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ii, Contents,1 Introduction 1,2 Univariate Stable Distributions 3. 2 1 Introduction 3, 2 1 1 Other Types of Stability 4. 2 2 Theoretical Background 4, 2 2 1 Introduction and Basic Definitions 4. 2 2 2 Interpretations of the Parameters 10, 2 2 3 Properties of Stable Distributions 11. 2 2 4 Closed Form Formulas for Stable Distributions 13. 2 2 5 Plots of Stable Densities and Distributions 13. 2 2 6 Other Parameterizations 13, 2 3 Numerical Techniques for Stable Distributions 17.

2 3 1 Computation of the Density 17, 2 3 2 Computation of the Distribution Function 21. 2 3 3 Computation of Stable Quantiles 21, 2 3 4 Generation of Stable Random Numbers 22. 2 4 Estimation of Stable Parameters 23, 2 4 1 Estimation via Maximum Likelihood 23. 2 4 2 Quantile Based Methods 24, 2 4 3 Methods based on Fitting the Characteristic Function 25. 2 4 4 An Estimator of the Stable Parameters Based on Q Q Plots 26. 2 4 5 Empirical Influence Functions 27, 2 5 Score and Fisher Information Matrix for Stable Laws 45.

3 Univariate t Distributions 49, 3 1 Introduction 49. 3 2 Score Function and Information Matrix for t Law 50. 3 3 Estimators 52,4 Performance of Estimators 57, 4 1 Estimator comparisons 57. 4 1 1 Estimators of stable distribution parameters 57. 4 1 2 Estimators of t distribution parameters 58, iii. iv CONTENTS,5 Tail Fatness and Firm Size 83, 5 1 Introduction 83. 5 2 Stable Law Parameters and Firm Size 83, 5 2 1 Data Setup 83.

5 2 2 Statistics 83, 5 2 3 Plots 90, 5 2 4 Models 100. 5 3 Cross Sectional Analysis 104, List of Notation. F G The convolution of F and G,N The natural numbers 1 2 3 . R The real numbers,C The complex numbers,iid independent and identically distributed. d, converges in distribution to, s x density of a stable distribution with parameters and .

S x distribution function of a stable distribution with parameters and . T Student s t distribution with degrees of freedom. v, vi CONTENTS, Chapter 1,Introduction, The term heavy tailed distribution refers to statistical distributions with more mass in. their tails than the standard normal a k a Gaussian distribution In essence greater. mass in the tails of these distribution means that under these models extreme events. are more likely to occur than under the normal model Heavy tailed distributions have. become important tools in modern financial work where the failure of the normal model. to adequately capture the observed frequency of extreme events such as market crashes . is or should be by now well known Tei71 MRP98 RM00 . One of the earliest uses of distributional modeling in finance was in the study of. changes in price between transactions in speculative markets Fam63 The seminal works. of Bachelier Bac67 and Osborne Osb59 gave theoretical arguments in support of the. hypothesis of approximately normally distributed price changes Empirical work by Kendall. KH53 and others supported this hypothesis while at the same time suggesting that the. tails of the empirical distributions of price changes where heavier than those of the normal. distribution Fam63 This lead to investigations by Mandelbrot Man63a Man63b and. Fama Fam63 into alternative models for price change data . Mandelbrot argued for distributional models that satisfied the Pareto law at least. in the tails of the distirbution Man63a The stable distributions of L vy L v24 were. natural candidates for models due to their closure under certain types of finite sums of. random variables and their status as the only possible limits of infinite sums of random. variables Unfortunately the lack of closed form formulas for the stable distributions. hindered their widespread use Numerical methods for stable distributions were available. by the early 70 s DuM71 but computers at the time were not fast enough to permit large. empirical studies It was not until the 90 s that faster algorithms and faster hardware made. the use of stable distributions practical In the meantime other heavy tailed distributions . such as Student s t and the various extreme value distributions served as computationally. feasible substitutes , Heavy tailed models have found their way into all areas of finance This work is far. from a comprehensive investigation of all the uses of heavy tailed distributions in finance . Rather we have limited our focus to some of the more practical concerns when using. heavy tailed distributions , How well do estimators of stable distribution parameters perform in the presence of. outliers , 1, 2 CHAPTER 1 INTRODUCTION, Is the extra effort required by stable distributions warranted Would a simpler to use. model such as the t model give results that are just as good . In the location scale model what is the correlation between the tail fatness parame . ter and scale , Is tail fatness related to firm size If so how .

