distribution yet yields finite variance is named truncated Levy flight.
This distribution is defined as [2]
f
TL
(x) ¼
0, jxj >‘
Cf
L
(x), x
(3:3:9)
In (3.3.9), f
L
(x) is the Levy distribution is the cutoff length, and C is
the normalization constant. Sometimes the exponential cut-off is used
at large distances [3]
f
TL
(x) exp ( l jx j), l > 0, jxj >‘ (3:3:10)
Since f
TL
(x) has finite variance, it converges to the normal distribu-
tion according to the central limit theorem.
3.4 REFERENCES FOR FURTHER READING
The Feller’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 1994–2003. What do you
think of the opposite signs for some of these coefficients?
2. Familiarize yourself with Microsoft Excel’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) ¼ 0:95 using the NORMSINV function; (c)
calculate x such that Pr(Z x) ¼ 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 ¼ 0 and s ¼ 1, respectively? (e) Do
the same for the standard uniform distribution as in (d).
Probability Distributions 27
(f) Generate 100 normally distributed random numbers x using
the function x ¼ 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 X
1
and X
2
be two independent copies of the normal distri-
bution X N(m, s
2
). Since X is stable, aX
1
þ bX
2
CX þ D.
Calculate C and D via given m, s, a, and b.
28 Probability Distributions

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