138
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
Fig. 9.13 Examples of the Gumbel, Fréchet and Weibull distributions
value larger than the abscissa x is given by 1 − G(x), or 1 − F(x), or 1 − W (x),
respectively. The Fréchet distribution, having an underlying power-law distribution
from which to draw the samples, rises very slowly at large values of x and is far from
unity, even at x = 10. This implies that there is a significant probability of finding
even more extreme values. In contrast, the Weibull distribution has an endpoint at
x = 2 and the Gumbel distribution approaches unity already for moderate values
of x.
After having discussed the distributions of potential day-to-day variations and
analyzed their behavior to some extent, we will now turn back to consider speculative
bubbles.
9.10 Finite-Time Divergence and Log-Periodic Oscillations
The time from the start of a speculative bubble to the time it grows without bounds
and crashes is finite. This is in contrast to an exponential growth, which needs an
infinitely long time to “reach infinity.” Thus bubbles and crashes are characterized by
a finite-time divergence and a larger-than-linear growth rate ν > 1. We investigate
the behavior by considering the following simple system for a dynamical variable s
ds
dt
=
1
τ
s
ν
ˆ
s ν−1 ,
(9.51)
where τ is a time constant and ˆ
s a scale factor. Rewriting the equation as ds/s
ν
=
dt/(ˆ s
ν−1
τ ) we can integrate it with the result
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
Fig. 9.13 Examples of the Gumbel, Fréchet and Weibull distributions
value larger than the abscissa x is given by 1 − G(x), or 1 − F(x), or 1 − W (x),
respectively. The Fréchet distribution, having an underlying power-law distribution
from which to draw the samples, rises very slowly at large values of x and is far from
unity, even at x = 10. This implies that there is a significant probability of finding
even more extreme values. In contrast, the Weibull distribution has an endpoint at
x = 2 and the Gumbel distribution approaches unity already for moderate values
of x.
After having discussed the distributions of potential day-to-day variations and
analyzed their behavior to some extent, we will now turn back to consider speculative
bubbles.
9.10 Finite-Time Divergence and Log-Periodic Oscillations
The time from the start of a speculative bubble to the time it grows without bounds
and crashes is finite. This is in contrast to an exponential growth, which needs an
infinitely long time to “reach infinity.” Thus bubbles and crashes are characterized by
a finite-time divergence and a larger-than-linear growth rate ν > 1. We investigate
the behavior by considering the following simple system for a dynamical variable s
ds
dt
=
1
τ
s
ν
ˆ
s ν−1 ,
(9.51)
where τ is a time constant and ˆ
s a scale factor. Rewriting the equation as ds/s
ν
=
dt/(ˆ s
ν−1
τ ) we can integrate it with the result