Is the Mandelbrot set as desired

Fractals in Financial Mathematics

A stock time series is not smooth, a boulder is not a sphere, and a coastline is not a straight line. In 1975 Benoît Mandelbrot coined the term “fractal” for this. This work has prepared a large field, which today can be summarized as the theory of chaos and fractals.

Chaos theory does not consider random dynamic systems, such as magnetic pendulums, which appear unpredictable over time and are referred to as deterministic chaos. In the case of fractals, on the other hand, the simplest mathematical replacement rules are often the basis. Self-similar structures can be created in both segments. The best-known example in this context is the so-called Mandelbrot set, which allows the structures to be enlarged infinitely and which in turn recursively contains the Mandelbrot set.

Fractal trends

But what does all of this have to do with financial market data? After his pioneering work in the above-mentioned area, Benoît Mandelbrot turned to the financial sector again in the 1990s and presented another model, namely the multi-fractals in price time series. He summed up his view of the markets in a large series of foundings for the quantitative finance journal, which is now the most renowned in the quantitative field. He described a fractal recursive construction process for price time series, which recursively breaks down trends into smaller trends. He considered this construction method to be incomplete and gave it the name "cartoons".

But what's a trend?

Colloquially, the answer seems to be downright trivial, so that no further clarification is required. Mathematically, however, it is anything but clear. Just as the term “chance” is not mathematically defined, but a philosophical term, so is trends. If you want to talk about trends mathematically, you have to define two things: the exact measurement method and the scale with which the measurements are made. This becomes clear using the example of the moving averages, which are often used: The methodology is the averaging of the prices themselves, and the period under consideration, for example 200 days, indicates the scale or granularity of the analysis. Every practitioner knows that this type of analysis is subject to imprecision, which increases as the analysis window is enlarged. Here we start with a method from signal theory: the wavelet trend decomposition.

Wavelets

The theory of wavelets was a “blockbuster” topic in mathematics in the 1980s and 1990s. If infinitely long oscillations, such as cosines, were conventionally used in signal analysis, wavelets are short and limited oscillations. Wavelets comes from English and means the diminution of waves. These vibrations are in turn self-similar by construction and have now found their way into many technical innovations, for example in image data compression, such as the JPEG 2000 and MPEG-4 standards.

With the help of this theory and another procedure from chaos theory, an extremely precise trend breakdown can be constructed, which breaks down any time series, including the DAX performance index, into trends or visibility structures on the basis of a scale:

The method has the property that it is optimal in terms of signal theory, which means that no other measurement method can measure more precisely. With the naked eye you can see that trends have fractal characteristics and that rough trends, i.e. large (time) scales, are composed of smaller trends, i.e. smaller scales. You can also see immediately that on a fixed scale, for example 20, trends are of different lengths. Mathematically, however, it is even harder: the trend lengths - but also other characteristics - follow a log-normal distribution almost everywhere in stocks. Trends are arbitrarily long or short as well as steep or flat, and these can be assumed to be coincidental. Furthermore, trends are self-similar and generate the power laws so typical for fractals.

Trend lengths and power laws

What insights do you get from this? If one concentrates on the question of why severe anomalies exist in the financial market, such as momentum, then the evaluation of the trend characteristics can be used to compare the mathematical model world with reality. In the following, we calculate the mean trend length of all trends seen in the DAX for each wavelet scale (or time scale) and compare this with the mean trend lengths of random processes that are assumed in classical financial mathematics (random walk). Both measured variables follow a power law, i.e. with increasingly larger scales, the mean trend lengths increase disproportionately (i.e. exponentially). In the following graphic, the wavelet scale and the determined mean trend lengths are therefore plotted logarithmically so that the exponential development follows a straight line:

If you compare the two straight lines with each other, you can see that the trends in the real data are significantly longer as the scale increases than in the purely random processes assumed in classical theory. For example, if you calculate the average trend length on a 30 scale for the DAX, you arrive at a length of approx. 54 trading days, while the model-theoretical counterpart (the random walk) has an average trend length of approx. 30 days. In other words: on this scale, the trends in the real data are on average more than a stock market month longer (~ 24 days) than one would theoretically assume.

Momentum, i.e. the structured e.g. monthly investment in stocks that have historically risen in price, now appears clear. Obviously, this effect takes advantage of excessively long trends in the real data. But what about the other capital market anomalies? As will be shown, it is even wilder in the best Mandelbrotsch sense ...

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