Continuous Probability Distributions Commonly Used in Financial Econometrics 353
The mean is given by=
−
EX >
n
n() n
2(^2) for2
2
2 (B.7)
while the variance equals
(^) σ^22
(^212)
12
2
2
22
24
==
+−
−−
var()Xnnn
nn n()
()()
fornn 2 > 4 (B.8)Note that according to equation (B.7), the mean is not affected by the
degrees of freedom n 1 of the first chi-square random variable, while the
variance in equation (B.8) is influenced by the degrees of freedom of both
random variables.
α-Stable Distribution
While many models in finance have been modeled historically using the nor-
mal distribution based on its pleasant tractability, concerns have been raised
that this distribution underestimates the danger of downturns of extreme
magnitude in stock markets that have been observed in financial markets.
Many distributional alternatives providing more realistic chances to severe
price movements have been presented earlier, such as the Student’s t. In the
early 1960s, Benoit Mandelbrot suggested as a distribution for commodity
price changes the class of Lévy stable distributions (simply referred to as the
stable distributions).^3 The reason is that, through their particular param-
eterization, they are capable of modeling moderate scenarios, as supported
by the normal distribution, as well as extreme ones.
The stable distribution is characterized by the four parameters α, β, σ,
and μ. In brief, we denote the stable distribution by S(α, β, σ, μ). Parameter α
is the so called tail index or characteristic exponent. It determines how much
probability is assigned around the center and the tails of the distribution.
The lower the value α, the more pointed about the center is the density and
the heavier are the tails. These two features are referred to as excess kurtosis
relative to the normal distribution. This can be visualized graphically as we
have done in Figure B.7 where we compare the normal density to an α-stable
(^3) Benoit B. Mandelbrot, “The Variation of Certain Speculative Prices,” Journal of
Business 36 (1963): 394–419.