Modeling and analysis of structured finance products [Elektronische Ressource] / vorgelegt von Florian Kramer

Sujets

• Mathematics

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2008
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Modeling and Analysis
of
Structured Finance Products
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakult¨at fu¨r Mathematik und Wirtschaftswissenschaften
der Universit¨at Ulm
vorgelegt von
Florian Kramer
aus
Kaufbeuren
Ulm, September 2008Amtierender Dekan: Prof. Dr. Frank Stehling
1. Gutachter: Prof. Dr. Gunter L¨offler
2. Gutachter: Prof. Dr. Ru¨diger Kiesel
Tag der Promotion: 12.12.2008Contents
List of Symbols and Notation vii
List of Abbreviations xi
List of Figures xiv
List of Tables xvi
1 Introduction 1
1.1 Motivation and Formulation of the Problem . . . . . . . . . . . . . 1
1.2 Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . 4
2 A Model for a Vector of Stopping Times 11
2.1 Definition of the Stopping Times Model. . . . . . . . . . . . . . . . 12
2.2 Single Survival and Jump Probabilities . . . . . . . . . . . . . . . . 17
2.3 Joint Survival Probabilities. . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Construction of the Stopping Times . . . . . . . . . . . . . . . . . . 28
2.5 Model-Implied Dependence Structure . . . . . . . . . . . . . . . . . 31
2.5.1 Conditional Independence and Contagion . . . . . . . . . . . 31
2.5.2 A Characterization Result for the Dependence Structure . . 35
2.6 The Loss Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.1 Formulation by Means of Orthogonal Point Processes and
Time-Change . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.2 Volatility Structure of the Point Process and a Measure of
Default Clustering . . . . . . . . . . . . . . . . . . . . . . . 49
2.7 Analytically Tractable Model Specifications . . . . . . . . . . . . . 54
2.7.1 AGeneralFormulaforDetermining theCharacteristic Func-
tion of a Process . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7.2 Calculation of Survival Probabilities and the Loss Distribution 59
2.8 Towards Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.8.1 Credit Portfolio Risk Modeling . . . . . . . . . . . . . . . . 66
2.8.2 Stochastic Mortality Modeling . . . . . . . . . . . . . . . . . 74
iiiiv CONTENTS
2.9 Summary and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 77
3 Modeling of Structured Credit Products 79
3.1 Estimation of the Firm’s Default Intensities . . . . . . . . . . . . . 80
3.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.3 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . 85
3.1.4 Comparing Model-implied and Observed Default
Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2 A Model for Default Intensities . . . . . . . . . . . . . . . . . . . . 101
3.3 Single Firm Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3.1 Model Properties and Estimation Methodology . . . . . . . 109
3.3.2 Estimation Results for the Single Firm Intensity Models . . 112
3.4 Assessing Model Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.4.1 Synthetic Structured Credit Products and their Valuation . 121
3.4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.4.3 Estimation Results and the Quantification of Model Risk . . 126
3.5 Portfolio Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.5.1 Estimation Methodology . . . . . . . . . . . . . . . . . . . . 135
3.5.2 Estimation Results for the Portfolio Models . . . . . . . . . 140
3.5.3 Simulating Transition Matrices of Structured Credit Products142
3.6 Summary and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 149
4 Analysis of Catastrophe Mortality Bonds 151
4.1 An Overview on Catastrophe Mortality Bonds . . . . . . . . . . . . 153
