142
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
Fitting the entire period shown on the left-hand plot in Fig. 9.14 requires a non-linear
theory (see [17] and references therein) that is, however, beyond our scope.
We have come a long way discussing speculative bubbles and crashes all the way
from a historic presentation, via behavioral concepts to extreme value theory, fractals
and eventually a theory that attempts to quantify and even predict the time of a crash.
But after all the havoc created by the bubbles and crashes we now turn back to the
quieter realm of option pricing, but use quantum mechanical tools and path integrals
for the task.
Exercises
1. Calculate the generating function of the probability distribution function R(x)
that is given by R(x) = 1/a for −a/2 < x < a/2 and R(x) = 0 otherwise.
2. Calculate the first four cumulants of R(x) from Exercise 1.
3. Calculate the convolution of two Cauchy distributions.
4. Generate 10
5 random numbers, uniformly distributed between −1 and 1, split
them into 10
4 groups of ten numbers, sum the ten numbers, and prepare a histogram of the sums. What do you observe? Can you reconcile the result with the
second cumulant, calculated in Exercise 2?
5. What is the fractal dimension of a modified Cantor set, where you always remove
the central quarter of each line segment?
6. Consider the distribution function D ν (x) = A(ν)e
−x
ν for 1/2 < ν < 2 in the
range 0 < x < ∞ and calculate the expected maximum value of the random
numbers after drawing n numbers and how it scales with n. Therefore,
a. determine A(ν) such that
∞
0 D ν (x
)dx
= 1;
b. calculate the cumulative distribution function C ν (x) =
x
0 D ν (x
)dx
;
c. numerically calculate x n from (9.33);
d. plot x n
ν versus log 10 (n) and convince yourself that this approximately
follows a straight line, just as the data in Fig. 9.10 does.
Hint: in (a) and (b) try the substitution t = x
ν .
7. The population N of a simplified biological system changes with time according
to the rate equation d N/dt = −δ N + β N
2 with the death rate δ and the birth rate
β. The latter enters quadratically, because two members need to meet in order to
procreate. The resemblance to (9.51) stimulates the question how δ affects the
divergence. Discuss and analyse how the time τ c until the singularity is affected.
8. Fat-tailed distributions: Where does the derivation of the Fokker-Planck equation in Sect. 4.5 break down if we use a fat-tailed distribution for what was
called φ(ξ )?
9. Research why the crash from 1929 led to a long depression, but the crash 1987
did not.
9 Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions
Fitting the entire period shown on the left-hand plot in Fig. 9.14 requires a non-linear
theory (see [17] and references therein) that is, however, beyond our scope.
We have come a long way discussing speculative bubbles and crashes all the way
from a historic presentation, via behavioral concepts to extreme value theory, fractals
and eventually a theory that attempts to quantify and even predict the time of a crash.
But after all the havoc created by the bubbles and crashes we now turn back to the
quieter realm of option pricing, but use quantum mechanical tools and path integrals
for the task.
Exercises
1. Calculate the generating function of the probability distribution function R(x)
that is given by R(x) = 1/a for −a/2 < x < a/2 and R(x) = 0 otherwise.
2. Calculate the first four cumulants of R(x) from Exercise 1.
3. Calculate the convolution of two Cauchy distributions.
4. Generate 10
5 random numbers, uniformly distributed between −1 and 1, split
them into 10
4 groups of ten numbers, sum the ten numbers, and prepare a histogram of the sums. What do you observe? Can you reconcile the result with the
second cumulant, calculated in Exercise 2?
5. What is the fractal dimension of a modified Cantor set, where you always remove
the central quarter of each line segment?
6. Consider the distribution function D ν (x) = A(ν)e
−x
ν for 1/2 < ν < 2 in the
range 0 < x < ∞ and calculate the expected maximum value of the random
numbers after drawing n numbers and how it scales with n. Therefore,
a. determine A(ν) such that
∞
0 D ν (x
)dx
= 1;
b. calculate the cumulative distribution function C ν (x) =
x
0 D ν (x
)dx
;
c. numerically calculate x n from (9.33);
d. plot x n
ν versus log 10 (n) and convince yourself that this approximately
follows a straight line, just as the data in Fig. 9.10 does.
Hint: in (a) and (b) try the substitution t = x
ν .
7. The population N of a simplified biological system changes with time according
to the rate equation d N/dt = −δ N + β N
2 with the death rate δ and the birth rate
β. The latter enters quadratically, because two members need to meet in order to
procreate. The resemblance to (9.51) stimulates the question how δ affects the
divergence. Discuss and analyse how the time τ c until the singularity is affected.
8. Fat-tailed distributions: Where does the derivation of the Fokker-Planck equation in Sect. 4.5 break down if we use a fat-tailed distribution for what was
called φ(ξ )?
9. Research why the crash from 1929 led to a long depression, but the crash 1987
did not.