Applied Quantitative Finance

Measuring Risk in Complex Stochastic Systems J. Franke, W. H¨ ardle, G. Stahl

Empirical Volatility

Parameter Estimates

http://www.xplore-stat.de/ebooks/ebooks.html

2

Preface Complex dynamic processes of life and sciences generate risks that have to be taken. The need for clear and distinctive definitions of different kinds of risks, adequate methods and parsimonious models is obvious. The identification of important risk factors and the quantification of risk stemming from an interplay between many risk factors is a prerequisite for mastering the challenges of risk perception, analysis and management successfully. The increasing complexity of stochastic systems, especially in finance, have catalysed the use of advanced statistical methods for these tasks. The methodological approach to solving risk management tasks may, however, be undertaken from many different angles. A financial institution may focus on the risk created by the use of options and other derivatives in global financial processing, an auditor will try to evaluate internal risk management models in detail, a mathematician may be interested in analysing the involved nonlinearities or concentrate on extreme and rare events of a complex stochastic system, whereas a statistician may be interested in model and variable selection, practical implementations and parsimonious modelling. An economist may think about the possible impact of risk management tools in the framework of efficient regulation of financial markets or efficient allocation of capital. This book gives a diversified portfolio of these scenarios. We first present a set of papers on credit risk management, and then focus on extreme value analysis. The Value at Risk (VaR) concept is discussed in the next block of papers, followed by several articles on change points. The papers were presented during a conference on Measuring Risk in Complex Stochastic Systems that took place in Berlin on September 25th - 30th 1999. The conference was organised within the Seminar Berlin-Paris, Seminaire Paris-Berlin. The paper by Lehrbass considers country risk within a no-arbitrage model and combines it with the extended Vasicek term structure model and applies the developed theory to DEM- Eurobonds. Kiesel, Perraudin and Taylor construct a model free volatility estimator to investigate the long horizon volatility of various short term interest rates. Hanousek investigates the failing of Czech banks during the early nineties. M¨ uller and Rnz apply a Generalized Partial Linear Model to evaluating credit risk based on a credit scoring data set from a French bank. Overbeck considers the problem of capital allocation in the framework of credit risk and loan portfolios. The analysis of extreme values starts with a paper by Novak, who considers confidence intervals for tail index estimators. Robert presents a novel approach to extreme value

3

calculation on state of the art α-ARCH models. Kleinow and Thomas show how in a client/server architecture the computation of extreme value parameters may be undertaken with the help of WWW browsers and an XploRe Quantlet Server. The VaR section starts with Cumperayot, Danielsson and deVries who discuss basic questions of VaR modelling and focus in particular on economic justifications for external and internal risk management procedures and put into question the rationale behind VaR. Slaby and Kokoschka deal with with change-points. Slaby considers methods based on ranks in an iid framework to detect shifts in location, whereas Kokoszka reviews CUSUM-type esting and estimating procedures for the change-point problem in ARCH models. Huschens and Kim concentrate on the stylised fact of heavy tailed marginal distributions for financial returns time series. They model the distributions by the family of α-stable laws and consider the consequences for β values in the often applied CAPM framework. Breckling, Eberlein and Kokic introduce the generalised hyperbolic model to calculate the VaR for market and credit risk. H¨ardle and Stahl consider the backtesting based on shortfall risk and discuss the use of exponential weights. Sylla and Villa apply a PCA to the implied volatility surface in order to determine the nature of the vola factors. We gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft, SFB ¨ 373 Quantification und Simulation Okonomischer Prozesse, Weierstra Institut f¨ ur Angewandte Analysis und Stochastik, Deutsche Bank, WestLB, BHF-Bank, Arthur Andersen, SachsenLB, and MD*Tech. The local organization was smoothly run by J¨org Polzehl and Vladimir Spokoiny. Without the help of Anja Bardeleben, Torsten Kleinow, Heiko Lehmann, Marlene M¨ uller, Sibylle Schmerbach, Beate Siegler, Katrin Westphal this event would not have been possible. J. Franke, W. H¨ardle and G. Stahl January 2000, Kaiserslautern and Berlin

4

Contributors Jens Breckling Insiders GmbH Wissensbasierte Systeme, Wilh.-Th.-R¨ omheld-Str. 32, 55130 Mainz, Germany Phornchanok J. Cumperayot Tinbergen Institute, Erasmus University Rotterdam Jon Danielsson London School of Economics Casper G. de Vries Erasmus University Rotterdam and Tinbergen Institute Ernst Eberlein Institut f¨ ur Mathematische Stochastik, Universit¨at Freiburg, Eckerstaße 1, 79104 Freiburg im Breisgau, Germany Wolfgang H¨ ardle Humboldt-Universit¨at zu Berlin, Dept. of Economics, Spandauer Str. 1, 10178 Berlin Jan Hanousek CERGE-EI, Prague Stefan Huschens Technical University Dresden, Dept. of Economics Bjorn N. Jorgensen Harvard Business School R¨ udiger Kiesel School of Economics, Mathematics and Statistics, Birkbeck College, University of London, 7-15 Gresse St., London W1P 2LL, UK Jeong-Ryeol Kim Technical University Dresden, Dept. of Economics Torsten Kleinow Humboldt-Universit¨at zu Berlin, Dept. of Economics, Spandauer Str. 1, 10178 Berlin Philip Kokic Insiders GmbH Wissensbasierte Systeme, Wilh.-Th.-R¨ omheld-Str. 32, 55130 Mainz, Germany Piotr Kokoszka The University of Liverpool and Vilnius University Institute of Mathematics and Informatics

Frank Lehrbass 01-616 GB Zentrales Kreditmanagement, Portfoliosteuerung, WestLB Marlene M¨ uller Humboldt-Universit¨at zu Berlin, Dept. of Economics, Spandauer Str. 1, 10178 Berlin

