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CHAPTER 13 Fat Tails, Scaling, and Stable Laws ost models of stochastic processes and time series examined thus far assume that distributions have ﬁnite mean and ﬁnite variance. In this chapter we describe fat tailed distributions with inﬁnite variance. Fat-tailed distributions have been found in many ﬁnancial economic variables ranging from forecasting returns on ﬁnancial assets to modeling recovery distributions in bankruptcies. They have also been found in numerous insurance applications such as catastrophic insurance claims and in value-at-risk measures employed by risk managers. In this chapter, we review the related concepts of fat-tailed, powerlaw and Levy-stable distributions, scaling and self-similarity, as well as explore the mechanisms that generate these distributions. We discuss the key intuition relative to the applicability of fat-tailed or scaling processes to ﬁnance: In a fat-tailed or scaling world (as opposed to an ergodic world), the past does not offer an exhaustive set of possible conﬁgurations. Adopting, as an approximation, a scaling description of ﬁnancial phenomena implies the belief that only a small space of possible conﬁgurations has been explored; vast regions remain unexplored. We begin with the mathematics of fat-tailed processes, followed by a discussion of classical Extreme Value Theory for independent and identically distributed sequences. We then explore the consequences of eliminating the assumption of independence and discuss different concepts of scaling and self similarity. Finally, we present evidence of fat tails in ﬁnancial phenomena and discuss applications of Extreme Value Theory. M 351 352 The Mathematics of Financial Modeling and Investment Management SCALING, STABLE LAWS, AND FAT TAILS Let's begin with a review of the different but related concepts and properties of fat tails, power laws, and stable laws. These concepts appear frequently in the ﬁnancial and economic literature, applied to both random variables and stochastic processes. Fat Tails Consider a random variable X. By deﬁnition, X is a real-valued function from the set Ω of the possible outcomes to the set R of real numbers, such that the set (X ≤ x) is an event. Recall from Chapter 6 that if P(X ≤ x) is the probability of the event (X ≤ x), the function F(x) = P(X ≤ x) is a well-deﬁned function for every real number x. The function F(x) is called the cumulative distribution function, or simply the distribution function, of the random variable X. Note that X denotes a function Ω → R, x is a real variable, and F(x) is an ordinary real-valued function that assumes values in the interval [0,1]. If the function F(x) admits a derivative dF ( x ) f ( x ) = --------------dx The function f(x) is called the probability density of the random variable X. The function F ( x ) = 1 – F ( x ) is the tail of the distribution F(x). The function F ( x ) is called the survival function. Fat tails are somewhat arbitrarily deﬁned. Intuitively, a fat-tailed distribution is a distribution that has more weight in the tails than some reference distribution. The exponential decay of the tail is generally assumed as the borderline separating fat-tailed from light-tailed distributions. In the literature, distributions with a power-law decay of the tails are referred to as heavy-tailed distributions. It is sometimes assumed that the reference distribution is Gaussian (i.e., normal), but this is unsatisfactory; it implies, for instance, that exponential distributions are fat-tailed because Gaussian tails decay as the square of an exponential and thus faster than an exponential. These characterizations of fat-tailedness (or heavy-tailedness) are not convenient from a mathematical and statistical point of view. It would be preferable to deﬁne fat-tailedness in terms of a function of some essential property that can be associated to it. Several proposals have been advanced. Widely used deﬁnitions focus on the moments of the distribution. Deﬁnitions of fat-tailedness based on a single moment focus either on the second moment, the variance, or the kurtosis, deﬁned as the fourth moment divided by the square of the variance. In fact, a distribution is often considered fat-tailed if its variance is inﬁnite or if it is leptokurtic Fat Tails, Scaling, and Stable Laws 353 (i.e., its kurtosis is greater than 3). However, as remarked by Bryson1 deﬁnitions of this type are too crude and should be replaced by more complete descriptions of tail behavior. Others consider a distribution fat-tailed if all its exponential moments sX are inﬁnite, E [ e ] = ∞ for every s ≥ 0. This condition implies that the moment-generating function does not exist. Some suggest weakening this condition, deﬁning fat-tailed distributions as those distributions that do not have a ﬁnite exponential moment of ﬁrst order. Exponential moments are particularly important in ﬁnance and economics when the logarithm of variables, for