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Stable Distributions



Microsoft Word - SFB DP Frontpage neu.doc SFB 649 Discussion Paper 2005-008 Stable Distributions Szymon Borak* Wolfgang Härdle* Rafal Weron** * CASE - Center for Applied Statistics and Economics, Humboldt-Universität zu Berlin, Germany **Hugo Steinhaus Center, Wroclaw University of Technology, Poland This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk". ISSN 1860-5664 SFB 649, Humboldt-Universität zu Berlin Spandauer Straße 1, D-10178 Berlin S FB 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 Ha¨rdle, and RafaÃl Weron JEL classification codes: C16 1.1 Acknowledgement We gratefully acknowledge financial support by the Deutsche Forschungsge- meinschaft and the Sonderforschungsbereich 649 “O¨konomisches Risiko”. 1.2 Introduction Many of the concepts in theoretical and empirical finance 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 financial asset returns are the cumulative outcome of a vast number of pieces of informati

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