5

Sergei Y. Novak EURANDOM PO Box 513, Eindhoven 5600 MB, Netherlands Ludger Overbeck Deutsche Bank AG, Group Market Risk Management, Methodology & Policy/CR, 60262 Frankfurt William Perraudin Birkbeck College, Bank of England and CEPR Christian Robert Centre de Recherche en Economie et Statistique (CREST), Laboratoire de Finance Assurance, Timbre J320 - 15, Bb G. Peri, 92245 MALAKOFF, FRANCE Bernd R¨ onz Humboldt-Universit¨at zu Berlin, Dept. of Economics, Spandauer Str. 1, 10178 Berlin Aleˇs Slab´ y Charles University Prague, Czech Republic Gerhard Stahl Bundesaufsichtsamt f¨ ur das Kreditwesen, Berlin Alpha Sylla ENSAI-Rennes, Campus de Ker-Lan, 35170 Bruz, France. Alex Taylor School of Economics, Mathematics and Statistics, Birkbeck College, University of London, 7-15 Gresse St., London W1P 2LL, UK Michael Thomas Fachbereich Mathematik, Universit¨at-Gesamthochschule Siegen Christophe Villa University of Rennes 1, IGR and CREREG, 11 rue jean Mac, 35019 Rennes cedex, France.

6

Contents

1 Allocation of Economic Capital in loan portfolios

15

Ludger Overbeck 1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2

Credit portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3

1.2.1

Ability to Pay Process . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.2

Loss distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Economic Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.1

Capital allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4

Capital allocation based on Var/Covar . . . . . . . . . . . . . . . . . . . . 19

1.5

Allocation of marginal capital . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6

Contributory capital based on coherent risk measures

1.7

1.8

. . . . . . . . . . . 21

1.6.1

Coherent risk measures . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.2

Capital Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.3

Contribution to Shortfall-Risk . . . . . . . . . . . . . . . . . . . . . 23

Comparision of the capital allocation methods . . . . . . . . . . . . . . . . 23 1.7.1

Analytic Risk Contribution . . . . . . . . . . . . . . . . . . . . . . 23

1.7.2

Simulation procedure

1.7.3

Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.7.4

Portfolio size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

. . . . . . . . . . . . . . . . . . . . . . . . . 24

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Estimating Volatility for Long Holding Periods

31

7

Contents R¨ udiger Kiesel, William Perraudin and Alex Taylor 2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2

Construction and Properties of the Estimator . . . . . . . . . . . . . . . . 32 2.2.1

Large Sample Properties . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.2

Small Sample Adjustments . . . . . . . . . . . . . . . . . . . . . . 34

2.3

Monte Carlo Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 A Simple Approach to Country Risk

43

Frank Lehrbass 3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2

A Structural No-Arbitrage Approach . . . . . . . . . . . . . . . . . . . . . 44

3.3

3.4

8

3.2.1

Structural versus Reduced-Form Models . . . . . . . . . . . . . . . 44

3.2.2

Applying a Structural Model to Sovereign Debt . . . . . . . . . . . 45

3.2.3

No-Arbitrage vs Equilibrium Term Structure . . . . . . . . . . . . 45

3.2.4

Assumptions of the Model . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.5

The Arbitrage-Free Value of a Eurobond . . . . . . . . . . . . . . . 48

3.2.6

Possible Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.7

Determination of Parameters . . . . . . . . . . . . . . . . . . . . . 54

Description of Data and Parameter Setting . . . . . . . . . . . . . . . . . 55 3.3.1

DM-Eurobonds under Consideration . . . . . . . . . . . . . . . . . 55

3.3.2

Equity Indices and Currencies . . . . . . . . . . . . . . . . . . . . . 56

3.3.3

Default-Free Term Structure and Correlation . . . . . . . . . . . . 57

3.3.4

Calibration of Default-Mechanism . . . . . . . . . . . . . . . . . . 58

Pricing Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.1

Test Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.2

Inputs for the Closed-Form Solution . . . . . . . . . . . . . . . . . 59

3.4.3

Model versus Market Prices . . . . . . . . . . . . . . . . . . . . . . 60

Contents 3.5

3.6

3.7

Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5.1

Static Part of Hedge . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5.2

Dynamic Part of Hedge . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.3

Evaluation of the Hedging Strategy . . . . . . . . . . . . . . . . . . 63

Management of a Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.6.1

Set Up of the Monte Carlo Approach . . . . . . . . . . . . . . . . . 64

3.6.2

Optimality Condition . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.6.3

Application of the Optimality Condition . . . . . . . . . . . . . . . 68

3.6.4

Modification of the Optimality Condition . . . . . . . . . . . . . . 69

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4 Predicting Bank Failures in Transition

73

Jan Hanousek 4.1

Motivation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2

Improving “Standard” Models of Bank Failures . . . . . . . . . . . . . . . 74

4.3

Czech banking sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4

Data and the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5 Credit Scoring using Semiparametric Methods

85

Marlene M¨ uller and Bernd R¨onz 5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2

Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3

Logistic Credit Scoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4

Semiparametric Credit Scoring . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5

Testing the Semiparametric Model . . . . . . . . . . . . . . . . . . . . . . 89

5.6

Misclassification and Performance Curves . . . . . . . . . . . . . . . . . . 89

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

9

Contents 6 On the (Ir)Relevancy of Value-at-Risk Regulation

103

Phornchanok J. Cumperayot, Jon Danielsson, Bjorn N. Jorgensen and Caspar G. de Vries 6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2

VaR and other Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2.1

VaR and Other Risk Measures . . . . . . . . . . . . . . . . . . . . 106

6.2.2

VaR as a Side Constraint . . . . . . . . . . . . . . . . . . . . . . . 108

6.3

Economic Motives for VaR Management . . . . . . . . . . . . . . . . . . . 109

6.4

Policy Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7 Backtesting beyond VaR

121

Wolfgang H¨ardle and Gerhard Stahl 7.1

Forecast tasks and VaR Models . . . . . . . . . . . . . . . . . . . . . . . . 121

7.2

Backtesting based on the expected shortfall . . . . . . . . . . . . . . . . . 123

7.3

Backtesting in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8 Measuring Implied Volatility Surface Risk using PCA

