Tractable multi-firm default models based on discontinuous processes [Elektronische Ressource] / vorgelegt von Matthias Scherer

Sujets

• Mathematics

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2007
• Nombre de lectures: 41
• Langue: English
• Poids de l'ouvrage: 1 Mo

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Universitat Ulm¨
Institute of Mathematical Finance
Tractable multi-firm default models
based on discontinuous processes
Dissertation
zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakultat fur Mathematik und Wirtschaftswissenschaften¨ ¨
der Universitat Ulm¨
vorgelegt von
Dipl.-Math. oec. Matthias Scherer, M.S.
aus
Biberach an der Riß
Ulm, im Februar 2007
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NAmtierender Dekan: Professor Dr. Frank Stehling
1. Gutachter: Professor Dr. Ru¨diger Kiesel, Universit¨at Ulm
2. Gutachter: Professor Dr. Ulrich Stadtmu¨ller, Universit¨at Ulm
3. Gutachter: Professor Dr. Wim Schoutens, K.U. Leuven
Tag der Promotion: 7. Mai 2007Preface
The aim of this thesis is to contribute to the ongoing research in structural credit-
risk models based on jump-diffusion processes. It concludes my doctorate research
which was carried out at the Institute of Mathematical Finance at the Universit¨at
Ulm intheperiodspring2005tospring2007. MyworkwassupervisedbyProfessor
Ru¨diger Kiesel and Professor Ulrich Stadtmu¨ller, both Universita¨t Ulm.
Acknowledgments
First and foremost I would like to express my deepest gratitude to Professor Ru¨di-
ger Kiesel for guiding and supporting me over the last two years. His friendliness
provides a working atmosphere athis institute which makes researching and teach-
ing enjoyable day-to-day. Moreover, it is always a pleasure and benefit to discuss
mathematical and other problems with him.
I would also like to kindly thank Professor Ulrich Stadtmu¨ller for being my co-
examiner. I am also deeply indebted to Professor Nick Bingham, University of
Sheffield, for his helpful comments on several of my working papers and to Dr.
Hartmut Lanzinger, Universit¨at Ulm, for discussing the important proofs of my
thesis in detail.
Furthermore, I would like to thank my colleagues Reik B¨orger, Ph.D. Stefan Kass-
berger, Gregor Mummenhoff, Clemens Prestele and Dr. Martin Riesner for nu-
merous discussions and social events. Also, I received several valuable ideas and
suggestions from Miglena Gavrilova, Alexei Ivanov, Johannes Ruf and Dmitry Za-
ykovskiy, who wrote their master or diploma thesis at our institute. I want to
address special thanks to Marius Hofert for helping me with the layout and for
proofreading the manuscript of this thesis. On a personal note, I want to thank
my parents for their enduring support and belief in me. Additionally, I am very
thankful for all the support and love of my partner Rebekka Meier.
34Contents
1 Introduction 9
1.1 The history of the bond market . . . . . . . . . . . . . . . . . . . . 9
1.2 Credit risk: Definition and models . . . . . . . . . . . . . . . . . . . 10
1.3 Our contribution and aim of this thesis . . . . . . . . . . . . . . . . 16
1.3.1 Our findings in Zhou’s univariate model . . . . . . . . . . . 16
1.3.2 Our new multidimensional model . . . . . . . . . . . . . . . 19
1.3.3 Organization of this thesis . . . . . . . . . . . . . . . . . . . 20
2 Technical background 21
2.1 An introduction to L´evy processes . . . . . . . . . . . . . . . . . . . 21
2.1.1 The probabilistic framework . . . . . . . . . . . . . . . . . . 21
2.1.2 Stopping times and martingales . . . . . . . . . . . . . . . . 24
2.1.3 General properties of L´evy processes . . . . . . . . . . . . . 25
2.1.4 Building a jump-diffusion process . . . . . . . . . . . . . . . 27
2.2 The Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Definition and basic properties . . . . . . . . . . . . . . . . . 30
2.2.2 The Gaver-Stehfest algorithm . . . . . . . . . . . . . . . . . 31
3 Products and pricing issues 35
3.1 Corporate bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Credit default swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 38
56 Contents
3.3 Collateralized debt obligations . . . . . . . . . . . . . . . . . . . . . 40
3.4 Portfolio CDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
th3.5 n -to default contracts . . . . . . . . . . . . . . . . . . . . . . . . 46
4 The univariate model 47
4.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 First-passage times . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1 First-passage times in a pure diffusion scenario . . . . . . . . 49
