Forschungsgemeinschaft through through Forschungsgemeinschaft **Hugo Steinhaus Center, Wroclaw University of Technology, of Technology, University Wroclaw Center, **Hugo Steinhaus SFB 649 Discussion Paper 2005-008 SFB 649DiscussionPaper2005-008 Stable Distributions * CASE - Center for Applied Statistics and Economics, Statisticsand Center forApplied - * CASE This research was supported by the Deutsche the Deutsche by was supported This research Humboldt-Universität zuBerlin,Germany SFB 649, Humboldt-Universität zu Berlin zu SFB 649,Humboldt-Universität Wolfgang Härdle* Spandauer Straße 1,D-10178 Berlin Spandauer Szymon Borak* http://sfb649.wiwi.hu-berlin.de http://sfb649.wiwi.hu-berlin.de Rafal Weron** ISSN 1860-5664 Poland Poland the SFB 649 "Economic Risk". "Economic the SFB649 SFB 6 4 9 E C O N O M I C R I S K B E R L I N 1 Stable distributions Szymon Borak, Wolfgang HÄardle,and RafaÃlWeron JEL classi¯cation codes: C16 1.1 Acknowledgement We gratefully acknowledge ¯nancial support by the Deutsche Forschungsge- meinschaft and the Sonderforschungsbereich 649 \OkonomischesÄ Risiko". 1.2 Introduction Many of the concepts in theoretical and empirical ¯nance developed over the past decades { including the classical portfolio theory, the Black-Scholes-Merton option pricing model and the RiskMetrics variance-covariance approach to Value at Risk (VaR) { rest upon the assumption that asset returns follow a normal distribution. However, it has been long known that asset returns are not normally distributed. Rather, the empirical observations exhibit fat tails. This heavy tailed or leptokurtic character of the distribution of price changes has been repeatedly observed in various markets and may be quan- titatively measured by the kurtosis in excess of 3, a value obtained for the normal distribution (Bouchaud and Potters, 2000; Carr et al., 2002; Guillaume et al., 1997; Mantegna and Stanley, 1995; Rachev, 2003; Weron, 2004). It is often argued that ¯nancial asset returns are the cumulative outcome of a vast number of pieces of information and individual decisions arriving almost continuously in time (McCulloch, 1996; Rachev and Mittnik, 2000). As such, since the pioneering work of Louis Bachelier in 1900, they have been modeled by the Gaussian distribution. The strongest statistical argument for it is based 2 1 Stable distributions on the Central Limit Theorem, which states that the sum of a large number of independent, identically distributed variables from a ¯nite-variance distribution will tend to be normally distributed. However, as we have already mentioned, ¯nancial asset returns usually have heavier tails. In response to the empirical evidence Mandelbrot (1963) and Fama (1965) pro- posed the stable distribution as an alternative model. Although there are other heavy-tailed alternatives to the Gaussian law { like Student's t, hyperbolic, nor- mal inverse Gaussian, or truncated stable { there is at least one good reason for modeling ¯nancial variables using stable distributions. Namely, they are supported by the generalized Central Limit Theorem, which states that sta- ble laws are the only possible limit distributions for properly normalized and centered sums of independent, identically distributed random variables. Since stable distributions can accommodate the fat tails and asymmetry, they often give a very good ¯t to empirical data. In particular, they are valuable models for data sets covering extreme events, like market crashes or natural catastrophes. Even though they are not universal, they are a useful tool in the hands of an analyst working in ¯nance or insurance. Hence, we devote this chapter to a thorough presentation of the computational aspects related to stable laws. In Section 1.3 we review the analytical concepts and basic characteristics. In the following two sections we discuss practical simulation and estimation approaches. Finally, in Section 1.6 we present ¯nancial applications of stable laws. 1.3 De¯nitions and basic characteristics Stable laws { also called ®-stable, stable Paretian or L¶evystable { were in- troduced by Levy (1925) during his investigations of the behavior of sums of independent random variables. A sum of two independent random variables having an ®-stable distribution with index ® is again ®-stable with the same index ®. This invariance property, however, does not hold for di®erent ®'s. The ®-stable distribution requires four parameters for complete description: an index of stability ® 2 (0; 2] also called the tail index, tail exponent or characteristic exponent, a skewness parameter ¯ 2 [¡1; 1], a scale parameter σ > 0 and a location parameter ¹ 2 R. The tail exponent ® determines the rate at which the tails of the distribution taper o®, see the left panel in Figure 1.1. When ® = 2, the Gaussian