133

Alpha Sylla and Christophe Villa 8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.2

PCA of Implicit Volatility Dynamics . . . . . . . . . . . . . . . . . . . . . 134

8.3

10

8.2.1

Data and Methodology

. . . . . . . . . . . . . . . . . . . . . . . . 135

8.2.2

The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Smile-consistent pricing models . . . . . . . . . . . . . . . . . . . . . . . . 139 8.3.1

Local Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . 139

8.3.2

Implicit Volatility Models . . . . . . . . . . . . . . . . . . . . . . . 140

8.3.3

The volatility models implementation . . . . . . . . . . . . . . . . 141

Contents 8.4

Measuring Implicit Volatility Risk using VaR . . . . . . . . . . . . . . . . 144 8.4.1

VaR : Origins and definition . . . . . . . . . . . . . . . . . . . . . . 144

8.4.2

VaR and Principal Components Analysis . . . . . . . . . . . . . . 145

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9 Detection and estimation of changes in ARCH processes

149

Piotr Kokoszka and Remigijus Leipus 9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9.2

Testing for change-point in ARCH . . . . . . . . . . . . . . . . . . . . . . 152

9.3

9.2.1

Asymptotics under null hypothesis . . . . . . . . . . . . . . . . . . 152

9.2.2

Asymptotics under local alternatives . . . . . . . . . . . . . . . . . 154

Change-point estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 9.3.1

ARCH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9.3.2

Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 10 Behaviour of Some Rank Statistics for Detecting Changes

161

Aleˇs Slab´ y 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 10.2 Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 10.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 10.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 11 A stable CAPM in the presence of heavy-tailed distributions

175

Stefan Huschens and Jeong-Ryeol Kim 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 11.2 Empirical evidence for the stable Paretian hypothesis . . . . . . . . . . . . 176 11.2.1 Empirical evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 11.2.2 Univariate und multivariate alpha-stable distributions . . . . . . . 178

11

Contents 11.3 Stable CAPM and estimation for beta-coefficients . . . . . . . . . . . . . . 180 11.3.1 Stable CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 11.3.2 Estimation of the beta-coefficient in stable CAPM . . . . . . . . . 182 11.4 Empirical analysis of bivariate symmetry test . . . . . . . . . . . . . . . . 183 11.4.1 Test for bivariate symmetry . . . . . . . . . . . . . . . . . . . . . . 183 11.4.2 Estimates for the beta-coefficient in stable CAPM . . . . . . . . . 185 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 12 A Tailored Suit for Risk Management: Hyperbolic Model

189

Jens Breckling, Ernst Eberlein and Philip Kokic 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 12.2 Advantages of the Proposed Risk Management Approach . . . . . . . . . 190 12.3 Mathematical Definition of the P & L Distribution . . . . . . . . . . . . . 191 12.4 Estimation of the P&L using the Hyperbolic Model . . . . . . . . . . . . . 192 12.5 How well does the Approach Conform with Reality . . . . . . . . . . . . . 195 12.6 Extension to Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 12.7 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 13 Computational Resources for Extremes

201

Torsten Kleinow and Michael Thomas 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 13.2 Computational Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 13.2.1 XploRe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 13.2.2 Xtremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 13.2.3 Extreme Value Analysis with XploRe and Xtremes . . . . . . . . . 203 13.2.4 Differences between XploRe and Xtremes . . . . . . . . . . . . . . 205 13.3 Client/Server Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . 205 13.3.1 Client/Server Architecture of XploRe . . . . . . . . . . . . . . . . 206

12

Contents 13.3.2 Xtremes CORBA Server . . . . . . . . . . . . . . . . . . . . . . . . 208 13.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 14 Confidence intervals for a tail index estimator

211

Sergei Y. Novak 14.1 Confidence intervals for a tail index estimator . . . . . . . . . . . . . . . . 211 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 15 Extremes of alpha-ARCH Models

219

Christian Robert 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 15.2 The model and its properties . . . . . . . . . . . . . . . . . . . . . . . . . 220 15.3 The tails of the stationary distribution . . . . . . . . . . . . . . . . . . . . 221 15.4 Extreme value results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 15.4.1 Normalizing factors

. . . . . . . . . . . . . . . . . . . . . . . . . . 224

15.4.2 Computation of the extremal index . . . . . . . . . . . . . . . . . . 225 15.5 Empirical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 15.5.1 Distribution of extremes . . . . . . . . . . . . . . . . . . . . . . . . 230 15.5.2 Tail behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 15.5.3 The extremal index

. . . . . . . . . . . . . . . . . . . . . . . . . . 233

15.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 15.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

13

Contents

14

1 Allocation of Economic Capital in loan portfolios Ludger Overbeck

1.1

Introduction

Since the seminal research of Markowitz (1952) and Sharpe (1964) capital allocation within portfolios is based on the variance/covariance analysis. Even the introduction of Value-at-Risk in order to measure risk more accurately than in terms of standard deviation, did not chance the calculation of a risk contribution of single asset in the portfolio or its contributory capital as a multiple of the asset’s β with the portfolio. This approach is based on the assumption that asset returns are normally distributed. Under this assumption, the capital of a portfolio, usually defined as a quantile of the distribution of changes of the portfolio value, is a multiple of the standard deviation of the portfolio. Since the βs yield a nice decomposition of the portfolio standard deviation and exhibit the interpretation as an infinitesimal marginal risk contribution (or more mathematically as a partial derivative of the portfolio standard deviation with respect to an increase of the weight of an asset in the portfolio), these useful properties also hold for the quantile, i.e. for the capital. In the case of the normal distributed assets in the portfolio, the though defined capital allocation rule also coincides with the capital allocation based on marginal economic capital, i.e. the capital difference between the portfolio with and without the single asset to which we want to allocate capital. Additionally it is equivalent to the expected loss in the single asset conditional on the event that the loss for the whole portfolio exceeds a quantile of the loss distribution. The purpose of the paper is to present and analyse these three capital allocation rules, i.e. the one based on conditional expectation, the one on marginal economic capital and the classical one based on covariances, in the context of a loan portfolio. The only method that gives analytic solutions of the (relative) allocation rule is the classical one based on covariances. All others have to be analysed by a Monte-Carlo-Simulation for real world portfolios. There is of course a possibility to quantify the other two approaches for highly uniformed and standardized portfolios. On the other hand in some situations also the calculation of the βs might be quicker in a Monte-Carlo-Simulation.

