Pricing Portfolio Credit Derivatios by Means of Evolutionary Algorithms

Svenja Hager Pricing Portfolio Credit Derivatives by Means of Evolutionary Algorithms GABLER EDITION WISSENSCHAFT S...

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Svenja Hager Pricing Portfolio Credit Derivatives by Means of Evolutionary Algorithms

GABLER EDITION WISSENSCHAFT

Svenja Hager

Pricing Portfolio Credit Derivatives by Means of Evolutionary Algorithms With a foreword by Prof. Dr.-Ing. Rainer Schöbel

GABLER EDITION WISSENSCHAFT

Bibliographic information published by Die Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

Dissertation Universität Tübingen, 2007

1st Edition 2008 All rights reserved © Betriebswirtschaftlicher Verlag Dr. Th. Gabler | GWV Fachverlage GmbH, Wiesbaden 2008 Editorial Office: Frauke Schindler / Anita Wilke Gabler-Verlag is a company of Springer Science+Business Media. www.gabler.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: Regine Zimmer, Dipl.-Designerin, Frankfurt/Main Printed on acid-free paper Printed in Germany ISBN 978-3-8349-0915-2

Dedicated to my parents and my beloved husband.

Foreword Collateralized Debt Obligations (CDOs) are the most prominent example of portfoliorelated credit derivatives. They make it possible to diversify and transfer credit risk by pooling and redistributing the risks of an underlying portfolio of defaultable assets. It comes as no surprise that the dependence structure of portfolio assets is crucial for the valuation of CDO tranches. The standard market model is the Gaussian copula model, which uses only one parameter to summarize the correlations of default times in the underlying credit portfolio. Comparable with the volatility smile from option pricing, this simplification leads to an implied correlation smile when the model is confronted with market data. There is a growing interest in literature searching for solutions of this problem. Dr. Svenja Hager contributes to this literature by extending the Gaussian copula model, allowing for a heterogeneous specification of the dependence structure of the underlying portfolio. She shows that heterogeneous correlation matrices are able to explain the correlation smile. Based on this discovery, she develops a method to find the implied correlation matrix which optimally reproduces the observed tranche spreads of a CDO structure. To overcome the complexity of the resulting optimization problems, Evolutionary Algorithms are applied successfully. This monograph puts a new complexion on the standard market model and should therefore be recognized for its substantial contribution in this fascinating field of research on credit derivatives.

Rainer Schöbel

Acknowledgements First and foremost, I would like to express gratitude and appreciation for my advisor Prof. Dr.-Ing. Rainer Schöbel. Prof. Schöbel gave me valuable academic advice and support during my three years at the University of Tübingen. I am thankful that he supervised my dissertation and that he provided me the freedom to do research in the exciting area of credit risk modeling. I would also like to thank the second referee of this thesis, Prof. Dr. Joachim Grammig, for his comments and suggestions in our internal seminars. Next, I record my gratitude to Dr. Axel Hager-Fingerle. His enthusiasm for science has always encouraged me to investigate new areas of research. Numerous discussions with him significantly improved the ideas expressed in this dissertation. I also thank Dr. Markus Bouziane for his companionship and for providing plenty of useful information, and Vera Klöckner for her organizational assistance. I am much obliged to Ute Hager for her literature search and to Clemens Dangelmayr for his computational assistance. I wish to acknowledge financial assistance of the Deutsche Forschungsgemeinschaft by funding my research at the University of Tübingen, and of the Stiftung Landesbank Baden-Württemberg by supporting the publication of this dissertation. Moreover, I thank JPMorgan Chase for providing the data sets used in this study. I am deeply grateful to my parents for being there for me every single day of my life. All the confidence and joy that I possess is born of your love and belief in me. I am wholeheartedly thankful for having a wonderful sister who has always been my long-distance confidante. Above all, I want to thank my beloved husband. I am grateful for your caring love and friendship. You make my life complete.

Svenja Hager

Table of Contents List of Tables

xvii

List of Figures

xix

List of Notations

xxiii

1 Introduction

1

2 Collateralized Debt Obligations: Structure and Valuation

7

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Credit Risk Transfer Instruments . . . . . . . . . . . . . . . . . . . . . .

9

2.2.1

Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.2

CDS Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.3

Collateralized Debt Obligations . . . . . . . . . . . . . . . . . . .

10

2.2.3.1

Arbitrage and Balance Sheet CDOs . . . . . . . . . . . .