distribution results. When ® < 2, the variance is in¯nite and the tails are asymptotically equivalent to a Pareto law, i.e. they 1.3 De¯nitions and basic characteristics 3 Dependence on alpha Tails of stable laws -2 -4 -5 -6 log(PDF(x)) log(1-CDF(x)) -8 -10 -10 -10 -5 0 5 10 0 1 2 x log(x) Figure 1.1: Left panel: A semilog plot of symmetric (¯ = ¹ = 0) ®-stable probability density functions (pdfs) for ® = 2 (black solid line), 1.8 (red dotted line), 1.5 (blue dashed line) and 1 (green long-dashed line). The Gaussian (® = 2) density forms a parabola and is the only ®-stable density with exponential tails. Right panel: Right tails of symmetric ®-stable cumulative distribution functions (cdfs) for ® = 2 (black solid line), 1.95 (red dotted line), 1.8 (blue dashed line) and 1.5 (green long-dashed line) on a double logarithmic paper. For ® < 2 the tails form straight lines with slope ¡®. STFstab01.xpl exhibit a power-law behavior. More precisely, using a central limit theorem type argument it can be shown that (Janicki and Weron, 1994; Samorodnitsky and Taqqu, 1994): ( ® ® limx!1 x P(X > x) = C®(1 + ¯)σ ; ® ® (1.1) limx!1 x P(X < ¡x) = C®(1 + ¯)σ ; where: µ Z 1 ¶¡1 ¡® 1 ¼® C® = 2 x sin(x)dx = ¡(®) sin : 0 ¼ 2 The convergence to a power-law tail varies for di®erent ®'s and, as can be seen in the right panel of Figure 1.1, is slower for larger values of the tail index. 4 1 Stable distributions Moreover, the tails of ®-stable distribution functions exhibit a crossover from an approximate power decay with exponent ® > 2 to the true tail with exponent ®. This phenomenon is more visible for large ®'s (Weron, 2001). When ® > 1, the mean of the distribution exists and is equal to ¹. In general, the pth moment of a stable random variable is ¯nite if and only if p < ®. When the skewness parameter ¯ is positive, the distribution is skewed to the right, i.e. the right tail is thicker, see the left panel of Figure 1.2. When it is negative, it is skewed to the left. When ¯ = 0, the distribution is symmetric about ¹. As ® approaches 2, ¯ loses its e®ect and the distribution approaches the Gaussian distribution regardless of ¯. The last two parameters, σ and ¹, are the usual scale and location parameters, i.e. σ determines the width and ¹ the shift of the mode (the peak) of the density. For σ = 1 and ¹ = 0 the distribution is called standard stable. 1.3.1 Characteristic function representation Due to the lack of closed form formulas for densities for all but three dis- tributions (see the right panel in Figure 1.2), the ®-stable law can be most conveniently described by its characteristic function Á(t) { the inverse Fourier transform of the probability density function. However, there are multiple pa- rameterizations for ®-stable laws and much confusion has been caused by these di®erent representations, see Figure 1.3. The variety of formulas is caused by a combination of historical evolution and the numerous problems that have been analyzed using specialized forms of the stable distributions. The most popular parameterization of the characteristic function of X » S®(σ; ¯; ¹), i.e. an ®-stable random variable with parameters ®, σ, ¯, and ¹, is given by (Samorodnitsky and Taqqu, 1994; Weron, 2004): 8 ® ® ¼® <>¡σ jtj f1 ¡ i¯sign(t) tan 2 g + i¹t; ® 6= 1; ln Á(t) = (1.2) > : 2 ¡σjtjf1 + i¯sign(t) ¼ ln jtjg + i¹t; ® = 1: For numerical purposes, it is often advisable to use Nolan's (1997) parameter- ization: 8 ® ® ¼® 1¡® <>¡σ jtj f1 + i¯sign(t) tan 2 [(σjtj) ¡ 1]g + i¹0t; ® 6= 1; ln Á (t) = 0 > : 2 ¡σjtjf1 + i¯sign(t) ¼ ln(σjtj)g + i¹0t; ® = 1: (1.3) 1.3 De¯nitions and basic characteristics 5 Dependence on beta Gaussian, Cauchy, and Levy distributions 0.3 0.4 0.25 0.2 0.3 0.15 PDF(x) PDF(x) 0.2 0.1 0.1 0.05 0 -5 0 5 -5 0 5 x x Figure 1.2: Left panel: Stable pdfs for ® = 1:2 and ¯ = 0 (black solid line), 0.5 (red dotted line), 0.8 (blue dashed line) and 1 (green long-dashed line). Right panel: Closed form formulas for densities are known only for three distributions { Gaussian (® = 2; black solid line), Cauchy (® = 1; red dotted line) and Levy (® = 0:5; ¯ = 1; blue dashed line). The latter is a totally skewed distribution, i.e. its support is R+. In general, for ® < 1 and ¯ = 1 (¡1) the distribution is totally skewed to the right (left). STFstab02.xpl 0 The S®(σ; ¯; ¹0) parameterization is a variant of Zolotariev's (M)-parameteri- zation (Zolotarev, 1986), with the characteristic function and hence the density and the distribution function jointly continuous in all four parameters, see the right panel in Figure 1.3. In particular, percentiles and convergence to the power-law tail vary in a continuous way as ® and ¯ vary. The location parame- ¼® ters of the two representations are related by ¹ = ¹0 ¡ ¯σ tan 2 for ® 6= 1 and 2 ¹ = ¹0 ¡ ¯σ ¼ ln σ for ® = 1.