Anon

356 The Basics of financial economeTrics

the “kurtosis”of the density), for the case of β = 0, μ = 0, and σ = 1. As the
values of α decrease, the distribution exhibits fatter tails and more peaked-
ness at the origin. Figure B.9 illustrates the influence of β on the skewness
of the density function for α = 1.5, μ = 0, and σ = 1. Increasing (decreasing)
values of β result in skewness to the right (left).
Only in the case of an α of 0.5, 1, or 2 can the functional form of the
density be stated. For our purpose here, only the case α = 2 is of interest
because, for this special case, the stable distribution represents the normal
distribution. Then, the parameter β ceases to have any meaning since the
normal distribution is not asymmetric.
A feature of the stable distributions is that moments such as the mean,
for example, exist only up to the power α. So, except for the normal case
(where α = 2), there exists no finite variance. It becomes even more extreme
when α is equal to 1 or less such that not even the mean exists any more.
The non existence of the variance is a major drawback when applying stable
distributions to financial data. This is one reason that the use of this family
of distribution in finance is still disputed.
This class of distributions owes its name to the stability property that we
described earlier for the normal distribution (Property 2): The weighted sum
of an arbitrary number of independent α-stable random variables with the
same parameters is, again, α-stable distributed. More formally, let X 1 ,... ,

FigURe B.9 Influence of β on the Resulting Stable Distribution

−5 −4 −3 −2 −1 0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

α = 1.5, σ = 1, μ = 0

β = 0

β = 0.25

β = 0.5

β = 0.75

β = 1

x

f(

x)