120
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
as an inverted parabola that actually tracks the data points quite well up to about
±5% or about five standard deviations, but beyond that point there is a clear surplus
of data points above the Gaussian. Moreover, the deviations on the negative side are
somewhat more pronounced. On the right-hand side in Fig. 9.3, we show the same
data points on double logarithmic scale, separately for positive and negative deviations, the latter simply flipped to the positive side, as well as the Gaussian. In this
representation the Gaussian does not fit the data very well. But at large horizontal
deviations the data in double logarithmic representation appears to be linear, which
indicates a power law of the form y ∝ x
−α
. We refrain from reporting the coefficients because of the limited statistics of the few data points at large deviations.
In general, however, power-law distributions appear in many contexts. An example
is the distribution of wealthy individuals; there are few extremely rich individuals
but very many more individuals of moderate wealth. Plotting a histogram of wealth
versus number of individuals in a particular wealth bracket in double logarithmic
fashion reveals a straight line, indicating that the distribution follows a power law.
This was first observed by V. Pareto, who analyzed the distribution of wealth in
Italy in the early 1900s. In the 1960s Mandelbrot [7] and Fama [8] found fat-tailed
distributions when analyzing returns of stocks. Other distributions that follow power
laws are moderately sized earthquakes when plotting a histogram of the number of
quakes versus their magnitude—the well-known Richter law and Zipf’s law of word
frequency in texts. We refer to [9] for further examples.
So, there is something special about power laws and we will therefore investigate
them further.
9.5 Power Laws
We follow [9] and write a generic power law in the form
p(x) = C x
−α
.
(9.1)
We point out that the power law behavior will probably be valid only for values of x
larger than some minimum value ˆ
x. This was the case for the day-to-day variations
of the Dow Jones index, which follows a power law only for values larger than
ˆ
x = 0.05. All distributions we discuss in the following section will cover the region
above ˆ
x up to infinity. Since divergences only arise in this region, we ignore the fact
that the normalization and all moments differ by an additive constant, depending on
whether we take the distribution below ˆ
x into account or not.
The normalization of the distribution p(x) from ˆ
x to infinity is given by
1 =
∞
ˆ
x
p(x)dx =
C
−α + 1
x
−α+1
∞
ˆ
x
=
C
α − 1
ˆ
x
1−α
(9.2)
or C = (α − 1) ˆ
x
α−1
, provided α > 1 to ensure convergence of the integral.
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
as an inverted parabola that actually tracks the data points quite well up to about
±5% or about five standard deviations, but beyond that point there is a clear surplus
of data points above the Gaussian. Moreover, the deviations on the negative side are
somewhat more pronounced. On the right-hand side in Fig. 9.3, we show the same
data points on double logarithmic scale, separately for positive and negative deviations, the latter simply flipped to the positive side, as well as the Gaussian. In this
representation the Gaussian does not fit the data very well. But at large horizontal
deviations the data in double logarithmic representation appears to be linear, which
indicates a power law of the form y ∝ x
−α
. We refrain from reporting the coefficients because of the limited statistics of the few data points at large deviations.
In general, however, power-law distributions appear in many contexts. An example
is the distribution of wealthy individuals; there are few extremely rich individuals
but very many more individuals of moderate wealth. Plotting a histogram of wealth
versus number of individuals in a particular wealth bracket in double logarithmic
fashion reveals a straight line, indicating that the distribution follows a power law.
This was first observed by V. Pareto, who analyzed the distribution of wealth in
Italy in the early 1900s. In the 1960s Mandelbrot [7] and Fama [8] found fat-tailed
distributions when analyzing returns of stocks. Other distributions that follow power
laws are moderately sized earthquakes when plotting a histogram of the number of
quakes versus their magnitude—the well-known Richter law and Zipf’s law of word
frequency in texts. We refer to [9] for further examples.
So, there is something special about power laws and we will therefore investigate
them further.
9.5 Power Laws
We follow [9] and write a generic power law in the form
p(x) = C x
−α
.
(9.1)
We point out that the power law behavior will probably be valid only for values of x
larger than some minimum value ˆ
x. This was the case for the day-to-day variations
of the Dow Jones index, which follows a power law only for values larger than
ˆ
x = 0.05. All distributions we discuss in the following section will cover the region
above ˆ
x up to infinity. Since divergences only arise in this region, we ignore the fact
that the normalization and all moments differ by an additive constant, depending on
whether we take the distribution below ˆ
x into account or not.
The normalization of the distribution p(x) from ˆ
x to infinity is given by
1 =
∞
ˆ
x
p(x)dx =
C
−α + 1
x
−α+1
∞
ˆ
x
=
C
α − 1
ˆ
x
1−α
(9.2)
or C = (α − 1) ˆ
x
α−1
, provided α > 1 to ensure convergence of the integral.