Source of some of this article is Wikipedia.
A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,”[1] a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning “broken” or “fractured.”
A fractal often has the following features:
* It has a fine structure at arbitrarily small scales.
* It is too irregular to be easily described in traditional Euclidean geometric language.
* It is self-similar (at least approximately or stochastically).
* It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
* It has a simple and recursive definition.
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics.
Benoit Mandelbrot, in one of his pioneering articles on the problems with linear based market predictions states:
“The risk-reducing formulas behind portfolio theory rely on a number of demanding and ultimately unfounded premises. First, they suggest that price changes are statistically independent of one another: for example, that today’s price has no influence on the changes between the current price and tomorrow’s. As a result, predictions of future market movements become impossible. The second presumption is that all price changes are distributed in a pattern that conforms to the standard bell curve. The width of the bell shape (as measured by its sigma, or standard deviation) depicts how far price changes diverge from the mean; events at the extremes are considered extremely rare. Typhoons are, in effect, defined out of existence.
Modern portfolio theory poses a danger to those who believe in it too strongly and is a powerful challenge for the theoretician. Though sometimes acknowledging faults in the present body of thinking, its adherents suggest that no other premises can be handled through mathematical modeling. This contention leads to the question of whether a rigorous quantitative description of at least some features of major financial upheavals can be developed. The bearish answer is that large market swings are anomalies, individual “acts of God” that present no conceivable regularity. Revisionists correct the questionable premises of modern portfolio theory through small fixes that lack any guiding principle and do not improve matters sufficiently. My own work – carried out over many years – takes a very different and decidedly bullish position”
Okay, okay, you are shaking your head what in the world does this have to do with actual investing? Yea, yea…there are lots of these little duplicating irregular shapes, but how does this help me?
I borrow again from Benoit Mandelbrots earlier article:
“In a detail of a graphic in which the features are higher than they are wide – as are the individual up-and-down price ticks of a stock – the transformation from the whole to a part must reduce the horizontal axis more than the vertical one. For a price chart, this transformation must shrink the time-scale (the horizontal axis) more than the price scale (the vertical axis). The geometric relation of the whole to its parts is said to be one of self-affinity.
The existence of unchanging properties is not given much weight by most statisticians. But they are beloved of physicists and mathematicians like myself, who call them invariances and are happiest with models that present an attractive invariance property. A good idea of what I mean is provided by drawing a simple chart that inserts price changes from time 0 to a later time 1 in successive steps. The intervals themselves are chosen arbitrarily; they may represent a second, an hour, a day or a year.
The process begins with a price, represented by a straight trend line (illustration 1). Next, a broken line called a generator is used to create the pattern that corresponds to the up-and-down oscillations of a price quoted in financial markets. The generator consists of three pieces that are inserted (interpolated) along the straight trend line. (A generator with fewer than three pieces would not simulate a price that can move up and down.) After delineating the initial generator, its three pieces are interpolated by three shorter ones. Repeating these steps reproduces the shape of the generator, or price curve, but at compressed scales. Both the horizontal axis (timescale) and the vertical axis (price scale) are squeezed to fit the horizontal and vertical boundaries of each piece of the generator.
Interpolations Forever
Only the first stages are shown in the illustration, although the same process continues. In theory, it has no end, but in practice, it makes no sense to interpolate down to time intervals shorter than those between trading transactions, which may occur in less than a minute. Clearly, each piece ends up with a shape roughly like the whole. That is, scale invariance is present simply because it was built in. The novelty (and surprise) is that these self-affine fractal curves exhibit a wealth of structure — a foundation of both fractal geometry and the theory of chaos.
A few selected generators yield so-called unifractal curves that exhibit the relatively tranquil picture of the market encompassed by modern portfolio theory. But tranquillity prevails only under extraordinarily special conditions that are satisfied only by these special generators. The assumptions behind this oversimplified model are one of the central mistakes of modern portfolio theory. It is much like a theory of sea waves that forbids their swells to exceed six feet.
The beauty of fractal geometry is that it makes possible a model general enough to reproduce the patterns that characterize portfolio theory’s placid markets as well as the tumultuous trading conditions of recent months. The just described method of creating a fractal price model can be altered to show how the activity of markets speeds up and slows down — the essence of volatility. This variability is the reason that the prefix “multi-” was added to the word “fractal.”