11

2.2.3.2

Cash Flow and Market Value CDOs . . . . . . . . . . .

12

2.2.3.3

Static Structures and Managed Structures . . . . . . . .

13

2.2.3.4

Cash Structures and Synthetic Structures . . . . . . . .

13

2.2.3.5

Single-Tranche Deals . . . . . . . . . . . . . . . . . . . .

15

2.2.3.6

Effect of Correlation . . . . . . . . . . . . . . . . . . . .

16

CDS Index Tranches . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2.4 2.3

7

Credit Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.3.1

18

Single-Name Credit Risk: Intensity-Based Models . . . . . . . . . 2.3.1.1

Stopping Times and the Hazard Rate Function . . . . .

19

2.3.1.2

Homogeneous Poisson Processes . . . . . . . . . . . . . .

22

2.3.1.3

Inhomogeneous Poisson Processes . . . . . . . . . . . . .

23

2.3.1.4

Cox Processes . . . . . . . . . . . . . . . . . . . . . . . .

24

2.3.2

Multi-Name Credit Risk: Copula Models . . . . . . . . . . . . . .

25

2.3.3

Valuation of Synthetic CDOs . . . . . . . . . . . . . . . . . . . .

29

Table of Contents

xii

2.3.3.1

Joint Distribution of Default Times in the Gaussian Copula Approach . . . . . . . . . . . . . . . . . . . . . . . .

2.4

30

2.3.3.2

Joint Distribution of Default Times in the Gaussian OneFactor Copula Approach . . . . . . . . . . . . . . . . . .

32

2.3.3.3

Pricing the Default Leg and the Margin Leg of a CDO .

33

2.3.3.3.1

The Default Leg . . . . . . . . . . . . . . . . .

33

2.3.3.3.2

The Margin Leg

34

. . . . . . . . . . . . . . . . .

2.3.3.4

Distribution of the Portfolio Loss in the One-Factor Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.3.3.5

Monte-Carlo Simulation of CDO Tranche Spreads . . . .

35

Valuation of CDOs: Literature . . . . . . . . . . . . . . . . . . . . . . . .

36

3 Explaining the Implied Correlation Smile

41

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2

Sensitivity of the Tranche Price to the Level of Correlation . . . . . . . .

42

3.3

The Implied Tranche Correlation . . . . . . . . . . . . . . . . . . . . . .

44

3.4

The Implied Correlation Smile . . . . . . . . . . . . . . . . . . . . . . . .

45

3.5

The Implied Base Correlation . . . . . . . . . . . . . . . . . . . . . . . .

47

3.6

Evolution of the Implied Correlation Smile . . . . . . . . . . . . . . . . .

49

3.7

Modeling the Correlation Smile: Literature . . . . . . . . . . . . . . . . .

56

3.8

Heterogeneous Dependence Structures

58

3.8.1

. . . . . . . . . . . . . . . . . . .

Heterogeneous Dependence Structures Can Cause Implied Correlation Smiles

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.8.1.1

The Existence Problem . . . . . . . . . . . . . . . . . .

60

3.8.1.2

The Uniqueness Problem . . . . . . . . . . . . . . . . . .

61

3.8.1.3

Exemplary Heterogeneous Matrices . . . . . . . . . . . .

62

3.8.2

Different Dependence Structures Can Lead to Identical Implied

3.8.3

Heterogeneous Dependence Structures Do Not Necessarily Lead to

Tranche Correlations . . . . . . . . . . . . . . . . . . . . . . . . . Implied Correlation Smiles . . . . . . . . . . . . . . . . . . . . . . 3.8.4 3.9

63 64

Heterogeneous Dependence Structures Allow for Flexible Portfolio Loss Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4 Optimization by Means of Evolutionary Algorithms

73

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.2

Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Table of Contents

xiii

4.3

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.4

Evolutionary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.4.1

4.4.2

4.4.3

4.5

4.6

4.7

Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.4.1.1

Elite Selection . . . . . . . . . . . . . . . . . . . . . . .

80

4.4.1.2

Tournament Selection . . . . . . . . . . . . . . . . . . .

80

4.4.1.3

Proportional Selection . . . . . . . . . . . . . . . . . . .

80

Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.4.2.1

Flat Crossover . . . . . . . . . . . . . . . . . . . . . . .

81

4.4.2.2

Discrete N-Point Crossover . . . . . . . . . . . . . . . .

81

4.4.2.3

Discrete Uniform Crossover . . . . . . . . . . . . . . . .

