122
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
using a double-logarithmic scale. It shows the linear dependence for large incomes,
consistent with a power-law dependence with a tail exponent α + 1 = 3.2 for the
integral, such that α = 2.2, which is slightly larger than the exponent α = 2.09 for
the US, reported in [9], and on the same order of magnitude as the exponent for Italy
(α = 2.5) or Australia (α = 2.3), reported in [11]. Note, however, that our determination of the tail exponent is very crude. It should only serve as an example an should
not be used for further comparison. The other data from [9] were determined using
much more sophisticated methods.
A special feature of power laws is their scale invariance in the sense that multiplying the argument will be equal to the same power law multiplied by a constant
p(ax) = q(a) p(x)
(9.5)
which, following [9], can be visualized by considering the scale factor a as a change
of units which preserves the overall shape of the distribution. Using this definition of
scale invariance we now show that p(x) must be a power law. We first differentiate
the left-hand side of (9.5) and then the right-hand side, which leads to
dp(ax)
da
= x p
(ax) = q
(a) p(x) =
p
(a)
p(1)
p(x)
a=1
−→ x p
(x) =
p
(1)
p(1)
p(x) .
(9.6)
where we used q(a) = p(a)/ p(1), which follows from setting x = 1 in (9.5). Separating variables and subsequently integrating yields
ln p(x) − ln p(1) =
p
(1)
p(1)
ln x = ln x
p
(1)/ p(1)
(9.7)
where the integration constant ln p(1) on the left-hand side follows from setting
x = 1, which causes both sides of the equation to vanish. Rewriting the previous
equation gives us
p(x) = p(1)x
−α
with
α = −p
(1)/ p(1) .
(9.8)
We have thus shown that the requirement for scale invariance in (9.5) directly implies
that p(x) follows a power law.
We point out that scale invariance is a very important concept in our context of
crashes, because it implies that there is no natural scale in the day-to-day variations
or the volatility. Conversely, Gaussian distributions possess their standard deviation
σ as a natural scale, whereas power the law distribution from (9.1) are scale-invariant
and can be responsible for very large variations.
In earlier chapters we described the temporal evolution of stock values by a random
walk in which the small day-to-day increments with a Gaussian distribution were led
to the evolution of the stock values over larger times. Stock values are therefore sums
of random numbers. In a later section we will follow this theme further, especially,
what happens if the random values are drawn from distributions different from a
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
using a double-logarithmic scale. It shows the linear dependence for large incomes,
consistent with a power-law dependence with a tail exponent α + 1 = 3.2 for the
integral, such that α = 2.2, which is slightly larger than the exponent α = 2.09 for
the US, reported in [9], and on the same order of magnitude as the exponent for Italy
(α = 2.5) or Australia (α = 2.3), reported in [11]. Note, however, that our determination of the tail exponent is very crude. It should only serve as an example an should
not be used for further comparison. The other data from [9] were determined using
much more sophisticated methods.
A special feature of power laws is their scale invariance in the sense that multiplying the argument will be equal to the same power law multiplied by a constant
p(ax) = q(a) p(x)
(9.5)
which, following [9], can be visualized by considering the scale factor a as a change
of units which preserves the overall shape of the distribution. Using this definition of
scale invariance we now show that p(x) must be a power law. We first differentiate
the left-hand side of (9.5) and then the right-hand side, which leads to
dp(ax)
da
= x p
(ax) = q
(a) p(x) =
p
(a)
p(1)
p(x)
a=1
−→ x p
(x) =
p
(1)
p(1)
p(x) .
(9.6)
where we used q(a) = p(a)/ p(1), which follows from setting x = 1 in (9.5). Separating variables and subsequently integrating yields
ln p(x) − ln p(1) =
p
(1)
p(1)
ln x = ln x
p
(1)/ p(1)
(9.7)
where the integration constant ln p(1) on the left-hand side follows from setting
x = 1, which causes both sides of the equation to vanish. Rewriting the previous
equation gives us
p(x) = p(1)x
−α
with
α = −p
(1)/ p(1) .
(9.8)
We have thus shown that the requirement for scale invariance in (9.5) directly implies
that p(x) follows a power law.
We point out that scale invariance is a very important concept in our context of
crashes, because it implies that there is no natural scale in the day-to-day variations
or the volatility. Conversely, Gaussian distributions possess their standard deviation
σ as a natural scale, whereas power the law distribution from (9.1) are scale-invariant
and can be responsible for very large variations.
In earlier chapters we described the temporal evolution of stock values by a random
walk in which the small day-to-day increments with a Gaussian distribution were led
to the evolution of the stock values over larger times. Stock values are therefore sums
of random numbers. In a later section we will follow this theme further, especially,
what happens if the random values are drawn from distributions different from a