Extreme Value Theory

Working Paper 2000-20 / Document de travail 2000-20

Steps in Applying Extreme Value Theory to Finance: A Review

by

Younes Bensalah

Bank of Canada

Banque du Canada

ISSN 1192-5434 Printed in Canada on recycled paper

Bank of Canada Working Paper 2000-20 November 2000

Steps in Applying Extreme Value Theory to Finance: A Review

by

Younes Bensalah Research and Risk Management Section Financial Markets Department Bank of Canada Ottawa, Ontario, Canada K1A 0G9 beny@bankofcanada.ca

The views expressed in this paper are those of the author. No responsibility for them should be attributed to the Bank of Canada.

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Contents

Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Abstract/Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. EVT: Some Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 The GEV distribution (Fisher-Tippett, Gnedenko result) . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The excess beyond a threshold (Picklands, Balkema-in Haan result) . . . . . . . . . . . . . . . 3 3. Steps in Applying EVT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1 Exploratory data analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1.1 Q-Q plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1.2 The mean excess function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Sampling the maxima and some data...