15

1

Allocation of Economic Capital in loan portfolios

1.2

Credit portfolios

Let us consider a portfolio of transactions with m counterparties. The time horizon at which the loss distribution is to be determined is fixed, namely 1 year. The random variable portfolio loss can than be written as L =

m X

Lk ,

(1.1)

k=1

where Lk is the loss associated with transaction k. There are now different models discussed in the literature and some of them are implemented by banks and software firms. From the growing literature the papers Baestaens & van den Bergh (1997), Credit Metrics (1997), Risk (1997), Artzner, Dealban, Eber & Heath (1997a), Kealhofer (1995), Overbeck & Stahl (1997), Schmid (1997), Vasicek (1997) and Wilson (1997) may be consulted in a first attempt. In the simplest model (pure default mode) Lk = lk 1Dk ,

(1.2)

where Dk is the default event and lk is the exposure amount, which is assumed to be known with certainty. More eloborate models (like Credit Metrics (1997)) assume Lk =

D X

lr,k 1Dr,k ,

(1.3)

r=AAA

where Dr,k the event that counterparty k is in rating class r and lr,k is the loss associated with the migration of k to rating r. The loss amount is usually deterministic given the total amount of exposure and the given migration, i.e. lr,k is a function of r, the exposure and the present rating of k. The straight asset value model Merton (1974), e.g. implemented by Kealhofer & Crosbie (1997), assumes Lk = L(k, A1 (k)),

(1.4)

where A(k) is the stochastic process governing the asset value process of counterparty k. In the default mode only model L(k, A1 (k)) = lk 1{A1 (k) 0.

The moment-generating function of gλ is given by: Z +∞ ϕλ (t) = etx gλ (x)dx −∞ +∞

=

Z

λS0 (λ)e(t+λS0 (λ))x+λm0 (λ) F¯Y (S0 (λ)x + m0 (λ))dx/C0 (λ)

−∞

=

C0 (λ + t/S0 (λ)) λ exp {−tm0 (λ)/S0 (λ)} . λ + t/S0 (λ) C0 (λ)

Using properties (15.8), (15.9) and (15.10), we have:    t m0 (λ) ϕλ (t) = exp ln λ − ln λ + −t + ln C0 (λ + t/S0 (λ)) S0 (λ) S0 (λ)  − ln C0 (λ)  2  t = exp + O(1/λ) 2  2 t → exp . λ→∞ 2

235

15

Extremes of alpha-ARCH Models

It follows that:

d

Gλ (x) → H(x). By the dominated convergence theorem and the inversion formula for characteristic functions, we have: Z +∞ 1 gλ (x) = e−itx ϕλ (it)dt 2π −∞  2  2 Z +∞ 1 t 1 x −itx → e exp dt = √ exp − . λ→∞ 2π −∞ 2 2 2π Choosing x = 0, we get: F¯Y (m0 (λ)) ∼

λ→∞

exp(−λm0 (λ))C0 (λ) √ , 2πλS0 (λ)

then, by making the substitution λ → m−1 0 (λ), we obtain (15.12). Invoking the equivalents, we establish the theorem. The following lemma will be very useful for ulterior proofs. LEMMA 15.1 Let ]a, b[ be an interval on R, bounded or not. Let ψ :]a, b[→ R be a C 2 -function, with an unique extremum in c. Moreover, we suppose that ψ 00 (c) < 0. Rb Let ϕ :]a, b[→ R be a continuous and strictly positive function, such that a ϕ(x)eψ(x) dx exists, then: s Z b 2π λψ(x) ϕ(x)e dx ∼ ϕ(c)eλψ(c) . 00 (c) λ→∞ −λψ a Rb Proof of lemma 15.1. Firstly, note that the integrals a ϕ(x)eλψ(x) dx, λ ∈ R+ , exist, since: Z b Z b ϕ(x)eλψ(x) dx ≤ e(λ−1)ψ(c) ϕ(x)eψ(x) dx. a

a

We look for an equivalent of the integral

Rb a

ϕ(x)eλψ(x) dx, when λ → ∞.

Step 1: The function ψ has the following development around c: ψ(x) = ψ(c) +

(x − c)2 00 ψ (c) + o(x − c)2 . 2

Let η ∈]0, 1[. There exists δ > 0 such that if 0 ≤ x − c < δ, then: (1 − η)ϕ(c) ≤ ϕ(x) ≤ (1 + η)ϕ(c) 1 1 (x − c)2 ψ 00 (c)(1 + η) ≤ ψ(x) − ψ(c) ≤ (x − c)2 ψ 00 (c)(1 − η), 2 2

236

15.6

Proofs

hence ∀λ ∈ R+ : (1 − η)ϕ(c)

Z

c+δ

1

e 2 λ(x−c)

2 ψ 00 (c)(1+η)

eλψ(c) dx ≤

Z

c

and Z

c+δ

ϕ(x)eλψ(x) dx,

c

c+δ

ϕ(x)eλψ(x) dx ≤ (1 + η)ϕ(c)

c+δ

Z

c

1

e 2 λ(x−c)

2 ψ 00 (c)(1−η)

eλψ(c) dx.

c

After a change of variable, the integral is undervalued by: q s Z −λψ00 (c)(1+η) δ 2 2 2 (1 − η)ϕ(c)eλψ(c) e−u du. 00 −λψ (c)(1 + η) 0 Moreover, we know that: q