instance logprices, are the primary quantity to be modeled.2 Fat-tailedness has a consequence of practical importance: the probability of extremal events (i.e., the probability that the random variable assumes large values) is much higher than in the case of normal distributions. A fat-tailed distribution assigns higher probabilities to extremal events than would a normal distribution. For instance, a six-sigma event (i.e., a realized value of a random variable whose difference from the mean is six times the size of the standard deviation) has a near zero probability in a Gaussian distribution but might have a nonnegligible probability in fat-tailed distributions. The notion of fat-tailedness can be made quantitative as different distributions have different degrees of fat-tailedness. The degree of fattailedness dictates the weight of the tails and thus the probability of extremal events. Extreme Value Theory attempts to estimate the entire tail region, and therefore the degree of fat-tailedness, from a ﬁnite sample. A number of indicators for evaluating the size of extremal events have been proposed; among these are the extremal claim index proposed in Embrechts, Kluppelberg, and Mikosch,3 which plays an important role in risk management. The Class L of Fat-Tailed Distributions Many important classes of fat-tailed distributions have been deﬁned; each is characterized by special statistical properties that are important in given application domains. We will introduce a number of such classes in order of inclusion, starting from the class with the broadest membership: the class L, which is deﬁned as follows. Suppose that F is a 1 M.C. Bryson, "Heavy-Tailed Distributions," in N.L. Kotz and S. Read (eds.), Encyclopedia of Statistical Sciences, Vol. 3 (New York: John Wiley & Sons, 1982), pp. 598–601. 2 See G. Bamberg and D. Dorfleitner, "Fat Tails and Traditional Capital Market Theory," Working Paper, University of Augsburg, August 2001. 3 P. Embrechts, C. Kluppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance (Berlin: Springer, 1999). 354 The Mathematics of Financial Modeling and Investment Management distribution function deﬁned in the domain (0,∞) with F < 1 in the entire domain (i.e., F is the distribution function of a positive random variable with a tail that never decays to zero). It is said that F ∈ L if, for any y > 0, the following property holds: F ( x – y) lim -------------------- = 1 , ∀y > 0 F(x) x→∞ We can rewrite the above property in an equivalent (and perhaps more intuitive from the probabilistic point of view) way. Under the same assumptions as above, it is said that, given a positive random variable X, its distribution function F ∈ L if the following property holds for any y > 0: F ( x + y) lim P ( X > x + y X > x) = lim --------------------- = 1 , ∀y > 0 x→∞ F(x) x→∞ Intuitively, this second property means that if it is known that a random variable exceeds a given value, then it will exceed any bigger value. Some authors deﬁne a distribution as being heavy-tailed if it satisﬁes this property. 4 It can be demonstrated that if a distribution F(x) ∈ L, then it has the following properties: ■ Inﬁnite exponential moments of every order: E[esX] = ∞ for every s ≥ 0 ■ lim F ( x ) e x→∞ λx = ∞ , ∀λ > 0 As distributions in class L have inﬁnite exponential moments of every order, they satisfy one of the previous deﬁnitions of fat-tailedness. However they might have ﬁnite or inﬁnite mean and variance. The class L is in fact quite broad. It includes, in particular, the two classes of subexponential distributions and distributions with regularly varying tails that are discussed in the following sections. Subexponential Distributions A class of fat-tailed distributions, widely used in insurance and telecommunications, is the class S of subexponential distributions. Introduced 4 See, for example, K. Sigman, "A Primer on Heavy-Tailed Distributions," Queueing Systems, 1999. 355 Fat Tails, Scaling, and Stable Laws by Chistyakov in 1964, subexponential distributions can be characterized by two equivalent properties: (1) the convolution closure property of the tails and (2) the property of the sums.5 The convolution closure property of the tails prescribes that the shape of the tail is preserved after the summation of identical and independent copies of a variable. This property asserts that, for x → ∞, the tail of a sum of independent and identical variables has the same shape as the tail of the variable itself. As the distribution of a sum of n independent variables is the n-convolution of their distributions, the convolution closure property can be written as n* F (x) lim ----------------- = n x→∞ F(x) Note that Gaussian distributions do not have this property although the sum of independent Gaussian distributions is again a Gaussian distribution. Subexponential distributions can be characterized by another important (and perhaps more intuitive) property, which is equivalent to the convolution closure property: