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Levy Processes in Credit Risk and Market Models
Dissertation zur Erlangung des Doktorgrades
der Mathematischen Fakultat
der Albert-Ludwigs-Universitat Freiburg i. Brsg.

vorgelegt von
Fehmi Ozkan
April 2002

Preface
During the last years banks have realized that default risk cannot be neglected.  The subjective assessment of credit worthiness has to be replaced with a more objective way of estimating default risk.  Mathematical credit risk models in the literature are mainly models based on Brownian motion although it is known that real-life nancial data provides a  different statistical behavior than that implied by these models. Levy processes are an appropriate tool to increase accuracy of models in nance. They have been
used to model stock prices, and term structures of interest rates, thus allowing more accurate derivative pricing and risk management. In this study we discuss how general Levy processes can be applied to credit risk and market models.
There are also a few jump-diffusion models where the jump-components have sample paths of bounded variation, but some of the very important subclasses are special types of Levy processes.

The emphasis in this study lies on the application of Levy processes in credit risk models. We will mainly deal with the question of pricing defaultable corporate bonds. However we will also consider a portfolio model where the main question is the modeling of the loss distribution. Furthermore, we present a Levy Libor model which contains the classical Libor model as a special case.
In Chapter 1 we briefly survey some aspects of credit risk, generalized hyperbolic distributions and Levy processes. In the overview of the structural approach, we show how Levy processes can be used to generalize the classical structural approach due to Merton (1974). In Chapter 2 we present and generalize the approach of the commercial software package CreditRisk+TM by Credit Suisse Financial Products. This chapter might be of some interest to practitioners since we introduce an alternative distribution to CreditRisk+TM which allows for heavier tails and still guarantees analytical tractability.  Then we introduce a credit risk Heath-Jarrow-Morton-framework based on Levy processes in Chapter 3, where we generalize the Gaussian approach in Bielecki and Rutkowski (1999, 2000). With a view to more realistic modeling we also include the topic of reorganization within this framework, which is based on an idea of Schonbucher (1998, 2000a).

In Chapter 4 we show how the market practice of pricing caplets with the Black formula can be pushed further to a model where the driving process is a Levy process.  At this point I want to express my warmest thanks to my academic teacher and supervisor Prof. Dr. Ernst Eberlein. I am most indebted to him for introducing me to nancial mathematics and to the theory of Levy processes and also for numerous encouraging discussions which gave me new ideas and led to new insights. Actually I have learned a lot more than this thesis contains.

I also want to thank Thomas Goll for studying and discussing together some chapters of Jacod and Shiryaev (1987) and other mathematical texts. Further thanks go to my other colleagues from the Institut fur Mathematische Stochastik, Albert-Ludwigs-Universitat Freiburg, and of the Freiburg Center for Data Analysis and Modeling, especially Thomas Gerds and Jan Beyersmann. Financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged.  I want to thank our secretary Monika "The Eye" Hattenbach for finding inconsistencies in the layout and the bibliography.  Furthermore, I want to thank my mother Ayse Ozkan and my sister Meltem Ozkan. Very special thanks go to Verena Trenkner.

Contents
1 Introduction 1
1.1 Mathematical approaches . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Structural models . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Intensity-based models . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Portfolio approaches . . . . . . . . . . . . . . . . . . . . . 7
1.2 Practical approaches . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Levy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Generalized hyperbolic distributions . . . . . . . . . . . . . . . . . 11
2 A Portfolio Approach 16
2.1 Default events with xed default rates . . . . . . . . . . . . . . . 16
2.2 Loss distribution with xed default rates . . . . . . . . . . . . . . 20
2.3 Sector analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Default losses with variable default rates . . . . . . . . . . . . . . 29
2.5 Generalized sector analysis, risk contribution, and correlation . . 32
2.6 The CreditRisk+TM example . . . . . . . . . . . . . . . . . . . . . 34
3 The Intensity-based Levy Credit Risk Model 38
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 The default-free term structure . . . . . . . . . . . . . . . . . . . 39
3.2.1 Absence of arbitrage . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 Construction of the density process . . . . . . . . . . . . . 49
3.3 Pre-default term structure . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Defaultable term structure . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Reorganization and multiple defaults . . . . . . . . . . . . . . . . 68
3.6 Credit spreads and default intensities . . . . . . . . . . . . . . . . 74
3.7 Market price of risk . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8 Outline of the implementation . . . . . . . . . . . . . . . . . . . . 80
3.9 Pure credit derivatives . . . . . . . . . . . . . . . . . . . . . . . . 82
3.10 A note on the Schonbucher approach . . . . . . . . . . . . . . . . 85
4 Market Models 92
4.1 Modeling instantaneous forward rates . . . . . . . . . . . . . . . . 93
4.1.1 The semimartingale setting . . . . . . . . . . . . . . . . . 93
4.1.2 The Levy setting . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.3 Relationship to the Jamshidian approach . . . . . . . . . . 107
4.2 A forward price model . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3 The discrete tenor Levy Libor rate model . . . . . . . . . . . . . . 115
4.3.1 Construction of the forward measures . . . . . . . . . . . . 117
4.3.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3.3 Forward measure modeling . . . . . . . . . . . . . . . . . . 125
4.3.4 Examples for the volatility structure . . . . . . . . . . . . 132
4.4 Pricing of caps and
oors . . . . . . . . . . . . . . . . . . . . . . . 133
A Construction of the Conditional Markov Process 141
IV

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