Sampling nested Archimedean copulas with applications to CDO pricing [Elektronische Ressource] / Jan Marius Hofert
175 pages
English

Sampling nested Archimedean copulas with applications to CDO pricing [Elektronische Ressource] / Jan Marius Hofert

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175 pages
English
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Ulm UniversityInstitute of Number Theoryand Probability TheoryHelmholtzstraße 1889081 UlmDissertationSampling Nested Archimedean Copulaswith Applications to CDO PricingSubmitted for the degree of Dr. rer. nat. in the Faculty ofMathematics and Economics at Ulm UniversityJan Marius HofertJanuary 2010Acting dean: Prof. Dr. Werner KratzFirst advisor: Prof. Dr. Ulrich StadtmüllerSecond Prof. Dr. Rüdiger KieselDate of defense: 2010-02-26I hereby declare that this dissertation was performed and written on my own and thatreferences and resources used within this work have been explicitly indicated. I am awarethat making a false declaration may have serious consequences.Ulm, January 2010 Jan Marius HofertTo the most important dependence in life,Love.PrefaceThis dissertation is based on the results I developed during my time at the Institute ofApplied Analysis and the Institute of Number Theory and Probability Theory at UlmUniversity. Ulm has always been a home to me and so this preface is dedicated to allpeople creating this friendly and pleasant atmosphere I have enjoyed since I came to thisvivid place.First of all, I would like to express my deepest gratitude to Prof. Dr. Ulrich Stadtmüllerfor introducing me to the fascinating world of copulas. I am fortunate to have an excellentadvisor who gives me the freedom to explore on my own but at the same time the supportto recover from setbacks.

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Publié le 01 janvier 2010
Nombre de lectures 25
Langue English
Poids de l'ouvrage 4 Mo