In a sum of n variables, the largest value will be of the same order of magnitude as the sum itself. For any n, deﬁne n Sn ( x ) = ∑ Xi i=1 as a sum of independent and identical copies of a variable X and call Mn their maxima. In the limit of large x, the probability that the tail of the sum exceeds x equals the probability that the largest summand exceeds x: P ( S n > x) lim -------------------------= 1 x → ∞P ( M > x) n The class S of subexponential distributions is a proper subset of the class L. Every subexponential distribution belongs to the class L while it can be demonstrated (but this is not trivial) that there are distributions 5 See, for example, C. M. Goldie and C. Kluppelberg, "Subexponential Distributions," in R.J. Adler, R.E. Feldman, and M.S. Taqqu (eds.), A Practical Guide to Heavy Tails: Statistical Techniques and Applications (Boston: Birkhauser, 1998), pp. 435–459 and Embrechts, Kluppelberg, and Mikosch, Modelling Extremal Events for Insurance and Finance. 356 The Mathematics of Financial Modeling and Investment Management that belong to the class L but not to the class S. Distributions that have both properties are called subexponential as it can be demonstrated that, as all distributions in L, they satisfy the property: lim F ( x )e x→∞ λx = ∞ , ∀λ > 0 Note, however, that the class of distributions that satisﬁes the latter property is broader than the class of subexponential distributions; this is because the former includes, for instance, the class L.6 Subexponential distributions do not have ﬁnite exponential sX moments of any order, that is, E [ e ] = ∞ for every s ≥ 0. They may or may not have a ﬁnite mean and/or a ﬁnite variance. Consider, in fact, that the class of subexponential distributions includes both Pareto and Weibull distributions. The former have inﬁnite variance but might have ﬁnite or inﬁnite mean depending on the index; the latter have ﬁnite moments of every order (see below). The key indicators of subexponentiality are (1) the equivalence in the distribution of the tail between a variable and a sum of independent copies of the same variable and (2) the fact that a sum is dominated by its largest term. The importance of the largest terms in a sum can be made more quantitative using measures such as the large claims index introduced in Embrechts, Kluppelberg, and Mikosch that quantiﬁes the ratio between the largest p terms in a sum and the entire sum. The class of subexponential distributions is quite large. It includes not only Pareto and stable distributions but also log-gamma, lognormal, Benkander, Burr, and Weibull distributions. Pareto distributions and stable distributions are a particularly important subclass of subexponential distributions; these will be described in some detail below. Power-Law Distributions Power-law distributions are a particularly important subset of subexponential distributions. Their tails follow approximately an inverse power law, decaying as x–α. The exponent α is called the tail index of the distribution. To express formally the notion of approximate power-law decay, we need to introduce the class ℜ(α), equivalently written as ℜα of regularly varying functions. A positive function f is said to be regularly varying with index α or f ∈ ℜ(α) if the following condition holds: 6 See Sigman, "A Primer on Heavy-Tailed Distributions." 357 Fat Tails, Scaling, and Stable Laws f ( tx ) α lim ------------ = t x → ∞ f(x) A function f ∈ ℜ(0) is called slowly varying. It can be demonstrated that a regularly varying function f(x) of index α admits the representation f(x) = xαl(x) where l(x) is a slowly varying function. A distribution F is said to have a regularly varying tail if the following property holds: –α F = x l(x) where l is a slowly varying function. An example of a distribution with a regularly varying tail is Pareto's law. The latter can be written in various ways, including the following: c F ( x ) = P ( X > x) = --------------- for x ≥ 0 α c+x Power-law distributions are thus distributions with regularly varying tails. It can be demonstrated that they satisfy the convolution closure property of the tail. The distribution of the sum of n independent variables of tail index α is a power-law distribution of the same index α. Note that this property holds in the limit for x → ∞. Distributions with regularly varying tails are therefore a proper subset of subexponential distributions. Being subexponential, power laws have all the general properties of fat-tailed distributions and some additional ones. One particularly important property of distributions with regularly varying tails, valid for every tail index, is the rank-size order property. Suppose that samples from a power law of tail index α are ordered by size, and call Sr the size of the rth sample. One then ﬁnds that the law S r = ar 1 – --α is approximately veriﬁed. The well-known Zipf's law is an example of this rank-size ordering. Zipf's law states that the size of an observation is inversely proportional to its rank. For example, the frequency of words in an English text is inversely proportional to their rank. The same is approximately valid for the size of U.S. cities. 