81

4.4.2.4

Intermediate Crossover . . . . . . . . . . . . . . . . . . .

81

4.4.2.5

Arithmetical Crossover . . . . . . . . . . . . . . . . . . .

81

Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.4.3.1

Standard Mutation . . . . . . . . . . . . . . . . . . . . .

82

4.4.3.2

Global Mutation Without Strategy Parameter . . . . . .

82

4.4.3.3

Global Mutation With Strategy Parameter . . . . . . . .

82

4.4.3.4

Local Mutation . . . . . . . . . . . . . . . . . . . . . . .

82

4.4.3.5

1/5-Rule . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Basic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.5.1

83

Evolution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1.1

Hill-Climber . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.5.1.2

The (μ, λ)-Strategy and the (μ + λ)-Strategy . . . . . .

84

4.5.2

Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.5.3

Monte-Carlo Search . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Parallel Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.6.1

Global Population Models . . . . . . . . . . . . . . . . . . . . . .

87

4.6.2

Local Population Model . . . . . . . . . . . . . . . . . . . . . . .

87

4.6.3

Regional Population Models . . . . . . . . . . . . . . . . . . . . .

87

Evolutionary Algorithms in Finance: Literature . . . . . . . . . . . . . .

88

5 Evolutionary Algorithms in Finance: Deriving the Dependence Structure 5.1

91 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.2

The Implied Correlation Structure . . . . . . . . . . . . . . . . . . . . . .

92

5.3

The Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.4

Description of the Genotypes

94

. . . . . . . . . . . . . . . . . . . . . . . .

xiv

Table of Contents

5.4.1

5.4.2

5.4.3

5.5

The Cholesky Approach . . . . . . . . . . . . . . . . . . . . . . .

95

5.4.1.1

Initialization . . . . . . . . . . . . . . . . . . . . . . . .

96

5.4.1.2

Repair Mechanism . . . . . . . . . . . . . . . . . . . . .

96

5.4.1.3

Evaluation and Program Termination . . . . . . . . . . .

97

5.4.1.4

Recombination . . . . . . . . . . . . . . . . . . . . . . .

98

5.4.1.5

Mutation . . . . . . . . . . . . . . . . . . . . . . . . . .

99

The One-Factor Approach . . . . . . . . . . . . . . . . . . . . . . 100 5.4.2.1

Initialization . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.2.2

Repair Mechanism . . . . . . . . . . . . . . . . . . . . . 100

The Cluster Approach . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4.3.1

Initialization . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.3.2

Repair Mechanism . . . . . . . . . . . . . . . . . . . . . 101

A Systematic Approach to Describe the Dependence Structure . . . . . . 101 5.5.1

The Nearest Neighbor Algorithm . . . . . . . . . . . . . . . . . . 103 5.5.1.1

5.6

General Scheme of the Nearest Neighbor Algorithm . . . 104

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Experimental Results

109

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2

Solution Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.2.1

Using the Expected Tranche Loss as Proxy for the Tranche Spread 110

6.2.2

Equivalent Information Content: Density of the Portfolio Loss and Expected Tranche Loss . . . . . . . . . . . . . . . . . . . . . . . . 112 6.2.2.1

Deriving the Expected Tranche Loss from the Density of

6.2.2.2

Deriving the Density of the Portfolio Loss from the Ex-

the Portfolio Loss . . . . . . . . . . . . . . . . . . . . . . 113 pected Tranche Loss . . . . . . . . . . . . . . . . . . . . 113 6.3

6.4

Performance Comparison: Basic Strategies . . . . . . . . . . . . . . . . . 113 6.3.1

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Performance Comparison: More Advanced Algorithms . . . . . . . . . . . 120 6.4.1

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.4.2.1

Monte-Carlo Search vs. (1+1)-ES . . . . . . . . . . . . . 125

6.4.2.2

(1+1)-ES vs. Multistart (1+1)-ES . . . . . . . . . . . . 125

6.4.2.3

(4,20)-ES vs. GA(40) . . . . . . . . . . . . . . . . . . . 126

xv

Table of Contents

6.4.2.3.1

Cholesky Approach . . . . . . . . . . . . . . . . 127

6.4.2.3.2

One-Factor Approach

. . . . . . . . . . . . . . 134

6.5

Implementation of a Parallel System . . . . . . . . . . . . . . . . . . . . 135

6.6

Performance Comparison: Parallel Algorithms . . . . . . . . . . . . . . . 136 6.6.1