4.2.2 The local default rate in a pure diffusion scenario . . . . . . 51
4.2.3 First-passage times in a jump-diffusion scenario . . . . . . . 52
4.2.4 The local default rate in a jump-diffusion scenario . . . . . . 54
4.3 Pricing corporate bonds . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.1 Pricing in a pure diffusion scenario . . . . . . . . . . . . . . 57
4.3.2 Pricing in a jump-diffusion scenario . . . . . . . . . . . . . . 58
4.3.3 Accelerating the algorithm . . . . . . . . . . . . . . . . . . . 62
4.3.4 The limit of credit spreads for short maturities . . . . . . . . 63
4.4 Pricing CDS contracts . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.1 Pricing CDS in a pure diffusion scenario . . . . . . . . . . . 65
4.4.2 Pricing CDS in a jump-diffusion scenario . . . . . . . . . . . 66
4.4.3 The limit of CDS spreads for short maturities . . . . . . . . 67
4.5 Generalizations of the model . . . . . . . . . . . . . . . . . . . . . . 69
4.5.1 Including a stochastic recovery rate . . . . . . . . . . . . . . 69
4.5.2 Including a short-rate model . . . . . . . . . . . . . . . . . . 70
4.5.3 Including a stochastic default threshold . . . . . . . . . . . . 72
4.5.4 Pricing based on reduced information . . . . . . . . . . . . . 74
4.6 The two-sided exponential distribution . . . . . . . . . . . . . . . . 80
4.6.1 Basic properties of Kou’s stock-price model. . . . . . . . . . 80Contents 7
4.6.2 The Laplace transform of first-passage times . . . . . . . . . 82
4.6.3 Bond and CDS pricing using the Laplace transform . . . . . 85
4.7 Sensitivity of the model parameters . . . . . . . . . . . . . . . . . . 87
4.7.1 Sensitivity with respect to the drift . . . . . . . . . . . . . . 87
4.7.2 Sensitivity with respect to the diffusion volatility . . . . . . 88
4.7.3 Sensitivity with respect to the jump intensity . . . . . . . . 89
4.7.4 Sensitivity with respect to the influence of jumps . . . . . . 90
4.7.5 Sensitivity with respect to the leverage ratio . . . . . . . . . 91
4.7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.8 A comparison of the different algorithms . . . . . . . . . . . . . . . 94
4.8.1 Run time and precision . . . . . . . . . . . . . . . . . . . . . 94
4.8.2 A closer comparison of both integral approximations . . . . 96
4.9 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.9.1 Calibration to corporate bonds . . . . . . . . . . . . . . . . 99
4.9.2 Calibration to iTraxx CDS quotes . . . . . . . . . . . . . . . 103
4.10 Summary of the univariate model . . . . . . . . . . . . . . . . . . . 107
5 The multidimensional model 109
5.1 The multivariate firm-value model . . . . . . . . . . . . . . . . . . . 110
5.1.1 A common market factor and common jumps . . . . . . . . 110
5.1.2 A common market factor and dependent jumps . . . . . . . 111
5.1.3 Segmentation by industry sector . . . . . . . . . . . . . . . . 113
5.1.4 Properties and applications of the model . . . . . . . . . . . 114
5.1.5 Existing structural portfolio models . . . . . . . . . . . . . . 115
5.2 Default, asset and implied correlation . . . . . . . . . . . . . . . . . 117
5.2.1 Default correlation . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.2 Asset-value correlation . . . . . . . . . . . . . . . . . . . . . 1198 Contents
5.2.3 Implied correlations . . . . . . . . . . . . . . . . . . . . . . . 123
5.3 Pricing CDOs via Monte Carlo simulation . . . . . . . . . . . . . . 126
5.3.1 Discussion of the pricing algorithm . . . . . . . . . . . . . . 132
5.3.2 The convergence of estimated CDO spreads . . . . . . . . . 134
5.3.3 Numerical experiments with fictitious portfolios . . . . . . . 136
5.3.4 Calibration to iTraxx quotes . . . . . . . . . . . . . . . . . . 140
th5.4 Pricing n -to default contracts via Monte Carlo simulation . . . . 150
5.4.1 Implementation of the pricing formula . . . . . . . . . . . . 150
5.4.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 151
5.5 Summary of the multivariate model . . . . . . . . . . . . . . . . . . 155
6 Appendix 157
6.1 The proof of Theorem 4.3.3 . . . . . . . . . . . . . . . . . . . . . . 157
6.2 The roots of a quartic polynomial . . . . . . . . . . . . . . . . . . . 160
6.3 Zhou’s bond-pricing algorithm . . . . . . . . . . . . . . . . . . . . . 161
6.4 Vasicek’s asymptotic single factor model . . . . . . . . . . . . . . . 162
7 Zusammenfassung 165Chapter 1
Introduction
1.1 The history of the bond market
The origin of organized markets dates back to the 12th century France and 13th
century Belgium. In France, the first brokers traded in debts of agricultural com-
munities. In Bruges, commodity traders met