To create a multifractal from a unifractal, the key step is to lengthen or shorten the horizontal time axis so that the pieces of the generator are either stretched or squeezed. At the same time, the vertical price axis may remain untouched. In illustration 2, the first piece of the unifractal generator is progressively shortened, which also provides room to lengthen the second piece. After making these adjustments, the generators become multifractal (M1 to M4). Market activity speeds up in the interval of time represented by the first piece of the generator and slows in the interval that corresponds to the second piece (illustration 3).
Such an alteration to the generator can produce a full simulation of price fluctuations over a given period, using the process of interpolation described earlier. Each time the first piece of the generator is further shortened — and the process of successive interpolation is undertaken — it produces a chart that increasingly resembles the characteristics of volatile markets (illustration 4).
What should a corporate treasurer, currency trader or other market strategist conclude from all this? The discrepancies between the pictures painted by modern portfolio theory and the actual movement of prices are obvious. Prices do not vary continuously, and they oscillate wildly at all timescales. Volatility — far from a static entity to be ignored or easily compensated for — is at the very heart of what goes on in financial markets. In the past, money managers embraced the continuity and constrained price movements of modern portfolio theory because of the absence of strong alternatives. But a money manager need no longer accept the current financial models at face value.
Instead multifractals can be put to work to “stress-test” a portfolio. In this technique the rules underlying multifractals attempt to create the same patterns of variability as do the unknown rules that govern actual markets. Multifractals describe accurately the relation between the shape of the generator and the patterns of up-and-down swings of prices to be found on charts of real market data.
On a practical level, this finding suggests that a fractal generator can be developed based on historical market data. The actual model used does not simply inspect what the market did yesterday or last week. It is in fact a more realistic depiction of market fluctuations, called fractional Brownian motion in multifractal trading time. The charts created from the generators produced by this model can simulate alternative scenarios based on previous market activity.
These techniques do not come closer to forecasting a price drop or rise on a specific day on the basis of past records. But they provide estimates of the probability of what the market might do and allow one to prepare for inevitable sea changes. The new modeling techniques are designed to cast a light of order into the seemingly impenetrable thicket of the financial markets. They also recognize the mariner’s warning that, as recent events demonstrate, deserves to be heeded: On even the calmest sea, a gale may be just over the horizon.
Many people believe that the markets are random. In fact, one of the most prominent investing books out there is “A Random Walk Down Wall Street” (1973) by Burton G. Malkiel, who argues that throwing darts at a dartboard is likely to yield results similar to those achieved by a fund manager (and Malkiel does have many valid points).
However, many others argue that although prices may appear to be random, they do in fact follow a pattern in the form of trends. One of the most basic ways in which traders can determine such trends is through the use of fractals. Fractals essentially break down larger trends into extremely simple and predictable reversal patterns. This article will explain what fractals are and how you might apply them to your trading to enhance your profits.
What Are Fractals?
When many people think of fractals in the mathematical sense, they think of chaos theory and abstract mathematics. While these concepts do apply to the market (it being a nonlinear, dynamic system), most traders refer to fractals in a more literal sense. That is, as recurring patterns that can predict reversals among larger, more chaotic price movements.
These basic fractals are composed of five or more bars. The rules for identifying fractals are as follows:
* A bearish turning point occurs when there is a pattern with the highest high in the middle and two lower highs on each side.
* A bullish turning point occurs when there is a pattern with the lowest low in the middle and two higher lows on each side.
The fractals shown in Figure 1 are two examples of perfect patterns. Note that many other less perfect patterns can occur, but the basic pattern should remain intact for the fractal to be valid.
Applying Fractals to Trading
Like many trading indicators, fractals are best used in conjunction with other indicators or forms of analysis. Perhaps the most common confirmation indicator used with fractals is the “Alligator indicator”, a tool that is created by using moving averages that factor in the use of fractal geometry. The standard rule states that all buy rules are only valid if below the “alligator’s teeth” (the center average), and all sell rules are only valid if above the alligator’s teeth.
