1 Stable distributions
on the Central Limit Theorem, which states that the sum of a large number of
independent, identically distributed variables from a finite-variance distribution
will tend to be normally distributed. However, as we have already mentioned,
financial asset returns usually have heavier tails.
In response to the empirical evidence Mandelbrot (1963) and Fama (1965) pro-
posed the stable distribution as an alternative model. Although there are other
heavy-tailed alternatives to the Gaussian law - like Student’s t, hyperbolic, nor-
mal inverse Gaussian, or truncated stable - there is at least one good reason
for modeling financial variables using stable distributions. Namely, they are
supported by the generalized Central Limit Theorem, which states that sta-
ble laws are the only possible limit distributions for properly normalized and
centered sums of independent, identically distributed random variables.
Since stable distributions can accommodate the fat tails and asymmetry, they
often give a very good fit to empirical data. In particular, they are valuable
models for data sets covering extreme events, like market crashes or natural
catastrophes. Even though they are not universal, they are a useful tool in
the hands of an analyst working in finance or insurance. Hence, we devote
this chapter to a thorough presentation of the computational aspects related
to stable laws. In Section 1.3 we review the analytical concepts and basic
characteristics. In the following two sections we discuss practical simulation and
estimation approaches. Finally, in Section 1.6 we present financial applications
of stable laws.
1.3 Definitions and basic characteristics
Stable laws - also called α-stable, stable Paretian or Levy stable - were in-
troduced by Levy (1925) during his investigations of the behavior of sums of
independent random variables. A sum of two independent random variables
having an α-stable distribution with index α is again α-stable with the same
index α. This invariance property, however, does not hold for different α’s.
The α-stable distribution requires four parameters for complete description:
an index of stability α ∈ (0, 2] also called the tail index, tail exponent or
characteristic exponent, a skewness parameter β ∈ [-1, 1], a scale parameter
σ > 0 and a location parameter μ ∈ R. The tail exponent α determines the
rate at which the tails of the distribution taper off, see the left panel in Figure
1.1. When α = 2, the Gaussian distribution results. When α < 2, the variance
is infinite and the tails are asymptotically equivalent to a Pareto law, i.e. they