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Credit portfolio modelling with elliptically contoured distributions [Elektronische Ressource] : approximation, pricing, dynamisation / vorgelegt von Clemens Prestele

211 pages
Universit¨at UlmInstitut fur¨ FinanzmathematikCredit Portfolio Modelling with EllipticallyContoured DistributionsApproximation, Pricing, DynamisationDissertationzur Erlangung des DoktorgradesDr. rer. nat.der Fakultat¨ fur¨ Mathematik und Wirtschaftswissenschaftender Universit¨at Ulmvorgelegt vonDipl.-Math. oec. Clemens Presteleaus ErlangenUlm, 2007iiAmtierender Dekan: Professor Dr. Frank Stehling1. Gutachter: Professor Dr. Rud¨ iger Kiesel, Universit¨at Ulm2. Gutachter: Professor Dr. Ulrich Stadtmul¨ ler, Universit¨at UlmTag der Promotion: 14. November 2007My ventures are not in one bottom trusted,Nor to one place; nor is my whole estateUpon the fortune of this present year;Therefore my merchandise makes me not sad.The Merchant of Venice, Act 1 Scene 1by William ShakespeareiiiivAcknowledgementsFirst of all, I wish to express my sincere appreciation to Prof. Dr. Ru¨diger Kiesel foroffering me the opportunity to write this thesis under his supervision at the Institute ofMathematical Finance. I am very thankful for his great guidance, valuable suggestionsandcomprehensivesupportduringtheyearsofmydoctoralresearch.Ialwaysenjoyedtocontributetotheteachingdutiesandthe“life”attheinstituteandhighlyappreciatedtheencouragement and financial assistance e.g. for taking part in seminars and conferences.I extend my great gratitude to Prof. Dr.
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Universit¨at Ulm
Institut fur¨ Finanzmathematik
Credit Portfolio Modelling with Elliptically
Contoured Distributions
Approximation, Pricing, Dynamisation
Dissertation
zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakultat¨ fur¨ Mathematik und Wirtschaftswissenschaften
der Universit¨at Ulm
vorgelegt von
Dipl.-Math. oec. Clemens Prestele
aus Erlangen
Ulm, 2007ii
Amtierender Dekan: Professor Dr. Frank Stehling
1. Gutachter: Professor Dr. Rud¨ iger Kiesel, Universit¨at Ulm
2. Gutachter: Professor Dr. Ulrich Stadtmul¨ ler, Universit¨at Ulm
Tag der Promotion: 14. November 2007My ventures are not in one bottom trusted,
Nor to one place; nor is my whole estate
Upon the fortune of this present year;
Therefore my merchandise makes me not sad.
The Merchant of Venice, Act 1 Scene 1
by William Shakespeare
iiiivAcknowledgements
First of all, I wish to express my sincere appreciation to Prof. Dr. Ru¨diger Kiesel for
offering me the opportunity to write this thesis under his supervision at the Institute of
Mathematical Finance. I am very thankful for his great guidance, valuable suggestions
andcomprehensivesupportduringtheyearsofmydoctoralresearch.Ialwaysenjoyedto
contributetotheteachingdutiesandthe“life”attheinstituteandhighlyappreciatedthe
encouragement and financial assistance e.g. for taking part in seminars and conferences.
I extend my great gratitude to Prof. Dr. Ulrich Stadtmu¨ller of the Institute of Number
Theory and Probability Theory for his readiness to advise the thesis as second assessor.
He has raised my interest in probability theory during his challenging and motivating
lecture in stochastics, which I attended early in my studies, and I am thankful for the
beneficial knowledge I gained from assisting him in a course in probability theory.
1GratefulthoughtsgotoProf.Dr.NickH.BinghamforproofreadingmyEnglish andfor
his valuable comments and suggestions. I want to express my warmest regards to Prof.
Dr. Frank Stehling for his great cordiality and for numerous precious discussions, and to
Prof. Dr. Werner Kratz for having been a great teacher and mentor to me from the early
beginning of my studies.
I would like to thank my friends and colleagues Dr. Bjorn¨ B¨ottcher, Gregor Mummen-
hoff, Dr. Hartmut Lanzinger, Dr. Martin Riesner, Matthias Lutz, Dr. Matthias Scherer,
Monika Thalmaier, Dr. Peter N. Posch, Reik B¨orger, Sebastian Singer and Dr. Stefan
Kassberger for great discussions, wonderful seminars and for their valuable assistance
whenever I asked them for advice.
I want to express my special gratitude to my parents, my sister Sabine and my twin
brother Benjamin for their loving empathy, continual support and perpetual encourage-
ment throughout the entire time of my studies.
Finally, I am very grateful for the scholarship, which I received from the federal state
of Baden-Wurt¨ temberg (LGFG Baden-Wurtt¨ emberg) during the time of my doctoral
studies.