Z

−λψ 00 (c)(1+η) δ 2

2

e−u du →

λ→∞

0

1√ π. 2

Then, there exists λ0 , such that ∀λ > λ0 : Z c+δ 1 2 00 (1 − η)ϕ(c) e 2 λ(x−c) ψ (c)(1+η) eλψ(c) dx c r π ≥ (1 − η) (1 − η)ϕ(c)eλψ(c) , 00 −2λψ (c)(1 + η) Z c+δ 1 2 00 (1 + η)ϕ(c) e 2 λ(x−c) ψ (c)(1−η) eλψ(c) dx c r π ≤ (1 + η) (1 + η)ϕ(c)eλψ(c) . 00 −2λψ (c)(1 − η) We deduce that for λ > λ0 : Z c+δ r (1 − η)2 π λψ(c) p ϕ(c)e ≤ ϕ(x)eλψ(x) dx, (1 + η) −2λψ 00 (c) c and: Z c

c+δ λψ(x)

ϕ(x)e

(1 + η)2 dx ≤ p (1 − η)

r

π ϕ(c)eλψ(c) . −2λψ 00 (c)

Step 2: Let ε > 0. Choose η in ]0, 1[, such that : (1 − η)2 p >1−ε (1 + η)

and

(1 + η)2 p < 1 + ε. (1 − η)

237

15

Extremes of alpha-ARCH Models

Choose also δ and λ0 . Step 3: We must now check that the second part is negligible. By the assumptions, ψ is strictly decreasing at the right of c, then ∀x > c + δ: ψ(x) − ψ(c) ≤ ψ(c + δ) − ψ(c) = −µ, where µ is strictly positive. We deduce that for λ ≥ 1: λψ(x) ≤ (λ − 1)ψ(c) − (λ − 1)µ + ψ(x), and: 0≤

Z

b

ϕ(x)eλψ(x) dx ≤ e(λ−1)ψ(c)−(λ−1)µ

c+δ

Z

b

ϕ(x)eψ(x) dx.

c+δ

Remark that e−(t−1)µ = o



1 √ t



, and there exists λ1 , such that ∀λ > λ1 then:

b

Z

ϕ(x)eλψ(x) dx < ε

c+δ

r

π ϕ(c)eλψ(c) . −2λψ 00 (c)

At least, we have for any λ > max(λ0 , λ1 ): r (1 − ε)

π ϕ(c)eλψ(c) < −2λψ 00 (c)

Z

b

ϕ(x)eλψ(x) dx r π < (1 + 2ε) ϕ(c)eλψ(c) . −2λψ 00 (c) c

Step 4: The same method used on ]a, c[ give the same results. Finally, we obtain: Z a

b λψ(x)

ϕ(x)e

dx ∼

λ→∞

s

2π ϕ(c)eλψ(c) . −λψ 00 (c)

Proposition 15.1  Let f0 be the stationary  density in (15.1)  when a = 0. We have R∞ q0 (λ) = ln E(exp λ ln X 2 ) = ln 2 0 exp λ ln x2 f0 (x)dx , then: h i α 1−α ) − 1 ln(2bα λ ln λ ln 2 q0 (λ) = + λ− ln λ + m + o(1) (1 − α) (1 − α) 2 ln α

238

15.6

Proofs

if and only if f0 (x) ∼

x→∞

with:

D exp(−exc ), xd α

α− 1−α , c = 2(1 − α), e= 2b(1 − α) "  (1−d)/c   # (1 − α) 2 π 1/2 d=α+ ln 2, m = ln 2D . ln(α) ec c

Proof of proposition 15.1. The theorem 15.2. gives the first implication. Reciprocally, we note f0 (x) = D|x|−d exp(−e|x|c )h(|x|). As f0 is equivalent to Dx−d exp(−exc ) and integrable, the function h has the following properties: limx→∞ h(x) = 1, and on any intervals [0, B] with B > 0, there exist two constants AB > 0 and βB ≤ 1 such that for each x ∈ [0, B]: h(x) < AB xd−βB . In particular, ∀x ∈ R+ , we have h(x) ≤ A1 xd−β1 + C where C = sup {h(x), x > 1}. By using lemma 15.1, we would like to obtain an equivalent of:  Z 
q0 (λ) = ln E exp λ ln X 2 = ln 2

0

∞

  exp λ ln x2 f0 (x)dx ,

We have:  D exp λ ln x2 f0 (x) = d exp {2λ ln x − exc } h(x). x The function x → 2λ ln x − exc reaches its maximum in x = Ωλ1/c , with Ω = We do the change of variable: x = Ωuλ1/c , and we obtain:


2 1/c . ec

Dx−d exp {2λ ln x − exc } n  o = exp 2λ ln Ωλ1/c Dλ−d/c u−d Ω−d exp {2λ (ln u − uc /c)} . The function: u → ln u − uc /c reaches its maximum in: um = 1. After the change of variable, h(x) becomes h(Ωuλ1/c ). We analyze its behavior in infinity and around 0: (i) behavior of h(Ωuλ1/c ) in infinity: let ε > 0 and u0 > 0 be fixed, then there exists λu0 such that ∀u > u0 and ∀λ > λu0 : h(Ωuλ1/c ) − 1 < ε.

239

15

Extremes of alpha-ARCH Models

(ii) behavior of h(Ωuλ1/c ) around 0: for δ > 0, we have: Z um −δ h(Ωuλ1/c )u−d exp {2λ (ln u − uc /c)} 0 Z um −δ   < A1 Ωd−β1 λ(d−β1 )/c u−β1 + Cu−d exp {2λ (ln u − uc /c)} du, 0

We can use lemma 15.1 (by taking account of (i) and (ii)), with the functions: ϕ(u) = u−d , ψ(u) = 2(ln u − uc /c). Remember that:  n  o q0 (λ) = ln 2D exp 2λ ln Ωλ1/c λ(1−d)/c Ω1−d  Z ∞ h(Ωuλ1/c )ϕ(u) exp {λψ(u)} du . 0