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.6.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.7

Deriving the Dependence Structure From Market Data . . . . . . . . . . 138

6.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7 Summary and Outlook

143

References

147

List of Tables 6.1

The parameters ω1e , ω2e , ω1g and ω2g in the Cholesky approach. . . . . . . . 118

6.2

The parameters ω1e , ω2e , ω1g and ω2g in the one-factor approach. . . . . . . 118

6.3

The parameters ω1e , ω2e , ω1g and ω2g in the cluster approach. . . . . . . . . 118

6.4

The parameters ω1e , ω2e , ω1g and ω2g in the Cholesky approach. . . . . . . . 123

6.5

The parameters ω1e , ω2e , ω1g and ω2g in the one-factor approach. . . . . . . 124

6.6

Speedup and efficiency obtained by parallel algorithms and parallel implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.7

Tranche spreads (in bps per year). . . . . . . . . . . . . . . . . . . . . . . 140

6.8

Implied tranche correlations. . . . . . . . . . . . . . . . . . . . . . . . . . 140

List of Figures 2.1

Structure of a CLN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

Structure of a CDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.3

CDO transaction with cash structure. . . . . . . . . . . . . . . . . . . . .

14

2.4

Structure of a partially funded synthetic CDO. . . . . . . . . . . . . . . .

14

2.5

Portfolio loss distribution for different levels of correlation. . . . . . . . .

17

3.1

Flat correlation structure. . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.2

Sensitivity of the tranche spread to the level of correlation. . . . . . . . .

45

3.3

The implied correlation smile on April, 4 2005. . . . . . . . . . . . . . . .

46

3.4

The implied base correlation skew on April, 4 2005. . . . . . . . . . . . .

48

3.5

CDX.NA.IG index spread. . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.6

CDX.NA.IG tranche spreads. . . . . . . . . . . . . . . . . . . . . . . . .

52

3.7

CDX.NA.IG implied tranche correlations. . . . . . . . . . . . . . . . . . .

53

3.8

Evolution of the CDX.NA.IG implied correlation smile. . . . . . . . . . .

54

3.9

Sensitivity of the tranche spread to the level of correlation for different index spreads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.10 The existence problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.11 The uniqueness problem. . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.12 Varying the intra sector correlation and the inter sector correlation influences the implied correlation smile. . . . . . . . . . . . . . . . . . . . . .

63

3.13 Correlation smiles produced by correlation matrices with 10 clusters of equal size. The intra sector correlation is 0.7, the inter sector correlation varies between 0.3 and 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.14 Correlation smiles produced by correlation matrices with 10 clusters of equal size. The inter sector correlation is 0.3, the intra sector correlation varies between 0.5 and 0.7. . . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.15 Varying the size and the number of sectors influences the correlation smile. The intra sector correlation is 0.7, the inter sector correlation is 0.3. . . .

67

List of Figures

xx

3.16 When there is only one sector of high correlation, the cluster size influences the correlation smile. The intra sector correlation is 0.8, the inter sector correlation is 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.17 Correlation matrices differing in shape but leading to identical implied tranche correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

3.18 The flat correlation matrix and the correlation matrix with 5 clusters of 20 assets each lead to identical tranche spreads and consequently to identical implied correlation parameters. These correlation matrices do not lead to correlation smiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.19 Portfolio loss distribution for heterogeneous correlation matrices. . . . . .

71

5.1

Different dependence structures can lead to identical tranche spreads. . . 102

5.2

Dependence structures can be ordered in a systematic way. . . . . . . . . 106

6.1

The expected tranche loss vs. the tranche premium, scaling factor k=0.04274. Note that we use a unique scaling factor for all tranches. . . . . . . . . . 112

6.2

Convergence behavior (mean value and quantiles): (1 + 1)-ES. . . . . . . 116

6.3

Convergence behavior (mean value and quantiles): (3, 12)-ES. . . . . . . 117

6.4

Convergence behavior (mean value): (1 + 1)-ES vs. (3, 12)-ES. . . . . . . 120

6.5

Dependence structures leading to the desired expected tranche losses. . . 121

6.6

Convergence behavior in the Cholesky approach (mean value and quantiles): (1 + 1)-ES and Monte-Carlo search. . . . . . . . . . . . . . . . . . 126

