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Unformatted text preview: An Application of Extreme Value Theory for Measuring Risk Manfred Gilli, Evis K ellezi ? Department of Econometrics, University of Geneva and FAME CH1211 Geneva 4, Switzerland Abstract Many fields of modern science and engineering have to deal with events which are rare but have significant consequences. Extreme value theory is considered to provide the basis for the statistical modelling of such extremes. The potential of ex- treme value theory applied to financial problems has only been recognized recently. This paper aims at introducing the fundamentals of extreme value theory as well as practical aspects for estimating and assessing statistical models for tail-related risk measures. Key words: Extreme Value Theory, Generalized Pareto Distribution, Generalized Extreme Value Distribution, Quantile Estimation, Risk Measures, Maximum Likelihood Estimation, Profile Likelihood Confidence Intervals 1 Introduction The last years have been characterized by significant instabilities in financial markets worldwide. This has led to numerous criticisms about the existing risk management systems and motivated the search for more appropriate metho- dologies able to cope with rare events that have heavy consequences. The typical question one would like to answer is: If things go wrong, how wrong can they go? The problem is then how can we model the rare phe- nomena that lie outside the range of available observations. In such a situation ? Supported by the Swiss National Science Foundation (projects 1252481.97 and 1214056900.99/1). We are grateful to an anonymous referee for corrections and comments and thank Elion Jani and Agim Xhaja for their suggestions. Email addresses: Manfred.Gilli@metri.unige.ch , Evis.Kellezi@metri.unige.ch (Manfred Gilli, Evis K ellezi). Preprint submitted to Elsevier Science 8 February 2003 it seems essential to rely on a well founded methodology. Extreme value theory (EVT) provides a firm theoretical foundation on which we can build statistical models describing extreme events. In many fields of modern science, engineering and insurance, extreme value theory is well established (see e.g. Embrechts et al. (1999), Reiss and Thomas (1997)). Recently, more and more research has been undertaken to analyze the extreme variations that financial markets are subject to, mostly because of currency crises, stock market crashes and large credit defaults. The tail behaviour of financial series has, among others, been discussed in Koedijk et al. (1990), Dacorogna et al. (1995), Loretan and Phillips (1994), Longin (1996), Danielsson and de Vries (1997b), Kuan and Webber (1998), Straet- mans (1998), McNeil (1999), Jondeau and Rockinger (1999), Rootz` en and Kluppelberg (1999), Neftci (2000) and McNeil and Frey (2000). An interest- ing discussion about the potential of extreme value theory in risk management is given in Diebold et al. (1998)....
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