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```PROJECT 1 IN MATH 889
Project 1:
Analyzing the Distribution of
Stock Price Changes
Minrong Zheng
1. Introduction of the project:
There are at least four schools of thought on the statistical distribution of stock
price differences, or more generally, stochastic models for sequences of stock prices.
In terms of followers, by far the most popular approach is that of the so-call
``technical analyst'', phrased in terms of short term trends, support and resistance
levels, technical rebounds, and so on.
Rejecting this technical viewpoint, two other schools agree that sequences of
prices describe a random walk, where price changes are independent of the
previous price history, but these schools disagree in their choice of the appropriate
probability distributions. Some authors find price changes to have a normal
distribution while the other group finds a distribution with ``fatter tail probabilities'',
and perhaps even an infinite variance. Finally, a fourth group, overlapping with the
previous two admits the random walk as a first-order approximation but notes sizable
second-order effects.
This project is to show compatibility between the middle two groups, i.e., under the
assumption of the price changes are independent of the previous price history;
consider the possibility of using distributions other than normal distribution in
modeling the price change. To reach the aim, we consider four type random variable
distributions: standard normal distribution, the so-called ``double exponential random
variable'', standard uniform random variable, and Cauchy random variable. First we
compare the “fatter tail “property of these four distributions, and then we will try to
evaluate the efficiency of “coefficient of excess” as a measurement of fatter tail
probability. At last, we will do a little further analysis about the factors that would
make it easy to use this random variable and it’s PDF in models of stock price
changes.
1
PROJECT 1 IN MATH 889
2. Theoretical framework:
2.1. Fatter tail probability:
The tails are the part of probability from a specific point a to infinity. We say that
one probability distribution x has a fatter tail than another probability distribution y if
Pr x a
Pr y a
(1)
for all a>K, for some K.
2.2. Definition of coefficient of excess:
In order to understand the concept better, we first regard to the term “kurtosis”,
m4
which is noted as Î²2 ( Î²2
m 22
), for standard normal distribution, it is 3. Taking the
term “coefficient of excess”, marked as Î³2 , it is Î²2 3 . So coefficient of excess is a
relative quantity, defined as
m4
Î³2
3
m22
(2)
Where mk is the kth moment of the random variable about its mean.
Î³2 is normally taken as the measurement of fatter tail ability, although sometimes
there will be some dispute on it.
2.3. Four type distributions considered in this project
2.3.1 Standard normal random variable Z, and its PDF is
f
z
x
x2
2
e
2 Ï€
(3)
2.3.2 Double exponential distribution, X, usually called Laplace distribution. It is the
distribution of differences between two independent variables with identical
exponential distributions. The PDF is
f x
x
e
(4)
2
2.3.3 Standard uniform random variable U with PDF
f x
1
x
0. 5,0 .5
0 elsewhere
2
(5)
PROJECT 1 IN MATH 889
2.3.4 Cauchy random variable Y: Suppose a random variable Y is defined as Y
X1
X2
Where X 1 and X 2 are independent standard normal random variables. It is a
stable distribution. The Cauchy distribution has probability density function as
f x
1
Ï€ 1 x 2
(6)
Figure 1 the comparison of PDF of four different type distributions
3 “Fatter tail probabilities'' and its relationship to the coefficient of
excess
According to the definition of the fatter tail probability, we compute the probability
for the four distribution with a=1, 2, 3, 4, through which we can easily compare the
four distributions’ fatter tail probability. The calculation results are listed in table 1.
Furthermore, I also compute the intersection point between PDFs of Standard
normal distribution and Double exponential distribution, which is 1.7406; between
PDFs of Standard normal distribution and Cauchy distribution, which is 1.8512;
between PDFs of Double exponential distribution and Cauchy distribution, which is
2.2643.
3
PROJECT 1 IN MATH 889
Table 1 the hazard probability comparison for the four type distributions
Pr(x>a)
Standard normal
Double
a
distribution
1
exponential
Standard uniform
Cauchy
distribution
distribution
distribution
.1587
.18394
0
.25
2
.0228
.0676676
0
.1475836
3
.0013
.0249
0
.1024164
4
0
.00916
0
.07797913
3.1. Comparison of random variables
,
,
and Y in terms of the fatness of
the tails
According to the definition of fatter tail probabilies, from table 1, we can see that tail
probability of standard normal distribution is fatter than that of uniform distribution;
tail probability of double exponential distribution is fatter than that of standard normal
distribution, and Cauchy distribution tail probability is fatter than double expotential
distribution. If we take K=2.2643, we surely arrive at the same conclusion.
3.2. Computation of coefficient of excess:
We easily find that the expectation values of the first three distributions are 0.
