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

191 pages
ECOD·ODNEICS·MLUTÄTUniversitat Ulm¨Institute of Mathematical FinanceTractable multi-firm default modelsbased on discontinuous processesDissertationzur Erlangung des DoktorgradesDr. rer. nat.der Fakultat fur Mathematik und Wirtschaftswissenschaften¨ ¨der Universitat Ulm¨vorgelegt vonDipl.-Math. oec. Matthias Scherer, M.S.ausBiberach an der RißUlm, im Februar 2007ISREVINU·ODNARUC·ODNAmtierender Dekan: Professor Dr. Frank Stehling1. Gutachter: Professor Dr. Ru¨diger Kiesel, Universit¨at Ulm2. Gutachter: Professor Dr. Ulrich Stadtmu¨ller, Universit¨at Ulm3. Gutachter: Professor Dr. Wim Schoutens, K.U. LeuvenTag der Promotion: 7. Mai 2007PrefaceThe 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 researchwhich was carried out at the Institute of Mathematical Finance at the Universit¨atUlm intheperiodspring2005tospring2007. MyworkwassupervisedbyProfessorRu¨diger Kiesel and Professor Ulrich Stadtmu¨ller, both Universita¨t Ulm.AcknowledgmentsFirst 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 friendlinessprovides a working atmosphere athis institute which makes researching and teach-ing enjoyable day-to-day. Moreover, it is always a pleasure and benefit to discussmathematical and other problems with him.
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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 inside the house of Van der Beurse.
These meetings where then institutionalized as Brugse Beurse. Later, organized
markets for debts, stocks and commodities were founded in Italy and the Nether-
lands. TheAmsterdamStock Exchange isconsideredtobethefirstmodernmarket,
allowing continuous trade in options and other derivatives already in the 17th cen-
tury. Today, corporate and government bonds are traded on every stock exchange
1in large volume. According to a recent study of the ECMI , the outstanding vol-
ume of corporate bonds in the US, Eurozone and Japan amounts to about 2.2, 1.2
and 0.6 trillion Euro in 2004, respectively. For government bonds, an outstanding
volume ofabout4.0, 5.0and4.6trillionEuro, respectively, isreportedforthesame
year.
Compared to the long history of bond markets, the global market of credit deriva-
tives is surprisingly young. It arose in the early 1990s in London and New York
and grew from virtually nothing to an outstanding notional amount of 26.0 trillion
US$, as reported by Bloomberg News on September 19, 2006. Today, the largest
2market share is still occupied by single-name instruments , but the importance
of multi-name derivatives such as collateralized debt obligations grew significantly
over the last years. Without doubt, the volume and growth of this market explain
and justify the scientific interest in sophisticated credit-risk models.
1 European capital market institute: Statistical package, 2006 edition.
2 August 31, 2006: The Wall Street Journal reports the notional in credit default swaps (CDS)
to exceed 17.0 trillion US$.
910 Chapter 1. Introduction
1.2 Credit risk: Definition and models
Today,debtisviewedasaninstrumentwhichallowscompaniestopursueeconomic
activities they could not finance from their own funds. Debtor and creditor agree
on the standard of deferred payments, which typically includes the principal sum
3plus interest. This interest is interpreted as the price of debt which has to be
determinedbasedoneconomicconsiderations. Substantialinfluenceontheamount
of interest is founded by the creditworthiness of the obligor. More abstractly, we
follow Sch¨onbucher (2003) and define credit risk as ”the risk that an obligor does
4not honour his payment obligations.” Over the last decades, several credit-risk
models have been set up to quantify this credit risk and to price bonds and credit
derivatives. The vast majority of all modern credit-risk models is based on one of
the following principles: The structural approach or the reduced-form approach.
Univariate structural default models
Structural default models aim to explain the economic cause of credit default of a
company. More precisely, default is assumed to be the consequence of insufficient
financial strength of a company. Solvency is linked to the ratio of the firm’s assets
and liabilities via the assumption that default is triggered when the value of the
5firm falls below a certain threshold . Consequently, the model of the firm-value
process implicitly specifies the term structure of default probabilities. Therefore,
this process plays the pivotal role in structural default models. Corporate bonds
and credit derivatives are then priced based on this implied term structure of de-
fault probabilities.
A natural criterion to distinguish structural default models is to classify them ac-
cording to the underlying firm-value process. This classification is closely related
to the historical development, as the model of the firm-value process has been
generalized over the years. The first structural default model was published by
Black and Scholes (1973), it relies on a geometric Brownian motion as firm-value
process. Originally, this model was designed to describe stock prices rather than
the value of a firm. Then, the observation ”It is not generally realized, that corpo-
6rate liabilities other than warrants may be viewed as options.” transformed their
3 A well written article about the history and criticism of interest can be found in: DIE ZEIT,
June 2003, ”Ein paar Prozent Streit”, http://www.zeit.de/2003/06/Zinsgeschichte.
4 Scho¨nbucher (2003), page 1.
5 The company’s total liabilities are often used as default threshold. Other popular interpre-
tations are weighted averages of short- and long-term liabilities, KMV: Crosbie and Bohn
(2003), or a minimum firm value which is required to operate the company, compare Black
and Cox (1976).
6 Black and Scholes (1973), Journal of Political Economy 81, page 649.

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