We have then a = 0, b = ∞, and c = 1. The steps 1, 2 and 3 are the same by taking account of the remark (i). For the step 4, we obtain that the integral is undervalued by (remark (ii)): Z 1−δ h(Ωuλ1/c )ϕ(u) exp {2λψ(u)} du 0

< e(λ−1)ψ(1)−(λ−1)µ Z 1  A1 Ωd−β1 λ(d−β1 )/c u−β1 + Cu−d exp {2 (ln u − uc /c)} du. 0


But, e−λµ = o λ(β1 −d)/c−1/2 ∨ λ−1/2 , and we can conclude in the same way. Finally, we obtain that: q0 (λ) =

λ ln λ + (1 − α)

h

i α ln(2bα 1−α ) − 1 (1 − α)

λ−

ln 2 ln λ + m + o(1). 2 ln α

Proof of theorem 15.3. We note: 

2



 Z ) = ln 2

∞



2



q(λ) = ln E(exp λ ln X exp λ ln x f (x)dx 0 

pa (αλ) = ln E exp (αλ) ln a/b + Xt2α /α  Z ∞  
2α = ln 2 exp (αλ) ln a/b + x /α f (x)dx . 0

240



15.6

Proofs

The following are equivalent: f (x) ∼

x→∞

f0 (x) ⇔ q(λ) = q0 (λ) + o(1)

(Proposition 15.1)

But, q0 (λ) = q0 (αλ) + λ ln λ + λ(ln 2b − 1) + ln 2/2 + o(1) q(λ) = pa (αλ) + λ ln λ + λ(ln 2b − 1) + ln 2/2 + o(1) and then, f (x) ∼

x→∞

f0 (x)

⇔

pa (αλ) − q0 (αλ) = o(1)

⇔

pa (αλ) − q(αλ) = o(1) 
R∞ λ ln a/b + x2α f (x)dx 0 exp R∞ → 1 2α λ→∞ 0 exp {λ ln x } f (x)dx  
R∞ 2α exp λ ln(1 + a/bx2α ) − 1 f (x)dx 0 exp λ ln x R → 0 ∞ 2α λ→∞ 0 exp {λ ln x } f (x)dx  
R∞ 2α exp λ ln(1 + a/bx2α ) − 1 k(x)f0 (x)dx 0 exp λ ln x R → 0, ∞ 2α λ→∞ 0 exp {λ ln x } k(x)f0 (x)dx

⇔ ⇔ ⇔

where k is a function such that limx→∞ k(x) = 1. To obtain an equivalent of the last one, we cut the integral in two parts: Z ∞ Z xλ Z ∞ = + . 0

0

xλ

The difficulty is to find a good speed for xλ . To do it, let xλ = λv . For x > xλ , we have:  exp λ ln(1 + a/bx2α ) − 1 < exp



aλ1−2αv b



− 1.

We suppose that we have the following condition: 1 − 2αv < 0

or

v > 1/2α.

(15.13)

Let ε > 0, there exists λ0 such that for each λ > λ0 :  
R∞ 2α exp λ ln(1 + a/bx2α ) − 1 k(x)f0 (x)dx ε xλ exp λ ln x R∞ < . 2α } k(x)f (x)dx 2 exp {λ ln x 0 0

241

15

Extremes of alpha-ARCH Models

We must now prove that the second part also tends to 0. To do it, we do the same operations as in the previous proof. We do the change of variable: x = Ωu(αλ)1/c . The boundaries of the integral are modified: Z

xλ

0

and if: v−

1 λ1 : ψλ (1 − δ) − ψ(1) < 0. And, ∀u < 1 − δ: ψλ (u) − ψλ (1 − δ) < 0, the end of the proof of lemma 15.1 (step 4) is valid and we can conclude in the same way. If α > 1/2, one can find a v which satisfies the constraints (15.13) and (15.14), and then: q(αλ) − pa (αλ) = o(1). On the contrary, if we suppose that α ≤ 12 , then there exists λm such that um − δ < u < um + δ and ∀λ > λm : 1−2α

λ ln(1 + a/b(Ωuλ

aλ (1−α) ) )> , 2b(um + δ)2α Ω2α

1/c 2α

and it follows that: q(αλ) − pa (αλ) 6→ 0. At least, we conclude that: f (x) ∼

x→∞

f0 (x) ⇔ 1/2 < α < 1.

An application of the Hospital rule yields the shape of F¯ .

243

15

Extremes of alpha-ARCH Models

Proof of theorem 15.6. We define the sequences: dZ n

=

cZ n

=

       ln n 2/c 2 ln n 2/c−1 f ln n ln 2D + − ln + , e c e ec e e   2 ln n 2/c−1 , ec e



such that if: Z Z uZ n (τ ) = cn (− ln τ ) + dn ,

then:
n 1 − FZ (uZ → τ. n (τ )) n→∞

Since (Xt )t∈N is geometrically strong mixing, (Zt )t∈N is geometrically strong mixing too, ln and then the condition D(uZ n (τ )) holds for (Zt ) with an upper bound αn,ln ≤ Constρ such that ρ < 1. We introduce now an auxiliary process (Yt )t∈N : Yt = ln Xt2 = ln Zt . We have then:

  a Yt = αYt−1 + ln(b2t ) + ln 1 + e−αYt−1 . b

We note: Ut = ln(b2t )

and

  a Pt = ln 1 + e−αYt−1 , b

and we define the autoregressive process of order 1 (Mt )t∈N , in the following way: M0 = Y0 , Mt = αMt−1 + Ut . We have then: Vt = Yt − Mt =

t−1 X

αj Pt−j .

j=0

Remark here that the random variables Pt and Vt are always positive and that:
n 1 − FY (uYn (τ )) → τ , with uYn (τ ) = ln(uZ n (τ )). n→∞

Let u be a threshold. We define Nu = inf {j ≥ 1|Yj ≤ u}.

244

15.6

Proofs

If we suppose that Y0 > u, then we have for any t ≤ Nu : Mt ≤ Yt ≤ Mt + κ(u), with: κ(u) =

a.s.,

a e−αu . b(1 − α)

Now, we want to check the condition D0 (uZ n (τ )), i.e.: lim n

n→∞

pn X j=1

 Z P Z0 > uZ n , Zj > un = 0.

