µ = µ0 ’ βσ π log σ for ± = 1.

The probability density function and the cumulative distribution function of ±-

stable random variables can be easily calculated in XploRe. Quantlets pdfstab

and cdfstab compute the pdf and the cdf, respectively, for a vector of values x

with given parameters alpha, sigma, beta, and mu, and an accuracy parameter

n. Both quantlets utilize Nolan™s (1997) integral formulas for the density and

the cumulative distribution function. The larger the value of n (default n=2000)

the more accurate and time consuming (!) the numerical integration.

Special cases can be computed directly from the explicit form of the pdf or

the cdf. Quantlets pdfcauch and pdflevy calculate values of the probability

density functions, whereas quantlets cdfcauch and cdflevy calculate values of

the cumulative distribution functions for the Cauchy and Levy distributions,

respectively. x is the input array; sigma and mu are the scale and location

parameters of these distributions.

1.2.2 Simulation of ±-stable variables

The complexity of the problem of simulating sequences of ±-stable random

variables results from the fact that there are no analytic expressions for the

inverse F ’1 of the cumulative distribution function. The ¬rst breakthrough

was made by Kanter (1975), who gave a direct method for simulating S± (1, 1, 0)

random variables, for ± < 1. It turned out that this method could be easily

adapted to the general case. Chambers, Mallows and Stuck (1976) were the

¬rst to give the formulas.

1.2 ±-stable distributions 17

The algorithm for constructing a random variable X ∼ S± (1, β, 0), in represen-

tation (1.1), is the following (Weron, 1996):

• generate a random variable V uniformly distributed on (’ π , π ) and an

22

independent exponential random variable W with mean 1;

• for ± = 1 compute:

(1’±)/±

cos{V ’ ±(V + B±,β )}

sin{±(V + B±,β )}

X = S±,β — — , (1.3)

{cos(V )}1/± W

where

arctan(β tan π± )

2

B±,β = ,

±

1/(2±)

π±

1 + β 2 tan2

S±,β = ;

2

• for ± = 1 compute:

π

2 W cos V

2 π

+ βV tan V ’ β log

X= . (1.4)

π

π 2 2 + βV

Given the formulas for simulation of a standard ±-stable random variable, we

can easily simulate a stable random variable for all admissible values of the

parameters ±, σ, β and µ using the following property: if X ∼ S± (1, β, 0) then

±

σX + µ, ± = 1,

Y= (1.5)

2

σX + π βσ log σ + µ, ± = 1,

is S± (σ, β, µ). Although many other approaches have been presented in the

literature, this method is regarded as the fastest and the most accurate.

Quantlets rndstab and rndsstab use formulas (1.3)-(1.5) and provide pseudo

random variables of stable and symmetric stable distributions, respectively.

Parameters alpha and sigma in both quantlets and beta and mu in the ¬rst

one determine the parameters of the stable distribution.

18 1 Stable distributions in ¬nance

1.2.3 Tail behavior

Levy (1925) has shown that when ± < 2 the tails of ±-stable distributions are

asymptotically equivalent to a Pareto law. Namely, if X ∼ S±<2 (1, β, 0) then

as x ’ ∞:

P (X > x) = 1 ’ F (x) ’ C± (1 + β)x’± ,

(1.6)

P (X < ’x) = F (’x) ’ C± (1 ’ β)x’± ,

where

’1

∞

1 π±

’±

C± = 2 x sin xdx = “(±) sin .

π 2

0

The convergence to a power-law tail varies for di¬erent ±™s (Mandelbrot, 1997,

Chapter 14) and, as can be seen in Figure 1.5, is slower for larger values of

the tail index. Moreover, the tails of ±-stable distribution functions exhibit

a crossover from an approximate power decay with exponent ± > 2 to the

true tail with exponent ±. This phenomenon is more visible for large ±™s

(Weron, 2001).

1.3 Estimation of parameters

The estimation of stable law parameters is in general severely hampered by the

lack of known closed“form density functions for all but a few members of the

stable family. Most of the conventional methods in mathematical statistics,

including the maximum likelihood estimation method, cannot be used directly

in this case, since these methods depend on an explicit form for the density.

However, there are numerical methods that have been found useful in practice

and are discussed in this section.

All presented methods work quite well assuming that the sample under con-

sideration is indeed ±-stable. However, if the data comes from a di¬erent

distribution, these procedures may mislead more than the Hill and direct tail

estimation methods. Since there are no formal tests for assessing the ±-stability

of a data set we suggest to ¬rst apply the ”visual inspection”or non-parametric

tests to see whether the empirical densities resemble those of ±-stable laws.

Given a sample x1 , ..., xn from S± (σ, β, µ), in what follows, we will provide

ˆˆˆ

estimates ±, σ , β and µ of ±, σ, β and µ, respectively.

ˆ

1.3 Estimation of parameters 19

Tails of stable laws

-5

log(1-CDF(x))

-10

0 1 2

log(x)

Figure 1.5: Right tails of symmetric ±-stable distribution functions for ± = 2