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Ulm University
Institute of Number Theory
and Probability Theory
Helmholtzstraße 18
89081 Ulm
Dissertation
Sampling Nested Archimedean Copulas
with Applications to CDO Pricing
Submitted for the degree of Dr. rer. nat. in the Faculty of
Mathematics and Economics at Ulm University
Jan Marius Hofert
January 2010Acting dean: Prof. Dr. Werner Kratz
First advisor: Prof. Dr. Ulrich Stadtmüller
Second Prof. Dr. Rüdiger Kiesel
Date of defense: 2010-02-26
I hereby declare that this dissertation was performed and written on my own and that
references and resources used within this work have been explicitly indicated. I am aware
that making a false declaration may have serious consequences.
Ulm, January 2010 Jan Marius HofertTo the most important dependence in life,
Love.Preface
This dissertation is based on the results I developed during my time at the Institute of
Applied Analysis and the Institute of Number Theory and Probability Theory at Ulm
University. Ulm has always been a home to me and so this preface is dedicated to all
people creating this friendly and pleasant atmosphere I have enjoyed since I came to this
vivid place.
First of all, I would like to express my deepest gratitude to Prof. Dr. Ulrich Stadtmüller
for introducing me to the fascinating world of copulas. I am fortunate to have an excellent
advisor who gives me the freedom to explore on my own but at the same time the support
to recover from setbacks. Further, I would like to cordially thank him for giving me the
opportunity to teach exercise classes at the Institute of Number Theory and Probability
Theory. I especially enjoyed assisting in the first lecture on copulas at Ulm University in
Summer 2009, incidentally taking place in the year this field of research celebrated its
50th anniversary.
I am also indebted to Prof. Dr. Werner Kratz for offering me the opportunity to teach
at the Institute of Applied Analysis and for fruitful suggestions, and to Prof. Dr. Rüdiger
Kiesel for being my second advisor.
I would further like to thank Prof. Dr. Ulrich Stadtmüller and Prof. Dr. Werner Kratz
for providing financial support for various conferences. Giving talks and meeting other
copula researchers are always exciting events and lead to invaluable input for further
research.
I would especially like to thank my best friends, Robin Nittka and Matthias Scherer,
for their support and for going through thick and thin with me over the last years. I
am indebted to Robin for many valuable discussions related to mathematics and to
Matthias for invitations to Munich and numerous athletic challenges. I greatly value their
close friendship and deeply appreciate their belief in me. I would also like to thank my
colleagues Matthias Degen, Catherine Donnelly, Christian Hering, Dominik Lambrigger,
and Matthias Lutz for proofreading this work.
Doing sports is the perfect balance for me and helps me staying fit and focussed. I
would like to thank my coach Manfred Straub for training me and guiding me like a
father guides his son.
Finally and most importantly, none of this would have been possible without the
enduring love and support of my parents Christine and Klaus, and my sister Caren, who
encourage me throughout all stages of my life.
5Preface
6Contents
Preface 5
List of Tables 11
List of Figures 13
Abstract 15
Einleitung und Zusammenfassung 19
1 Copulas 23
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.2 Definition and basic properties of subcopulas and copulas . . . . . . . . . 28
1.3 Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4 Random vectors and copulas . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5 Survival copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.6 Symmetries of copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.7 Measures of association . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.7.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.7.2 Measures of dependence . . . . . . . . . . . . . . . . . . . . . . . . 39
1.7.3 of concordance . . . . . . . . . . . . . . . . . . . . . . . . 40
1.7.4 Tail dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.8 Sampling copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.9 Elliptical copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2 Archimedean and nested Archimedean copulas 51
2.1 Archimedean copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.1 Archimedean copulas in two dimensions . . . . . . . . . . . . . . . 52
2.1.2 Arc in arbitrary dimensions . . . . . . . . . . . . 53
2.2 Nested Archimedean copulas . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3 Properties of Archimedean and nested Archimedean copulas . . . . . . . . 60
2.4 Parametric Arc families . . . . . . . . . . . . . . . . . . . . . . . 63
2.5 Sampling Archimedean and nested Archimedean copulas . . . . . . . . . . 67
3 Numerically sampling nested Archimedean copulas 71
3.1 Numerical inversion of Laplace transforms . . . . . . . . . . . . . . . . . . 72
3.1.1 The Fixed Talbot algorithm . . . . . . . . . . . . . . . . . . . . . . 72
7Contents
3.1.2 The Gaver-Stehfest and Gaver-Wynn-rho algorithm . . . . . . . . 73
3.1.3 The Laguerre-series algorithm . . . . . . . . . . . . . . . . . . . . . 76
3.2 Sampling Archimedean copulas using numerical inversion algorithms . . . 77
3.3 Experiments and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.1 A word concerning the implementation . . . . . . . . . . . . . . . . 80
3.3.2 Precision and run-time comparison for Clayton’s copula . . . . . . 82
3.3.3 A Clayton copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.4 An outer power Clayton copula . . . . . . . . . . . . . . . . . . . . 85
3.3.5 A nested outer power Clayton copula . . . . . . . . . . . . . . . . 86
3.3.6 Nested Gumbel and Clayton copulas . . . . . . . . . . . . . . . . . 87
3.3.7 An Ali-Mikhail-Haq(Clayton) copula . . . . . . . . . . . . . . . . . 89
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Directly sampling nested Archimedean copulas 93
4.1 Nested Ali-Mikhail-Haq, Frank, and Joe copulas . . . . . . . . . . . . . . 94
4.2 Archimedean copulas based on generator transformations . . . . . 98
4.2.1 Exponentially tilted generators and a fast sampling algorithm . . . 99
4.2.2 Sampling exponentially tilted stable distributions . . . . . . . . . . 101
4.2.3 A general nesting transformation . . . . . . . . . . . . . . . . . . . 104
4.3 Nested Archimedean copulas based on different Archimedean families . . . 107
4.4 Experiments and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4.1 Joe copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4.2 Nested Frank and Joe copulas . . . . . . . . . . . . . . . . . . . . . 113
4.4.3 Comparison of algorithms for nested Clayton copulas . . . . . . . . 115
4.4.4 Overall precision and run-time comparison . . . . . . . . . . . . . 118
4.4.5 An Ali-Mikhail-Haq(Clayton(20)) copula . . . . . . . . . . . . . . 119
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5 CDO pricing with nested Archimedean copulas 123
5.1 Motivation and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.1 The copulas considered . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.2 Default correlations in a nested framework . . . . . . . . . . . . . 128
5.3 Portfolio CDSs and CDOs . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.1 The payment streams . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.2 The pricing approach . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4 Calibration to market spreads . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.4.1 Data and setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4.2 Calibration of the model . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4.3 Results of the calibration . . . . . . . . . . . . . . . . . . . . . . . 137
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A Supplementary material 145
A.1 Conditional distribution functions . . . . . . . . . . . . . . . . . . . . . . 145
8Contents
A.2 Marshall-Olkin copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A.3 Laplace-Stieltjes transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B Supplementary CDO pricing results 155
Bibliography 163
Index 171
9Contents
10

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