Z u Since uZ n → ∞ when n → ∞, we suppose that un > e . We have:

n

pn X j=1



P Z0 >

uZ n , Zj

>

uZ n



≤n

pn X

 Z P j > N u , Z 0 > uZ n , Z j > un

j=1 pn X

+n

j=1

 Z P j ≤ N u , Z 0 > uZ n , Z j > un

≤ I1 + I2 To get an upper bound of I1 , we show first, as Borkovec (1999), that there exist constants C > 0 and n0 ∈ N such that for any n > n0 , z ∈ [0, eu ], k ∈ N∗ : nP (Zk > uZ n |Z0 = z) ≤ C. Assume that it does not hold. Choose C and N > 0 arbitrary and η > 0 small. There exist n > N , z ∈ [0, eu ] and δ(η) > 0, such that for any y ∈]z − δ, z + δ[∩[0, eu ], we have: nP (Zk > uZ n |Z0 = y) > C − η. But, we have also:
lim n 1 − FZ (uZ n (τ )) = τ,

n→∞

for any τ as small as you want, and: Z ∞
Z n 1 − FZ (un (τ )) = nP (Zk > uZ n |Z0 = y)dFZ (y) Z0 ≥ nP (Zk > uZ n |Z0 = y)dFZ (y) ]z−δ,z+δ[∩[0,eu ]

> (C − η)P (Z0 ∈]z − δ, z + δ[∩[0, eu ]) ≥ (C − η)D

245

15

Extremes of alpha-ARCH Models

where D = inf z∈[0,eu ] (FZ (z + δ) − FZ (z)) > 0 because FZ is continuous. Since C > 0 is arbitrary, there is a contradiction. Now, we have: I1 ≤

pn X j−1 X

 Z nP Nu = l, Z0 > uZ n , Z j > un

pn X j−1 X

 u Z nP Z0 > uZ n , Z l < e , Z j > un

j=1 l=1

≤

j=1 l=1

u Let Cn =]uZ n , ∞[ and D = [0, e ]. We note : X1 = Z0 , X2 = Zl , X3 = Zj . We have :  u Z P Z0 > uZ n , Z l < e , Z j > un Z = I{x1 ∈Cn ,x2 ∈D,x3 ∈Cn } PX1 ,X2, X3 (x1 , x2 , x3 )dx1 dx2 dx3 R3+

=

Z

I{x1 ∈Cn ,x2 ∈D,x3 ∈Cn } PX3 |X2 =x2 ,X1 =x1 (x3 )dx3 PX2 |X1 =x1 (x2 )dx2

R3+

PX1 (x1 )dx1 =

Z

I{x1 ∈Cn ,x2 ∈D,x3 ∈Cn } PX3 |X2 =x2 (x3 )dx3 PX2 |X1 =x1 (x2 )dx2 PX1 (x1 )dx1

Z

P (X3 ∈ Cn |X2 = x2 )I{x1 ∈Cn ,x2 ∈D} PX2 |X1 =x1 (x2 )dx2 PX1 (x1 )dx1

R3+

=

R2+

Z

≤

C n

≤

C C Cτ P (X1 ∈ Cn , X2 ∈ D) ≤ P (X1 ∈ Cn ) ∼ . n→∞ n2 n n

R2+

I{x1 ∈Cn ,x2 ∈D} PX2 |X1 =x1 (x2 )dx2 PX1 (x1 )dx1

At least, we have: I1 ≤

pn X 2Cτ p2 j ≤ Cτ n n n j=1

Furthermore, we have: I2 ≤ n

pn X j=1

 P M0 > uYn , Mj > uYn − κ(u) .

Moreover, Mk = Uk + ... + αk−1 U1 + αk M0 and then: o  n Mk > uYn − κ(u) ⊂ Uk + ... + αk−1 U1 > (uYn − κ(u))/2 n o ∪ αk M0 > (uYn − κ(u))/2 ,

246

15.6

Proofs

hence:  P M0 > uYn , Mj > uYn − κ(u) n o ≤ P M0 > uYn , αk M0 > (uYn − κ(u))/2 o  n + P M0 > uYn P Uk + ... + αk−1 U1 > (uYn − κ(u))/2 . Note that α > 1/2. We choose ε > 0 such that 1 + ε < 1/(2α). It exists nu such that for all n > nu , we have : (uYn − κ(u)) (uYn − κ(u)) > > (1 + ε)uYn > uYn . 2α 2αk We deduce that : n o Y k Y P M0 > un , α M0 > (un − κ(u))/2 n o  (1+ε) ≤ P M0 > (1 + ε)uYn = P Z0 > (uZ (τ )) . n But, we have : (1+ε) (uZ n (τ ))

=



ln n e

2(1+ε)/c

A(1+ε) (τ ), n

o` u lim An (τ ) = 1, n→∞

and then : n o (1+ε) P Z0 > (uZ (τ )) n n   o 2(1+ε)/c (1+ε) ≤ exp −(ln n)(1+ε) Ac/2 (τ ) − f ln (ln n/e) A (τ ) + ln(2D) n n o n ≤ exp −2(ln n)(1+ε) for any n big enough. Furthermore, it is easy to see that the bigger k, the heavier the distribution tail of the random variable Uk + ... + αk−1 U1 , by using lemma 1.6.1 for example. And in the same way as before, we have that for any n big enough: n o  P Uk + ... + αk−1 U1 > (uYn − κ(u))/2 ≤ P Y0 > (uYn − κ(u))/2 n o ≤ exp −2(ln n)1/2−ε It follows that: o  n  I2 ≤ pn const exp −2(ln n)1+ε + const exp −2(ln n)1/2−ε n o I2 ≤ pn const exp −2(ln n)1/2−ε

247

15

Extremes of alpha-ARCH Models

where const is a generic constant. Finally, by choosing: h n oi pn = exp (ln n)1/4

h n oi ln = exp (ln n)1/8

and

then all the conditions for D0 (uZ n (τ )) are verified, the statement follows: the extremal index of (Zt ), θZ , exists and is equal to one.

