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PROJECT 1 IN MATH 889 Project 1: Analyzing the Distribution of Stock Price Changes Minrong Zheng Advisor: Steven Dunbar 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 http://www.riskglossary.com/link/stable_paretian_distributions.htm 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