or higher density of values at the extreme end of the probability curve. Fat tails imply greater risk and suggest a nonlinear stochastic process. Assets that exhibit price jumps also display fat tail distributions. 

Mandelbrot’s analysis led him to coin the term “fractal,” although he did not provide a concise definition. Fractals are not limited to geometric patterns found in nature (some common fractals include seashells, snowflakes,  ferns, coastlines and broccoli), but can also describe processes in time. 

Fractals exhibit two quantifiable characteristics: Self-similarity and the fractal dimension. Self-similarity means that the parts are related to the whole. Peters puts it best: “The object or the process is similar at different scales, spatial or temporal, statistically. Each scale resembles other scales, but is not identical.”

An object is said to be self-similar if it looks “roughly” the same on any scale. For this discussion, we assert that the trends found on a four-hour spot euro candlestick chart are fractal shapes: Each trend roughly resembles other trends, but they are never the same. 

The fractal dimension measures how, in our particular case, a time series (a set of historical data) deviates. A line has dimension of 1, a plane has a dimension of 2, and a cube has a dimension of 3. A random line has a fractal dimension of 1.5. If a fractal dimension of a time series is greater than 1 but less than 1.5, then this particular time series exists between a straight line and a Gaussian random walk. Again, Peters proposes an excellent definition: “Regarding a time series, the fractal dimension measures how jagged the time series is.”

We accept the fractal market hypothesis as stated by Peter and discussed below. Various empirical studies show that financial assets produced skewed and fat tail return distributions (Mandelbrot, 1963; Fama, 1965; Hols, et al., 1991). In fact, the frequency distribution of currency returns has a higher peak and fatter tails than U.S. stocks or bonds. We define a short-term investment horizon as a period of less than five years and a long-term investment horizon of greater than four years.

Portraying the market in five basic points: 

  1. The market is stable when it consists of investors covering a large number of investment horizons. This ensures that there is ample liquidity for traders.
  2. The information set is more related to market sentiment and technical factors in the short-term than in the longer-term. As investment horizons increase, longer-term fundamental information dominates. Thus, price changes may reflect information important only to that investment horizon. 
  3. If an event occurs that makes the validity of fundamental information questionable, long-term investors either stop participating in the market or begin trading based on short-term information. When the overall investment horizon of the market shrinks to a uniform level, the market becomes unstable. There are no long-term investors to stabilize the market by offering liquidity to short-term traders.
  4. Prices reflect a combination of short-term technical trading and long-term fundamental valuation. Thus, short-term price changes are likely to be more volatile or “noisier” than long-term trades. The underlying trend in the market reflects changes in the fundamental (economic) environment. There is no reason to believe that the length of short-term trends is related to the long-term economic trend.
  5. If a security has no tie to the economic cycle, then there will be no long-term trend. Trading, liquidity and short-term information will dominate. 

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