15.7

Conclusion

We observe quite different extremal behaviors depending on whether α = 1 or α < 1. In the first case, we observe Pareto-like tails and an extremal index which is strictly less than one. In the second case, for α > 1/2, the tails are Weibull–like and the extremal index is equal to one.

APPENDIX Appendix 1: We define the functions: g1 (λ) = λ ln λ,

g2 (λ) = λ,

and

g3 (λ) = ln λ.

We have then: g1 (λ) − g1 (αλ) = (1 − α)λ ln λ − λα ln α, g2 (λ) − g2 (αλ) = (1 − α)λ, g3 (λ) − g3 (αλ) = − ln α. There exist three constants a1 , a2 et a3 such that: a1 (g1 (λ) − g1 (αλ)) + a2 (g2 (λ) − g2 (αλ)) + a3 (g3 (λ) − g3 (αλ)) = λ ln λ + λ(ln 2b − 1) + ln 2/2, which are given by: a1 =

1 , (1 − α)

a2 =

h

  i α ln 2bα 1−α − 1 (1 − α)

et

a3 = −

ln 2 . 2 ln α

And then, q0 (λ) − q0 (αλ) =a1 (g1 (λ) − g1 (αλ)) + a2 (g2 (λ) − g2 (αλ)) + a3 (g3 (λ) − g3 (αλ)) + O(1/λ).

248

15.7

Conclusion

Remark now that if s is a function such that s(λ) ∼ c/λ with c different from 0, then: λ→∞

s(λ) − s(αλ) ∼

λ→∞

c(1 − 1/α) λ

At least, we note C the set of continuous fonctions from R+ to R, and we define the application Ψ : C → C such that Ψ(f ) = fα and fα (λ) = f (λ) − f (αλ), ∀λ ∈ R+ . The kernel of this linear application is the set of the constants. With all these elements, we deduce that: h   i α 1−α ln 2bα − 1 ln 2 λ ln λ + λ− ln λ + m + O(1/λ). q0 (λ) = (1 − α) (1 − α) 2 ln α Appendix 2: The process (Yt ) is defined by: Yt = [Xt ] + Ut − 0.5. We have the following inequalities: −0.5 ≤ Ut − 0.5 ≤ 0.5

and

Xt − 1 < [Xt ] ≤ Xt ,

and then, Xt − 1.5 < Yt ≤ Xt + 0.5. We deduce that: P (Xt > x + 1.5) P (Yt > x) P (Xt > x − 0.5) ≤ ≤ . P (Xt > x) P (Xt > x) P (Xt > x) But, we have also: P (Xt > x − 0.5) ∼ x→∞ P (Xt > x) P (Xt > x + 1, 5) ∼ x→∞ P (Xt > x)

   1 f ecxc−1 exp → 1, 1− x→∞ 2x 2     3 f 3ecxc−1 1+ exp − → 1, x→∞ 2x 2



and finally we obtain: P (Yt > x) ∼ P (Xt > x) . x→∞

Bibliography Beirlant, J., Broniatowski, M., Teugels, J. & Vynckkier, P. (1995). The mean residual life function at great age: Applications to tails estimation, J. Stat. Plann. Inf 45: 21–48.

249

15

Extremes of alpha-ARCH Models

Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity, J. Econometrics 31: 307–327. Bollerslev, T., Chou, R. & Kroner, K. (1992). Arch modelling in finance, J. Econometrics 52: 5–59. Borkovec, M. (1999). Extremal behavior of the autoregressive process with arch(1) errors, Technical report, University of M¨ unchen. Breidt, F. & Davis, R. (1998). Extremes of stochastic volatility models, Ann. Appl. Probab 8: 664–675. Cont, R., Potters, M. & Bouchaud, J. (1997). Scaling in stock market data: Stable law and beyond, in D. et al. (ed.), Scale Invariance and Beyond, Springer, Berlin. Davis, R., Mikosch, T. & Basrak, C. (1999). Sample acf of stochastic recurrence equations with application to garch, Technical report, University of Groningen. Diebolt, J. & Guegan, D. (1991). Le mod`ele de s´erie chronologique autor´egressive β-arch, Acad. Sci. Paris 312: 625–630. Embrechts, P., C., K. & Mikosch, T. (1997). Modelling Extremal Events, Springer, Berlin. Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation, Econometrica 50: 987–1008. Gourieroux, C. (1997). ARCH Models and Financial Applications, Springer, Berlin. Gourieroux, C., Jasiak, J. & Le Fol, G. (1999). Intraday market activity, Journal of Financial Markets 2: 193–226. Haan, L. D., Resnick, S., Rootzen, H. & Vries, C. D. (1989). Extremal behaviour of solutions to a stochastic difference equation with application to arch processes, Stoch. Proc Appl. 32: 213–224. Leadbetter, M., Lindgren, G. & Rootzen, H. (1983). Extremes and Related Properties of Random Sequences and Processes, Springer, Berlin. Leadbetter, M. & Rootzen, H. (1988). Extremal theory for stochastic processes, Ann. Probab. 16: 431–478. Mandelbrot, B. (1963). The variation of certain speculative prices, J. Business 36: 394– 419. Mikosch, T. & Starica, C. (1998). Limit theory for the sample autocorrelations and extremes of a garch(1,1) process, Technical report, University of Groningen. Perfect, C. (1994). Extremal behavior of stationary markov chains with applications, Ann. Appl. Probab 4: 529–548.

250

15.7

Conclusion

Resnick, S. (1987). Extreme Values, Regular Variation, and Point Processes, Springer, Berlin. Robert, C. (2000). Mouvement extrˆemes des s´eries financi`eres haute fr´equence, Finance . Smith, R. & Weismann, L. (1994). Estimating the extremal index, Journal of the Royal Statistical Society B 56: 515–528.

251