6.7

Convergence behavior in the Cholesky approach (mean value): (1 + 1)-ES

6.8

Convergence behavior in the one-factor approach (mean value and quan-

6.9

Convergence behavior in the one-factor approach (mean value): (1+1)-ES

vs. Monte-Carlo search. . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 tiles): (1 + 1)-ES and Monte-Carlo search. . . . . . . . . . . . . . . . . . 127 vs. Monte-Carlo search. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.10 Exemplary solution matrix in the Cholesky approach. . . . . . . . . . . . 128 6.11 Exemplary solution matrix in the one-factor approach. . . . . . . . . . . 128 6.12 Convergence behavior in the Cholesky approach (mean value and quantiles): Evolution Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.13 Convergence behavior in the Cholesky approach (mean value): Evolution Strategies in comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.14 Convergence behavior in the one-factor approach (mean value and quantiles): Evolution Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . 130

List of Figures

xxi

6.15 Convergence behavior in the one-factor approach (mean value): Evolution Strategies in comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.16 Convergence behavior in the Cholesky approach (mean value and quantiles): Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.17 Convergence behavior in the Cholesky approach (mean value): Genetic Algorithms in comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.18 Convergence behavior in the one-factor approach (mean value and quantiles): Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.19 Convergence behavior in the one-factor approach (mean value): Genetic Algorithms in comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.20 Cholesky approach: performance (mean value) based on evaluations. . . . 133 6.21 Cholesky approach: performance (mean value) based on generations.

. . 133

6.22 One-factor approach: performance (mean value) based on evaluations. . . 135 6.23 One-factor approach: performance (mean value) based on generations. . . 135 6.24 The implied correlation matrix. . . . . . . . . . . . . . . . . . . . . . . . 140 6.25 Fitting the implied correlation smile. . . . . . . . . . . . . . . . . . . . . 141

List of Notations Single-Name Credit Risk F (t) F (t, T ) f (t) f (t, T ) G (Gt )t≥0 h(t) h(t, T )

probability distribution function conditional probability distribution function probability density function conditional probability density function background information background filtration hazard rate function conditional hazard rate function

N(t)

counting process

S(t)

survival function

S(t, T ) λ(t) τ (Ω, (Ft )(t≥0) , P )

conditional survival function intensity of the counting process N(t) default time filtered probability space, the filtration Ft describes the information structure, P describes the risk-neutral probability measure

xxiv

List of Notations

Multi-Name Credit Risk C(u1 , . . . , un ) c(u1 , . . . , un ) CΣGa cGa Σ D(t) Ej F (x) F (x) F(x) ˆ F X (x) ˆ X F (x) Ij (t) L(t) Rj sA,B t1 , . . . , tI tp(τj ) U X1 , ..., Xn X 1, . . . , X n ˆ X Σ Φ ΦnΣ ωA,B (L)

n-dimensional copula function copula density Gaussian copula density of the Gaussian copula discount factor for maturity t exposure of asset j probability distribution function joint distribution function of x = (x1 , . . . , xn ) vector of marginal probabilities (F1 (x1 ), . . . , Fn (xn )) ˆ probability distribution function conditional on X ˆ joint distribution function conditional on X default indicator process of asset j cumulative portfolio loss at time t fractional loss given default of asset j spread of tranche (A, B) regular payment dates last regular payment date before default τj random variable with uniform distribution on [0, 1] standard normally distributed random variables independent standard normally distributed idiosyncratic risk factors standard normally distributed common risk factor correlation matrix univariate standard normal distribution function multivariate standard normal distribution function cumulative default of tranche (A, B)

List of Notations

xxv

Evolutionary Algorithms GA(μ) I m P (t) = {p1 (t), ..., pμ (t)} p = (x1 , ..., xn , σ1 , ..., σn ) r s Z ι λ μ (μ, λ)-ES (μ + λ)-ES Φ Ψ Ω (1 + 1)-ES

Genetic Algorithm with population size μ search space containing all individuals mutation operator population at time t individual having one strategy variable σi for each object variable xi recombination operator selection operator standard normally distributed random variable termination criterion number of offspring individuals number of parent individuals Evolution Strategy selecting the μ best individuals out of λ offspring individuals to form the next parent generation Evolution Strategy selecting the μ best individuals out of the union of μ parents and λ offspring to form the next parent generation objective function assigning a certain fitness value to the individuals generation transition function set of probabilistic genetic operators Hill-Climber

xxvi

List of Notations

Implied Correlation Matrix C f1 (Σ), f2 (Σ), f3 (Σ) se , sm , ss se (Σ), sm (Σ), ss (Σ) z(t) ρ Σ σe (Σ), σm (Σ), σs (Σ)

