Anon

Continuous Probability Distributions Commonly Used in Financial Econometrics 355

Possible values of the parameters are listed below:

α (0,2]

β [–1,1]

σ (0,∞)

μ any real number

Depending on the parameters α and β, the distribution has either support on
the entire real line or only the part extending to the right of some location.
In general, the density function is not explicitly presentable. Instead,
the distribution of the α-stable random variable is given by its characteristic
function which we do not present here.^5
Figure B.8 shows the effect of α on tail thickness of the density as well
as peakedness at the origin relative to the normal distribution (collectively

(^5) There are three possible ways to uniquely define a probability distribution: the
cumulative distribution function, the probability density function, and the char-
acteristic function. The precise definition of a characteristics function needs some
advanced mathematical concepts and is not of major interest for this book. At this
point, we just state the fact that knowing the characteristic function is mathemati-
cally equivalent to knowing the probability density or the cumulative distribution
function. In only three cases does the density of a stable distribution have a closed-
form expression.
FigURe B.8 Influence of α on the Resulting Stable Distribution
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
β = 0, σ = 1, μ = 0
α = 0.5
α = 1
α = 1.5
α = 2
f(
x)
x