248 Frequently Asked Questions In Quantitative Finance
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Levy
α = 0.5
μ = 0
β = 0
ν = 1
wherei=
√
−1. For the Levy distribution ́
ln(M(z))=iμz−να|z|α( 1 −iβsgn(z)tan(πa/2)),forα = 1
or
ln(M(z))=iμz−ν|z|
(
1 +
2 iβ
π
sgn(z)ln(|z|)
)
,forα= 1.
The normal distribution is a special case of this with
α=2andβ=0, and with the parameterνbeing one
half of the variance. The Levy distribution, or Pareto ́
Levy distribution, is increasingly popular in finance ́
because it matches data well, and has suitable fat tails.
It also has the important theoretical property of being
a stable distribution in that the sum of independent
random numbers drawn from the Levy distribution will ́
itself be L ́evy. This is a useful property for the distribu-
tion of returns. If you add upnindependent numbers
from the Levy distribution with the above parameters ́
