Assuming daily logarithmic return of the stock market follows a stable distribution, simulate and visualize stock prices over a period of 5 years:
Assuming stock logarithmic return follows a stable distribution, find the value at risk at the 95% level:
Compute the 95% value at risk point loss of the current S&P 500 index value, assuming the above distribution:
Find the expected shortfall of logarithmic return:
Compute the associated point loss:
Fit the daily logarithm return of the Nikkei 225 index since January 1, 2005 to a stable distribution:
Compute logarithmic returns:
Fit logarithmic returns to a stable distribution:
Compare the estimated distribution to a data histogram:
The product of a symmetric stable random variate and the power of an exponential random variate follows a Linnik distribution:
Generate random variates and show the histogram:
Map-Airy distribution [] is a member of the stable family:
Its probability density is known in closed form:
Find the location of the mode:
Estimate the parameter of stable distribution from a sample characteristic function:
Plot absolute values of a sample characteristic function and a population characteristic function:
Compare with the maximum likelihood estimation:
Generalized central limit theorem gives sequences
and
such that the distribution of the shifted and rescaled sum
of
i.i.d. random variates
whose distribution function
has asymptotes
as
and
as
weakly converges to the stable distribution
:
Illustrate the generalized central limit theorem using a two-sided Pareto distribution:
Define the mean and variance of the two-sided Pareto distribution for future use:
Define a routine to generate
-variates:
Define a function to visualize a density plot and data histogram:
Case of
:
Case of
:
Case of
:
Case of
:
Case of standard central limit theorem, i.e.
:
Holtsmark distribution is the distribution of forces acting on a particle in an infinite Poisson system. The
-component of the gravitational force follows symmetric stable distribution:
Simulate absolute value of the force:
Distribution of the absolute value is known in closed form:
Compare it to the histogram:
with index of stability 
, skewness parameter 
, location parameter 
, and scale parameter 
.
, which satisfies a functional equation where for any 
and 
there exist 
and 
such that 
. The general solution to the functional equation has four parameters. 
, 
is continuous in 
.
is discontinuous in 
.
.
is equivalent to StableDistribution
.
is equivalent to StableDistribution
. 
:
:
of an 
stable random variate is manifested in the sensitivity of mode to small changes in 
:
is again a stable variate:
is not 1:
:
and 
:
:
the following equality holds:
EXAMPLES
Basic Examples 
Scope 