Stable Distributions

1 Stable distributions

Szymon Borak, Wolfgang Hardle, and Rafal Weron

JEL classification codes: C16

1.1 Acknowledgement

We gratefully acknowledge financial support by the Deutsche Forschungsge-
meinschaft and the Sonderforschungsbereich 649 “Okonomisches 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 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

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