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distribution) has a finite tail.

25

Probability Distributions

3.3 STABLE DISTRIBUTIONS AND SCALE

INVARIANCE

The principal property of stable distribution is that the sum of

variates has the same distribution shape as that of addends (see,

e.g., [6] for details). Both the Cauchy distribution and the normal

distribution are stable. This means, in particular, that the sum of

two normal distributions with the same mean and variance is also the

normal distribution (see Exercise 2). The general definition for

the stable distributions was given by Levy. Therefore, the stable

distributions are also called the Levy distributions.

Consider the Fourier transform F(q) of the probability distribution

function f(x)

Ã°

F(q) Â¼ f(x)eiqx dx (3:3:1)

The function F(q) is also called the characteristic function of the

stochastic process. It can be shown that the logarithm of the charac-

teristic function for the Levy distribution has the following form

(

imq Ã€ gjqja [1 Ã€ ibd tan (pa=2)], if a 6Â¼ 1

ln FL (q) Â¼ (3:3:2)

imq Ã€ gjqj[1 Ã¾ 2ibd ln (jqj)=p)], if a Â¼ 1

In (3.3.2), d Â¼ q=jqj and the distribution parameters must satisfy the

following conditions

2, Ã€ 1

0<a 1, g > 0 (3:3:3)

b

The parameter m corresponds to the mean of the stable distribution

and can be any real number. The parameter a characterizes the

distribution peakedness. If a Â¼ 2, the distribution is normal. The

parameter b characterizes skewness of the distribution. Note that

skewness of the normal distribution equals zero and the parameter

b does not affect the characteristic function with a Â¼ 2. For the

normal distribution

ln FN (q) Â¼ imq Ã€ gq2 (3:3:4)

The non-negative parameter g is the scale factor that characterizes the

spread of the distribution. In the case of the normal distribution,

g Â¼ s2 =2 (where s2 is variance). The Cauchy distribution is defined

26 Probability Distributions

with the parameters a Â¼ 1 and b Â¼ 0. Its characteristic function

equals

ln FC (q) Â¼ imq Ã€ gjqj (3:3:5)

The important feature of the stable distributions with a < 2 is that

they exhibit the power-law decay at large absolute values of the

argument x

fL (jxj) $ jxjÃ€(1Ã¾a) (3:3:6)

The distributions with the power-law asymptotes are also named the

Pareto distributions. Many processes exhibit power-law asymptotic

behavior. Hence, there has been persistent interest to the stable distri-

butions.

The power-law distributions describe the scale-free processes. Scale

invariance of a distribution means that it has a similar shape on

different scales of independent variables. Namely, function f(x) is

scale-invariant to transformation x ! ax if there is such parameter

L that

f(x) Â¼ Lf(ax) (3:3:7)

The solution to equation (3.3.7) is simply the power law

f(x) Â¼ xn (3:3:8)

where n Â¼ Ã€ln (L)= ln (a). The power-law function f(x) (3.3.8) is scale-

free since the ratio f(ax)=f(x) Â¼ L does not depend on x. Note that the

parameter a is closely related to the fractal dimension of the function

f(x). The fractal theory will be discussed in Chapter 6.

Unfortunately, the moments of stable processes E[xn ] with power-

law asymptotes (i.e., when a < 2) diverge for n ! a. As a result, the

mean of a stable process is infinite when a 1. In addition, variance

of a stable process is infinite when a < 2. Therefore, the normal

distribution is the only stable distribution with finite mean and finite

variance.

The stable distributions have very helpful features for data analysis

such as flexible description of peakedness and skewness. However, as it

was mentioned previously, the usage of the stable distributions in

financial applications is often restricted because of their infinite vari-

ance at a < 2. The compromise that retains flexibility of the Levy

27

Probability Distributions

distribution yet yields finite variance is named truncated Levy flight.

This distribution is defined as [2]

jxj > â€˜

0,

fTL (x) Â¼ (3:3:9)

CfL (x), Ã€â€˜ x â€˜

In (3.3.9), fL (x) is the Levy distribution â€˜ is the cutoff length, and C is

the normalization constant. Sometimes the exponential cut-off is used

at large distances [3]

fTL (x) $ exp ( Ã€ ljxj), l > 0, jxj > â€˜ (3:3:10)

Since fTL (x) has finite variance, it converges to the normal distribu-

tion according to the central limit theorem.

3.4 REFERENCES FOR FURTHER READING

The Fellerâ€™s textbook is the classical reference to the probability

theory [1]. The concept of scaling in financial data has been advocated

by Mandelbrot since the 1960s (see the collection of his work in [7]).

This problem is widely discussed in the current Econophysics litera-

ture [2, 3, 8].

3.5 EXERCISES

1. Calculate the correlation coefficients between the prices of

Microsoft (MSFT), Intel (INTC), and Wal-Mart (WMT). Use

monthly closing prices for the period 1994â€“2003. What do you

think of the opposite signs for some of these coefficients?

2. Familiarize yourself with Microsoft Excelâ€™s statistical tools. As-

suming that Z is the standard normal distribution: (a) calculate

Pr(1 Z 3) using the NORMSDIST function; (b) calculate x

such that Pr(Z x) Â¼ 0:95 using the NORMSINV function; (c)

calculate x such that Pr(Z ! x) Â¼ 0:15; (d) generate 100 random

numbers from the standard normal distribution using Tools/

Data Analysis/Random Number Generation. Calculate the

sample mean and standard variance. How do they differ from

the theoretical values of m Â¼ 0 and s Â¼ 1, respectively? (e) Do

the same for the standard uniform distribution as in (d).

28 Probability Distributions

(f) Generate 100 normally distributed random numbers x using

the function x Â¼ NORMSINV(z) where z is taken from a sample

of the standard uniform distribution. Explain why it is possible.

Calculate the sample mean and the standard deviation. How do

they differ from the theoretical values of m and s, respectively?

3. Calculate mean, standard deviation, excess kurtosis, and skew

for the SPY data sample from Exercise 2.1. Draw the distribu-

tion function of this data set in comparison with the standard

normal distribution and the standard Cauchy distribution.

Compare results with Figure 3.1.

Hint: (1) Normalize returns by subtracting their mean and divid-

ing the results by the standard deviation. (2) Calculate the histo-

gram using the Histogram tool of the Data Analysis menu. (3)

Divide the histogram frequencies with the product of their sum and

the bin size (explain why it is necessary).

4. Let X1 and X2 be two independent copies of the normal distri-

bution X $ N(m, s2 ). Since X is stable, aX1 Ã¾ bX2 $ CX Ã¾ D.

Calculate C and D via given m, s, a, and b.

Chapter 4

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