126
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
Fig. 9.8 The same type of plots as in Fig. 9.7 but here the three subintervals are randomly scrambled.
The differences on the third plot now appear like white noise with a number of large outliers sticking
out
three segments for the next iteration. This produces significantly more random traces,
which can be seen in the second plot in Fig. 9.8. The third plot shows the increments
and they appear to consist of a rather uniform background noise with superimposed
large outliers. This is also apparent in the histogram, shown on the bottom plot in
Fig. 9.8. Note that here the vertical axis of the histogram is logarithmic in order to
make the “fat tails” more noticeable.
We observe that Mandelbrot’s algorithm to generate scrambled fractals in fact generates traces and increments that somewhat resemble stock traces and the histogram
with the increments shows fat tails. Note that the stretching and folding procedure
can also be applied to the horizontal, the time-axis. In that case the resulting curve
shows more pronounced periods of tranquility and bursts of activity, very much like
trading progresses in time. The resulting traces are called multi-fractals as opposed
to the uni-fractal traces we discussed above.
A particular feature of the fractal curve generation is the repeated application of
a simple “rule” resulting in a self-similar structure, which is what we earlier called
“scaling.” The structure found on a large scale re-appears on a much reduced scale.
We will see in Sect. 9.10 that this self-similarity with discrete scales causes additional
large scale features that can be observed in stock charts.
The share values evolving in time, as shown in Fig. 9.1, or the synthetic trace from
Fig. 9.8, can be interpreted as adding a random increment to the previous value, much
in the spirit of (4.7), only here the increments are drawn from a fat-tailed distribution.
This motivates the topic of the next section—adding multiple random numbers.
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
Fig. 9.8 The same type of plots as in Fig. 9.7 but here the three subintervals are randomly scrambled.
The differences on the third plot now appear like white noise with a number of large outliers sticking
out
three segments for the next iteration. This produces significantly more random traces,
which can be seen in the second plot in Fig. 9.8. The third plot shows the increments
and they appear to consist of a rather uniform background noise with superimposed
large outliers. This is also apparent in the histogram, shown on the bottom plot in
Fig. 9.8. Note that here the vertical axis of the histogram is logarithmic in order to
make the “fat tails” more noticeable.
We observe that Mandelbrot’s algorithm to generate scrambled fractals in fact generates traces and increments that somewhat resemble stock traces and the histogram
with the increments shows fat tails. Note that the stretching and folding procedure
can also be applied to the horizontal, the time-axis. In that case the resulting curve
shows more pronounced periods of tranquility and bursts of activity, very much like
trading progresses in time. The resulting traces are called multi-fractals as opposed
to the uni-fractal traces we discussed above.
A particular feature of the fractal curve generation is the repeated application of
a simple “rule” resulting in a self-similar structure, which is what we earlier called
“scaling.” The structure found on a large scale re-appears on a much reduced scale.
We will see in Sect. 9.10 that this self-similarity with discrete scales causes additional
large scale features that can be observed in stock charts.
The share values evolving in time, as shown in Fig. 9.1, or the synthetic trace from
Fig. 9.8, can be interpreted as adding a random increment to the previous value, much
in the spirit of (4.7), only here the increments are drawn from a fat-tailed distribution.
This motivates the topic of the next section—adding multiple random numbers.