358 The Mathematics of Financial Modeling and Investment Management Many properties of power-law distributions are distinctly different in the three following ranges of α: 0 < α ≤ 1, 1 < α ≤ 2, α > 2. The threshold α = 2 for the tail index is important as it marks the separation between the applicability of the standard Central Limit Theorem; the threshold α = 1 is important as it separates variables with a ﬁnite mean from those with inﬁnite mean. Let's take a closer look at the Law of Large Numbers and the Central Limit Theorem. The Law of Large Numbers and the Central Limit Theorem There are four basic versions of the Law of the Large Numbers (LLN), two Weak Laws of Large Numbers (WLLN), and two Strong Laws of Large Numbers (SLLN). The two versions of the WLLN are formulated as follows. 1. Suppose that the variables Xi are IID with ﬁnite mean E[Xi] = E[X] = µ. Under this condition it can be demonstrated that the empirical average tends to the mean in probability: n ∑ Xi P i=1 X n = ---------------n → n→∞ E[X] = µ 2. If the variables are only independently distributed (ID) but have ﬁnite means and variances (µi,σi), then the following relationship holds: n n ∑ i=1 Xi X n = ---------------n P → n→∞ ∑ i=1 n Xi ∑ µi i=1 ---------------- = -------------n n In other words, the empirical average of a sequence of ﬁnite-mean ﬁnitevariance variables tends to the average of the means. The two versions of the SLLN are formulated as follows. 1. The empirical average of a sequence of IID variables Xi tends almost surely to a constant a if and only if the expected value of the variables is ﬁnite. In addition, the constant a is equal to µ. Therefore, if and only if E [ X i ] = E [ X ] = µ < ∞ the following relationship holds: 359 Fat Tails, Scaling, and Stable Laws n ∑ Xi A.S. i=1 → X n = ---------------n n→∞ E[X] = µ where convergence is in the sense of almost sure convergence. 2. If the variables Xi are only independently distributed (ID) but have ﬁnite means and variances (µi,σi) and n 1 2 lim -----σi < ∞ n→∞ 2 n i=1 ∑ then the following relationship holds: n n ∑ Xi i=1 X n = ---------------n ∑ Xi A.S. i=1 → n ∑ µi i=1 ---------------- = -------------n n n→∞ Suppose the variables are IID. If the scaling factor n is replaced with n , then the limit relation no longer holds as the normalized sum n ∑ Xi i=1 ---------------n diverges. However, if the variables have ﬁnite second-order moments, the classical version of the Central Limit Theorem (CLT) can be demonstrated. In fact, under the assumption that both ﬁrst- and second-order moments are ﬁnite, it can be shown that S n – nµ D ---------------------- → Φ σ n n Sn = ∑ Xi i=1 360 The Mathematics of Financial Modeling and Investment Management where µ, σ are respectively the expected value and standard deviation of X, and Φ the standard normal distribution. If the tail index α > 1, variables have ﬁnite expected value and the SLNN holds. If the tail index α > 2, variables have ﬁnite variance and the CLT in the previous form holds. If the tail index α ≤ 2, then variables have inﬁnite variance: The CLT in the previous form does not hold. In fact, variables with α ≤ 2 belong to the domain of attraction of a stable law of index α. This means that a sequence of properly normalized and centered sums tends to a stable distribution with inﬁnite variance. In this case, the CLT takes the form S n – nµ D ---------------------- → G α , if 1 < α ≤ 2 n 1 --α Sn D ------ → G α , if 0 < α ≤ 1 n 1 --α where G are stable distributions as deﬁned below. Note that the case α = 2 is somewhat special: variables with this tail index have inﬁnite variance but fall nevertheless in the domain of attraction of a normal variable, that is, G2. Below the threshold 1, distributions have neither ﬁnite variance nor ﬁnite mean. There is a sharp change in the normalization behavior at this tail-index threshold. Stable Distributions Stable distributions are not, in their generality, a subset of fat-tailed distributions as they include the normal distribution. There are different, equivalent ways to deﬁne stable distributions. Let's begin with a key property: the equality in distribution between a random variable and the (normalized) independent sum of any number of identical replicas of the same variable. This is a different property than the closure property of the tail insofar as (1) it involves not only the tail but the entire distribution and (2) equality in distribution means that distributions have the same functional form but, possibly, with different parameters. Normal distributions have this property: The sum of two or more normally distributed variables is again a normally distributed variable. But this property holds for a more general class of distributions called stable dis-

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