Cholesky decomposition functions indicating the quality of Σ target spreads tranche spreads as a function of Σ objective function indicating the quality of the best individual found so far correlation parameter correlation matrix implied tranche correlations as a function of Σ

List of Notations

xxvii

Abbreviations ABS bps CBO CDO CDO2 CDS cf. CLN CLO DJ CDX.NA.IG 5yr DJ iTraxx EUR 5yr EA e.g. ES GA i.e. ISDA MBS SF CDO SPV STCDO

asset-backed security basis points collateralized bond obligation collateralized debt obligation CDO squared, CDO of CDOs credit default swap confer to credit linked note collateralized loan obligation CDS index comprising a portfolio of 125 investment grade US companies, maturity 5 years CDS index comprising a portfolio of 125 investment grade European companies, maturity 5 years Evolutionary Algorithm for example Evolution Strategy Genetic Algorithm that is International Swaps and Derivatives Association mortgage-backed security structured finance CDO special purpose vehicle single-tranche CDO

Chapter 1 Introduction In recent years, the credit derivatives market has become extremely active. Especially credit default swaps (CDSs) and collateralized debt obligations (CDOs) have contributed to what has been an amazing development. The most important benefit of credit derivatives is their ability to transfer the credit risk of an arbitrary number of obligors in a simple, efficient, and standardized way, giving rise to a liquid market for credit risk that can be easily accessed by many market participants. Credit risk management, the traditional area of application for credit derivatives, has gained additional importance due to the high number of downgrades and credit events over recent years. For financial service institutions, such as banks, the exchange of credit risk is an interesting means of risk management, as long as it allows for maintenance of the client relationship. Eliminating the credit risk of a client by simply selling the risky item is generally not an option. By the use of credit derivatives, the credit risk can be removed from the balance sheet without having to sell the risky asset. This general tendency in the credit derivatives market was intensified by a number of economic factors including low interest rates, which focused the attention of market participants on high-yield sectors. Compared to traditional investment products, credit derivatives provide relatively high yields. Consequently, they are not only used to manage credit risk, they are also established as investment instruments. Another contributing factor to the growth in the credit derivatives market is the high regulatory demand on the credit risk control side. Major groups participating in the market for credit derivatives are banks, insurance and reinsurance companies, hedge funds, investment funds, and industrials. These market participants use credit derivatives for various purposes. Banks generally buy credit protection. They use credit derivatives to manage their regulatory capital requirements

2

Chapter 1. Introduction

(regulatory arbitrage) and to exploit funding opportunities by the securitization of loan portfolios (funding arbitrage). In addition, banks use credit derivatives for trading reasons. Insurance companies, reinsurance companies, and investment funds are essential providers of credit protection, buying credit risk in order to obtain a competitive yield. They also use credit derivatives for credit risk management of their bond portfolios. Hedge funds invest in credit derivatives so as to gain on relative value trades between different markets, e.g. the bond market and the CDS market. They are also attracted by the high levels of leverage that can be achieved in many credit derivative transactions. The market for single-name CDSs has become rather liquid with entire term structures of CDS margins available for an increasing number of reference assets. As a result, CDSs are considered to be “plain vanilla” instruments, while CDS premia are used to calibrate pricing models for more complex derivative products. Even more than in other markets, it is important to fully understand the basic pricing model, since credit markets are incomplete. Since 2003, the credit risk of standardized portfolios has been summarized in CDS indices. On these index portfolios CDO tranches are now actively quoted. Thus, the portfolio loss distribution and the asset correlation as seen by market participants have become an observable variable in as much as quoted tranche premia can be interpreted as aggregating the market’s view of dependence among the underlying assets. According to e.g. Burtschell et al. (2005a), Duffie (2004), Finger (2004), Friend and Rogge (2004), Hull and White (2004), and Schönbucher (2003) the market has converged to a standard model for the pricing of index tranches. A one-factor Gaussian copula approach, essentially along the lines of Vasicek (1987) and Li (2000), is considered to be the benchmark in a manner reminiscent of the Black and Scholes (1973) model for option pricing. The standard market model assumes in its simplest version that the copula correlations between each pair of assets are the same. This assumption is frequently used in the market according to Burtschell et al. (2005a), Duffie (2004), Finger (2004), Hull and White (2004), and McGinty et al. (2004). We denote such a correlation structure as homogeneous or flat (see Burtschell et al. (2005b)). Using this approach, an implied correlation parameter which is not directly observed in the market can be derived from quoted CDO tranche prices, similar to the implied volatility parameter in option markets. CDO tranches are often quoted in terms of this correlation parameter. However, different correlation levels are needed to reproduce the market prices of different tranches on the same underlying portfolio (see Andersen and Sidenius (2004) and Gregory and Laurent (2004)). This phenomenon illustrates that the standard market model is not able to consistently reproduce market prices. Even more than in the case