4.1.1 Structure of the Securities . . . . . . . . . . . . . . . . . . . 154
4.1.2 Market Development . . . . . . . . . . . . . . . . . . . . . . 157
4.1.3 Modeling Approaches in Practice . . . . . . . . . . . . . . . 161
4.2 Our Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.3 Calibration of the Model . . . . . . . . . . . . . . . . . . . . . . . . 167
4.3.1 Backtesting the Model and Historical Parametrizations . . . 168
4.3.2 Risk-Adjusted Calibration Based on Insurance Prices . . . . 176
4.3.3 Parameters Implied by Market Prices . . . . . . . . . . . . . 180
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5 Conclusion 193
A Appendix to Chapter 3 195
B Appendix to Chapter 4 203CONTENTS v
Bibliography 207
Zusammenfassung 217
Danksagung 223
Erkl¨arung 225vi CONTENTSList of Symbols and Notation
We list all important expressions occurring in the text according to their order of
first appearance. In order to discern between constants, stochastic processes and
functions, we use the following notation:
Constants and sets are denoted by plain symbols, primary stochastic processes are
followed by “(t)”, whereas other stochastic processes as well as functions (which
may map stochastic processes) are followed by brackets either explicitly showing
their dependencies or just their dependence on time t.
Inthetextwewillusevector-andmatrix-notationthroughout. Thismeansthatfor
T2 Iafunctionf :R →R ,f(x,y)= (f (x,y),...,f (x,y)) denotesanI-dimensional1 I
Tcolumnvectorandf (x,y)itstranspose. Wewritef (x,y)forthe|Π|-dimensional(Π)
sub-vector of f(x,y) that contains all coordinates f(x,y), i∈Π⊆{1,...,I}, andi
f (x,y) denotes the vector which consists of all components of the original vector\i
f(x,y) except the i-th component. Comparisons between vectors f and constants
Ic∈ R are to be understood component-wise, e.g. f > c corresponds to f > ci i
for all i∈{1,...,I}; if c is one-dimensional, f > c corresponds to f > c for alli
i∈{1,...,I}.
Chapter 2
F=(F ) General model filtrationt ∗0≤t≤T
X XF = F Subfiltration generated by a process X
∗t 0≤t≤T
τ I-dimensional vector of F-stopping times
I Dimension of τ
N(t) I-dimensional indicator process associated with τ
E I-dimensional vector of mutually independent,
Exp(1)-distributed random variables
viiviii LIST OF SYMBOLS AND NOTATION
Λ(t) I-dimensional jump-trigger process
1X (t) d -dimensional background process1
N2X (t) d -dimensional contagion process, adapted toF2
R Positive real numbers+
R Positive real numbers including{0}0+
ν M-dimensional vector of L´evy measures
d|ς| Euclidian norm of ς∈R
P (t,T) Survival probability of i over [t,T]i
Q (t,T) Jump probability of N over [t,T]i i
p Conditional survival probability of i over [t,T]T,t i
q Conditional jump probability of N over [t,T]T,t i i
Π Subset of{1,...,I}, i.e. Π⊆{1,...,I}
P (t,T) Joint survival probability of Π over [t,T]Π
Q (t,T) Joint jump probability of N over [t,T]Π (Π)
p Conditional joint survival probability of Π over [t,T]T,t Π
q Conditional joint jump probability of N over [t,T]T,t Π (Π)
A (t) Compensator of Ni i
1 λ (t) Intensity process of Nτ >t i ii
ΠdPΣ (t) Density process of the change of measureΠ dP
1 h (t) Intensity process of 1−1τ >t Π τ >t(Π) (Π)
1 λ (t) Intensity process of N (t) := 1τ >t Π Π ∀i=j∈Π:τ =τ ≤t(Π) i j
∗ ∗1 λ (t) Intensity process of N (t) := 1τ >t ∀i=j∈Π,l∈/Π: τ =τ ≤t,τ =τ(Π) Π Π i j i l
F Dependence function of N over [t,T]T,t Π (Π)
ϕ Laplace transform of the incrementsT,t X
of a process X over [t,T]
ρ Linear correlation between N(T) and N (T) givenT,t ij i j
the information at t<T
L(t) The loss process
A(t) Compensator of the loss process
H (t) Process that counts jumps of the loss process L with size kk
⊥λ (t) Intensity process of Hkk
666