Ulm, 17. September 2007 Clemens Prestele
1
Of course, all the remaining errors are entirely the author’s responsibility.
vviContents
1 Introduction 5
1.1 Recent developments of the credit derivatives markets . . . . . . . . . . . 5
1.2 Collateralized Debt Obligations . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Overview over the existing literature . . . . . . . . . . . . . . . . . . . . . 7
1.4 The contribution and the aim of this thesis . . . . . . . . . . . . . . . . . 11
1.5 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Collateralized Debt Obligations and the pricing problem 19
2.1 The basic structure of a CDO . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Challenges with modelling and pricing CDOs . . . . . . . . . . . . . . . . 22
3 Fundamental theoretical concepts 25
3.1 Elliptical distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Regular variation and Karamata’s Theorem . . . . . . . . . . . . . . . . . 39
3.3 Copulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 A limit theorem for martingales . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Credit portfolio models 51
4.1 Basic setup of structural models . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 The Merton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 A multivariate Merton model . . . . . . . . . . . . . . . . . . . . . . . . . 53
1Contents 2
4.4 One-period credit portfolio models . . . . . . . . . . . . . . . . . . . . . . 54
4.5 One-period factor models . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5.1 Gaussian factor models: Vasicek, KMV and CreditMetrics . . . . . 55
4.5.2 Gaussian one-factor model, implied and base correlations . . . . . 58
4.5.3 Double t-distribution copula model . . . . . . . . . . . . . . . . . . 61
4.5.4 Normal Inverse Gaussian factor model . . . . . . . . . . . . . . . . 61
5 An elliptical distributions credit portfolio model 63
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 The setup - extension of the Gaussian case . . . . . . . . . . . . . . . . . 66
5.3 Introducing tail dependence in the elliptical distributions model . . . . . . 72
5.3.1 Tail dependence in mixtures of Normal distributions . . . . . . . . 73
5.3.2 Examples for mixtures of Normal distributions . . . . . . . . . . . 81
5.3.3 Effects of the distributions on the dependence structure . . . . . . 88
5.3.4 Tail dependence in linear factor models . . . . . . . . . . . . . . . 90
6 Large portfolio approximation in the elliptical distributions model 93
6.1 Credit ratings and credit losses . . . . . . . . . . . . . . . . . . . . . . . . 94
6.1.1 Credit ratings and rating thresholds . . . . . . . . . . . . . . . . . 94
6.1.2 Losses due to rating migrations and defaults . . . . . . . . . . . . 96
6.2 Large portfolio approximation . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 First example of the large portfolio approximation . . . . . . . . . . . . . 102
7 Application to the valuation of credit derivatives 107
7.1 The modelling of the CDO structure . . . . . . . . . . . . . . . . . . . . . 108
7.2 The valuation of a CDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3 Defaults and their probabilities . . . . . . . . . . . . . . . . . . . . . . . . 112Contents 3
7.4 Approximation of the portfolio losses . . . . . . . . . . . . . . . . . . . . . 113
7.5 Key quantities revisited within the models based on mixture distributions 117
7.5.1 Default probabilities and default thresholds . . . . . . . . . . . . . 117
7.5.2 Expected tranche losses . . . . . . . . . . . . . . . . . . . . . . . . 120
7.5.3 The number of defaults . . . . . . . . . . . . . . . . . . . . . . . . 123
7.6 Application of the models on CDO data . . . . . . . . . . . . . . . . . . . 124
7.6.1 Sequence of computations for the valuation of CDOs . . . . . . . . 124
7.6.2 The data basis we use . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.6.3 Calibrating the models . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.6.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.6.5 Performance of the models on the iTraxx data . . . . . . . . . . . 129
8 Dynamic elliptical distributions model 139
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.2 Dynamic setup of elliptical distributions factor model . . . . . . . . . . . 140
8.3 Effects of the dynamical setup. . . . . . . . . . . . . . . . . . . . . . . . . 141
8.4 The discrete-time case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4.1 Employing time-series models for the scaling process . . . . . . . . 145
8.4.2 Applying discrete-time volatility models . . . . . . . . . . . . . . . 151
8.4.3 Continuous-time results for the discrete-time case . . . . . . . . . . 155
8.5 The continuous-time case . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.5.1 Continuous-time interest-rate models for the scaling process . . . . 158
8.5.2 Dynamic setup via time-changed Brownian motions . . . . . . . . 164
8.5.3 Dynamic setup via subordinated L´evy processes . . . . . . . . . . 168
8.5.4 Construction of a L´evy process with the Exp-Exp Law? . . . . . . 172Contents 4
A Supplementary lemma 179
A Supplementary data 181
List of Abbreviations 183
Bibliography 185
List of Tables 193
List of Figures 195
Zusammenfassung 197

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