3

of the Black and Scholes model, the shortcomings of the one-factor Gaussian copula approach are acknowledged by practitioners and academics alike. Trading correlation was originally considered as simply trading a new asset class. But still, market participants are far from fully understanding the correlation concept and the measurement issues it entails. Aside from liquidity and transparency, the variety of derivative products is increasing. Products range from single-name credit default swaps and basket credit default swaps to more sophisticated products like derivatives on credit derivatives. Examples of exotic multi-name credit derivatives are so-called bespoke single-tranche CDOs (STCDOs), forward starting STCDOs, STCDOs with embedded options, outright options on STCDOs or on indices, and CDOs of CDOs (CDO2 ) as mentioned in Schönbucher (2006). A comparable evolution towards exotic products could formerly be observed in option markets. Although substantial progress in the pricing of portfolio credit derivatives is being made, pricing capacities are still not where they were during the emergence of exotic options. With the recent development of non-standard credit derivatives, it has become increasingly important to develop pricing models for these illiquid products that are consistent with the pricing models and the market quotes of related liquid instruments. In our study, we aim at pricing non-standard illiquid portfolio credit derivatives which are related to standard CDO tranches in as much as they have the same underlying portfolio of obligors. Now standard CDO tranches are actively traded in the market. To ensure consistent pricing, we have to calibrate our pricing model to the observed market prices of these standard CDO tranches. We have to make sure that the calibrated model comprises all market information of relevance to the pricing of the particular illiquid instrument. Then we use the calibrated model to determine the price of the non-standard contract. As mentioned above, the standard market approach is generally not able to reproduce all quoted tranche spreads of a CDO structure simultaneously. The inability of the standard market model to fit market prices led to the development of numerous extended and generalized pricing methods. Examples are Gregory and Laurent (2004), Andersen and Sidenius (2004), Hull and White (2004), Kalemanova et al. (2005), Joshi and Stacey (2005), Moosbrucker (2006), Baxter (2006), Hull et al. (2005), Willemann (2005b), Hull and White (2006), Tavares et al. (2004), Mashal et al. (2004), van der Voort (2005), and Schönbucher (2006). We discuss these models in Section 3.7. In this study we hold on to the Gaussian copula approach. But instead of assuming a flat correlation structure as in the standard market model, we focus on the use of heterogeneous correlation structures. Even if a flat correlation structure is appealing because of its intuitive simplicity,

4


we allow for non-flat structures because actual default time correlations are generally characterized by high levels of complexity. We aim at finding a correlation matrix sufficiently flexible for all tranche spreads of a CDO structure to be reproduced simultaneously, when we use this correlation matrix in conjunction with the Gaussian copula approach. Using the appropriate correlation structure allows for consistent pricing, because only one correlation structure is used for the pricing of all CDO tranches - which agrees with the fact that the underlying portfolio is the same. The price paid for a better fit is the increased complexity of model calibration. As there is no standard optimization technique to derive the correlation structure from market prices, we use Evolutionary Algorithms. Evolutionary Algorithms (EAs) have been studied for almost 40 years in the area of computer science. However, their application to economics in general and to finance in particular has a much shorter history. The idea of EAs was applied to game theory in the late 1980s by Axelrod (1987) and shortly after to agent-based economic modeling by Arifovic (1995) and Birchenhall (1995). Applications of EAs were further extended to econometrics (Bearse et al. (1997)) and to financial engineering (Alemdar and Ozyildirim (1998) and Bullard and Duffy (1998)). EAs also ventured into an especially active research area in financial engineering: option pricing. Compared to the purely analytical option pricing model of Black and Scholes (1973), EAs allow for a data-driven approach. Early studies are Jay White (1998), Chidambaran et al. (1998), and Keber (1999). Keber has applied EAs to the implied volatility problem in option markets. Using EAs, accurate approximations for the implied volatility of American put options can be obtained. A recently published study that addresses this topic is Hamida and Cont (2005). However, to the best of our knowledge there are no publications connecting the area of EAs to the pricing of portfolio credit derivatives. Therefore, our contribution to the current academic debate about CDO pricing is twofold. First, we illustrate in detail that heterogeneous correlation structures are able to explain a phenomenon that is observed in the market: the implied correlation smile. Second, we use EAs in the area of portfolio credit derivatives. Using this optimization approach, we derive a suitable correlation structure which reproduces quoted market spreads.