Chapter 2,Univariate Stable Distributions,2 1 Introduction. Stable distributions play an important role in the theory of probability they are the only. possible limiting distributions of infinite sums of independent identially distributed iid . random variables When the random variables all have finite variances this result is more. commonly known as the Central Limit Theorem and the limiting distribution is the familar. normal or Gaussian distribution The other members of the stable family the limiting. distributions when the restriction of finite variance is removed are just as interesting as. the normal but much harder to study due to the lack of closed form formulas for their. densities and distribution functions , The concept of a stable distribution was first introduced by L vy around 1924 L v24 . in his studies of sums of independent random variables Hal81 Some of the basic details. of stable distributions remained unclear though until the 1936 paper of Khintchine and. Levy KL36 , According to L vy s original definition a distribution F is stable if for each pair of. positive real numbers a1 and a2 there exists another positive real number a such that. F a1 x F a2 x F ax 2 1 1 , Here denotes convolution This definition however has a significant drawback . in certain cases stability is not preserved under translation i e F x b is not stable. even though F x is This led L vy to introduce the weaker notion of quasi stability . a distribution F is quasi stable if for all real numbers a1 b1 a2 and b2 with a1 and a2. positive there exist real numbers a and b with a positive such that. F a1 x b1 F a2 x b2 F ax b 2 1 2 , In the modern literature L vy s quasi stable distributions are known as stable distrib .

utions while his stable distributions are referred to as strictly stable distributions Hal81 . Stable distributions have popped up in numerous scientific fields over the years The. earliest known occurrence of a stable distribution is generally agreed to be the 1919 paper. of the Danish astronomer Holtsmark HC73 Zol86 Hol19 In his studies of the gravita . tional field of stars he derived via Fourier transform methods a probability distribution. 3, 4 CHAPTER 2 UNIVARIATE STABLE DISTRIBUTIONS, for the gravitational force exerted by a group of stars at a point in space This distribution . now known in astrophysics as the Holtsmark distribution corresponds to a symmetric. stable distribution with index 3 2 see below 1, Stable distributions have also proven useful in the study of Brownian motion Fel71 . in economics and finance Man63a Man63b Fam63 in electrical engineering HC73 . and in telecommunications Kur01 We will discuss their applications in finance in a later. chapter the interested reader can refer to the citations for more information about other. applications ,2 1 1 Other Types of Stability, In this work we shall only be concerned with stability under nonrandom summation . often called Paretian stability due to the Pareto like behavior of the tails of such stable. distributions There are however other types of stability and the resulting distributions . that could be discussed For instance we could consider stability with respect to the max . imum operation i e distributions for which max Xi i 1 n is equal in distribution. to X1 after suitable translation and rescaling whenever the random variables Xi are iid . We could also consider stability under random summation etc We refer the reader to the. work of Rachev and Mittnik RM00 for more information on such matters . 2 2 Theoretical Background,2 2 1 Introduction and Basic Definitions. First let us start with a definition of a stable distribution that is more modern than that of. L vy , Definition 2 2 1 A non degenerate distribution F is stable if for all n N there exist.