3.2.1 Find the coefficient of excess Î³2 Z for a normal r.v. with pdf. f
m4
x
1
2 Ï€
m2
2 Ï€
2 Ï€
limb
x
1
4e
x2
2
2e
limb
1
dx
2 Ï€
3 xe
x2
2
2 Ï€
dx
xe
b
limb
x2
2 b
b
x
2
b
b
1
b
3e
b
2 Ï€
x2
2 b
limb
b
4
x e
x
2
2
dx
1
2 Ï€
2
x e
3
x
2
1
2 Ï€
b
b
3e
x
2
e
x
2
x e
2
dx 3
2
dx
2
b
limb
2
dx
3
1
dx
2 Ï€
4
e
x
2
2
dx 1
x
2
z
x2
2
e
x
2 Ï€
2
b
b
b
b
2
3x e
x
2
2
dx
PROJECT 1 IN MATH 889
So Î³2 Z
0
3.2.2 Find the coefficient of excess Î³2 X for a double exponential r.v. with
: From the synmetricity of the PDF,
2
x4
m4
x
e
PDF f x
x
e
2
4 x3 e
limb
x2
m2
0
2
x
12 x 2 e x dx
b
x
e
x4e
b
xb
0
b
dx limb
0
x2 e
dx limb
limb
x
dx limb
b
x4e
x b
0
12 x 2 e
xb
0
b
x2 e
xb
0
b
0
b
0
4 x3 e
x
dx
24 xe
x
dx
2 xe
x
dx
24 e
0
0
x
m 4 m2 3 3
3.2.3
Find
the
1
PDF f x
x
0. 5,0 .5
0 elsewhere
4
m4
x f x dx
m2
x f x dx
4
2
coefficient
m4 m2 3
.5
.5
.5
.5
4
x dx
2
x dx
of
excess
Î³2 U for
a
uniform
r.v.
with
.
1
80
1
12
1. 2
3.2.4. Cauchy distribution:
The distribution's coefficient of excess is undefined. Actually, its mean and
standard deviation are also not defined.
3.3. Relationship of ``fatter tail probabilities'' and the coefficient of excess
From the above computation, we easily find that Î³2 U
Î³2 Z
Î³2 X .
Combining it with the fatter tail probabilities comparison results, we can find
coefficient of excess is really a good measurement of fatness, as long as it exists.
Furthermore, it is bigger, the tail is fatter. For Cauchy distribution, coefficient of
excess does not exist, we can judge this property according to the value of Pr(x>a).
Furthermore, the variance of the distribution is bigger the “tail” is fatter.
4. Factors affecting the use of these random variables in models of
stock price changes
5
dx 24
2 e x dx 2
2
Î³2 X
Î³2 U
b
dx limb
PROJECT 1 IN MATH 889
Normal distribution is the earliest and widely used in modeling the stock price
change, for example, the famous Black-Scholes model takes normal distribution as
the stock price change step distribution. I searched papers using the key words
“fatter tail probabilities”, “stock price” and plus “normal double exponential
distribution”, “normal uniform distribution” and “Cauchy distribution” respectively. I
found a lot and skimmed some of them. I get information from the papers listed
below. I found it was to adapt to the real stock price change figures, other random
distributions and models (other than black-scholes model) are becoming used in the
simulations
In the middle two groups in the first part in this report, the common idea of them
is that price changes are independent of the previous price history, the same as the
property of Brownian motion, in which model, standard normal distribution is used to
model the stock price change.
Firstly, the variance of the distribution will affect the application of these three
distributions. Since Cauchy distribution dose not have variance, mean value and
coefficient of excess. Since the concept of variance is crucial in modeling stock price,
it makes it difficult to use this distribution into modeling price change. The other three
including the normal distributions seem to be ok considering this aspect.
The second factor is the fatter tail property. Overwhelming empirical evidence
shows that the distribution of stock price change has tails fatter than those of the
normal distributions. The normal double exponential distribution and Cauchy
distribution both have fatter tails than normal distribution. On the contrary, uniform
distribution has thinner tail than normal distribution.
Third, we will consider asymmetric property. All of the three distributions have
asymmetric distribution, similar to the standard normal distribution, which makes
them be applied to modeling stock price change easily.
Forth, we arrive to Generalised Central Limit Theorem, which states that the only
possible non-trivial limit of normalised sums of independent identically distributed
terms is stable, which can be use to model the sum of independent innovations the
price of a stock. Since Cauchy distribution is one of stable distribution, from this
aspect, it may be used to model price change. This also should be fine for other two
distributions.
5. Conclusion
In this report, three random variable distributions are discussed in this report as
6
PROJECT 1 IN MATH 889
far as the concept “fatter tail probabilities” are concerned. The relationship between
the concepts of “fatter tail probabilities” and “coefficient of excess”, as well as the
possibility of using the three random distributions to model stock price, is discussed.
Reference: papers or web pages:
2.1
2.2
http://www.math.unl.edu/~sdunbar1/Teaching/MathematicalFinance/Projects/fattails
2.3
Antonis Parapantoken, Tino Senge, “ Option Pricing in a jump diffusion model with double
exponential jumps”, Quantitative Research ,June, 6, 2002.
2.4
http://www.institute.mathfinance.de/colloquium/abstracts/Antonis.pdf
2.5
http://mathworld.wolfram.com/LaplaceDistribution.html
2.6
http://www.columbia.edu/~sk75/MagSci02.pdf ( an very important paper)
2.7
http://www.math.uvic.ca/faculty/reed/NL.draft.1.pdf (*)
2.8
http://www.bos.frb.org/economic/neer/neer1997/neer697b.pdf
2.9
http://www.essex.ac.uk/ccfea/Seminarpapers/Spring0405/Slides/Elias_Dinenis.doc
2.10
http://www.santafe.edu/research/publications/workingpapers/04-02-006.pdf
7
```
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