pth moment of a stable random variable is ¬nite if and only if p < ±. When

the skewness parameter β is positive, the distribution is skewed to the right,

i.e. the right tail is thicker, see Figure 1.2. When it is negative, it is skewed to

the left. When β = 0, the distribution is symmetric about µ. As ± approaches

2, β loses its e¬ect and the distribution approaches the Gaussian distribution

14 1 Stable distributions in ¬nance

Gaussian, Cauchy and Levy distributions

0.4

0.3

PDF(x)

0.2

0.1

0

-5 0 5

x

Figure 1.3: Closed form formulas for densities are known only for three distri-

butions: Gaussian (± = 2; thin black), Cauchy (± = 1; red) and

Levy (± = 0.5, β = 1; thin, dashed blue). The latter is a totally

skewed distribution, i.e. its support is R+ . In general, for ± < 1

and β = 1 (’1) the distribution is totally skewed to the right (left).

STFstab03.xpl

regardless of β. The last two parameters, σ and µ, are the usual scale and

location parameters, i.e. σ determines the width and µ the shift of the mode

(the peak) of the distribution.

1.2.1 Characteristic function representation

Due to the lack of closed form formulas for densities for all but three distri-

butions (see Figure 1.3), the ±-stable law can be most conveniently described

by its characteristic function φ(t) “ the inverse Fourier transform of the prob-

1.2 ±-stable distributions 15

S parameterization S0 parameterization

0.5

0.5

0.4

0.4

0.3

0.3

PDF(x)

PDF(x)

0.2

0.2

0.1

0.1

0

0

-5 0 5 -5 0 5

x x

Figure 1.4: Comparison of S and S 0 parameterizations: ±-stable probability

density functions for β = 0.5 and ± = 0.5 (thin black), 0.75 (red),

1 (thin, dashed blue), 1.25 (dashed green) and 1.5 (thin cyan).

STFstab04.xpl

ability density function. However, there are multiple parameterizations for

±-stable laws and much confusion has been caused by these di¬erent represen-

tations, see Figure 1.4. The variety of formulas is caused by a combination

of historical evolution and the numerous problems that have been analyzed

using specialized forms of the stable distributions. The most popular param-

eterization of the characteristic function of X ∼ S± (σ, β, µ), i.e. an ±-stable

random variable with parameters ±, σ, β and µ, is given by (Samorodnitsky

and Taqqu, 1994; Weron, 1996):

±

’σ ± |t|± {1 ’ iβsign(t) tan π± } + iµt, ± = 1,

2

log φ(t) = (1.1)

2

’σ|t|{1 + iβsign(t) π log |t|} + iµt, ± = 1.

16 1 Stable distributions in ¬nance

For numerical purposes, it is often useful (Fofack and Nolan, 1999) to use

a di¬erent parameterization:

±

’σ ± |t|± {1 + iβsign(t) tan π± [(σ|t|)1’± ’ 1]} + iµ0 t, ± = 1,

2

log φ0 (t) =

2

’σ|t|{1 + iβsign(t) π log(σ|t|)} + iµ0 t, ± = 1.

(1.2)

0

The S± (σ, β, µ0 ) parameterization is a variant of Zolotariev™s (M)-parameteri-

zation (Zolotarev, 1986), with the characteristic function and hence the den-

sity and the distribution function jointly continuous in all four parameters,

see Figure 1.4. In particular, percentiles and convergence to the power-law

tail vary in a continuous way as ± and β vary. The location parameters of

the two representations are related by µ = µ0 ’ βσ tan π± for ± = 1 and

2