constants cn 0 and dn such that whenever X1 X2 Xn and X are independently and. identically distributed with distribution function F the sum X1 Xn is distributed as. cn X dn DuM71 , A random variable X is termed stable if its distribution function is stable . If dn 0 for all n the distribution random variable etc is said be strictly stable . We will show below that the coefficient cn must take the form n1 for some . 0 2 The exponent is called the index or characteristic exponent of the distribution . The easiest way to develop stable distributions is via characteristic functions Fel71 . Lam96 Let k t and t be the characteristic functions of Xk and X respectively The. characteristic function of the sum X1 Xn is simply the product of the character . istic functions of the summands i e nk 1 k t Since the Xk are iid the characteristic. functions k t are all identical so this product equals t n . 1 Although Holtsmark s work predates that of L vy Holtsmark only studied the specific case mentioned . It was L vy who first introduced the notion of stability and who did the first in depth work on the stable. family thus he is commonly credited with their invention . 2 2 THEORETICAL BACKGROUND 5, The characteristic function of cn X dn on the other hand is eidnt cnt Since the. two quantities X1 Xn and cn X dn are equal in distribution their characteristic. functions must agree everywhere so we must have, t n eidn cnt 2 2 3 . Let us consider first the case of F symmetric about x 02 The characteristic function. of a symmetric distribution is real valued3 Furthermore it is a continuous even function. of its argument Finally we always have 0 1 for a characteristic function . Next notice that by symmetry we have, d d d, X1 Xn X1 Xn cn X dn cn X dn . which forces dn 0 Thus t n cnt If cn 1 for all n then t is identically 1 . and F is a degenerate distribution Since we have excluded that case cn 6 1 for at least. one n , We now show that t is supported on the entire real line Suppose to the contrary.

that t vanishes somewhere in R Then because it is a continous function that attains the. value 1 at 0 the zero set of t restricted to the positive real line must have a smallest. element t0 Using our previous observation we see that. 0 t0 n cnt0 ,and, 0 t0 cn n t0 , Hence both cnt0 and t0 cn are also zeros of t But since cn is positive and not 1 one. of these numbers is strictly smaller than t0 which contradicts the choice of t0 Hence we. conclude that t 0 for all t R , Now that we know t is a positive function we may safely work with its logarithm . which we will call t Since t is real valued and bounded above by 1 t is a real . valued continuous nonpositive function In terms of t our functional equation 2 2 3 . for the characteristic function of a stable distribution is. n t cnt 2 2 4 , At this point it is not entirely obvious that the constant cn is unique for a fixed n Suppose. that for a given n there are two constants cn and c0n for which 2 2 4 holds Without. loss of generality we may assume that cn c0n It is clear that. 0 , c, n t t , cn, 2 Our argument follows that of Lamperti Lam96 . 3 The characteristic function of X is the complex conjugate of that of X By symmetry however X and. X are identically distributed and hence have identical characteristic functions Thus imaginary part of. the characteristic function of X must vanish The same reasoning can be used to show that the characteristic. function of X is an even function , 6 CHAPTER 2 UNIVARIATE STABLE DISTRIBUTIONS.

By repeated use of this relation we can actually establish the relation. 0 k , cn, t t , cn, for every k 1 Since cn c0n c0n cn 1 and as k c0n cn 0 Since t is. continuous we must have t 0 0 for all t This means that F is a degenerate. distribution which contradicts our assumption about F Thus cn c0n i e cn is unique. Lam96 , Our functional relation 2 2 4 is multiplicative i e it implies that for any two inte . gers m n, cmnt mn t m cnt cm cnt , Since the constants cn cm and cmn are unique this forces cmn cm cn for all pairs of. positive integers m and n , Our functional relation 2 2 4 will be easier to solve if we allow n to take values in. the positive real numbers To this end we must extend our coefficient identity cmn cm cn. to positive real indices We switch to a more traditional notation c y for the coefficients. here to avoid confusion for any integer n we will have c n cn It is clear that our. argument establishing the uniqueness of the coefficients still holds as well . We can easily see that c 1 1 We define the extension of the coefficient identit. Contents 1 Introduction 1 2 Univariate Stable Distributions 3 2 1 Introduction 3 2 1 1 Other Types

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