5

The dissertation is structured as follows: The first chapter is an introduction outlining the scope of the thesis. The second chapter introduces credit derivatives. First, we consider CDSs as exemplifying the concept of single-obligor credit derivatives. We explain the intensity-based model, which is a well-established approach to the pricing of single-name credit risk. Then we discuss multi-obligor credit derivatives, especially the structure and valuation of CDOs. We explain the Gaussian copula model that can be used to price portfolio credit derivatives. In the third chapter, we describe a phenomenon that can be observed in the market for CDOs: the implied correlation smile. First, we present possible explanations for the correlation smile that are discussed in the literature. Then we outline an additional explanation for the appearance of the smile. We illustrate how a heterogeneous specification of dependence can generate a correlation smile. Thus, we conclude that modeling heterogeneous dependence structures may have the potential to reproduce market prices. We aim at deriving a correlation matrix that reproduces all observed tranche spreads of a CDO structure. Thus, we have to solve an optimization problem. The fourth chapter addresses optimization methods, in particular EAs. We introduce the basic idea of EAs, with several evolutionary operators and algorithms being presented. In the fifth chapter, we outline systematically how to derive a correlation matrix from observed tranche spreads using EAs. We explain why EAs are appropriate for this kind of optimization problem. In the sixth chapter, we compare the performance of different serial variants of EAs in light of our optimization problem. Furthermore, we analyze the performance of parallel EAs and parallel implementations. We show that parallel algorithms and implementations can significantly accelerate the optimization process. Finally, we analyze both the potential and the tractability of our approach on the basis of market data. We show that we obtain a very good fit to real-world prices. The last chapter concludes the thesis with a summary of our results.

Chapter 2 Collateralized Debt Obligations: Structure and Valuation 2.1

Introduction

Derivatives are financial instruments that transfer risk from one party to another. They can be classified according to the kind of risk transmitted. For instance, we distinguish between interest rate derivatives, foreign exchange derivatives, commodity derivatives, equity derivatives, and credit derivatives. In the following, we focus on credit derivatives which transfer the credit risk exposure between two parties. By means of credit derivatives it is possible to acquire or reduce credit risk exposure. Credit risk includes default risk, downgrade risk, and credit spread risk (see Lucas et al. (2006)). In this study we concentrate on default risk, which is the risk that the issuer of a bond or the debtor of a loan will not repay the outstanding debt in full. The default can be either complete, when no amount of the bond or the loan is repaid, or partial, when only a part of the original debt is recovered. Unlike the other forms of derivatives, where there are both exchange traded and overthe-counter contracts, credit derivatives are merely over-the-counter products. That is, they are individually negotiated financial contracts. However, there are virtual meeting places where buyers and sellers of credit protection can get together. There are two electronic exchange platforms (Creditex and CreditTrade) for the trading of credit default swaps and brokers that are specialized in the trading of credit risk (e.g. GFInet). Furthermore, large investment banks regularly quote the prices of credit risk on their internet pages and on Bloomberg. See Schönbucher (2003) for further details about the structure of the credit derivative market.

8

Chapter 2. Collateralized Debt Obligations: Structure and Valuation

Since credit derivatives are over-the-counter trades, transparency is an issue. Market transparency has improved since the above-mentioned firms that are specialized in credit derivatives are providing internet trading platforms. Nowadays, documentation of most credit derivative transactions is based on documents and definitions provided by the International Swaps and Derivatives Association (ISDA)1 . This documentation is recognized as the most flexible framework for over-the-counter derivatives and is therefore the most widely used (see Franzen (2000)). It also contributed to an enhanced level of transparency in the market (see Bielecki and Rutkowski (2002)). Credit derivatives are grouped into funded and unfunded instruments (see Fabozzi et al. (2004)). In a funded credit derivative, typified by a credit-linked note (CLN, see Figure 2.1), the protection seller makes an advance payment to the protection buyer when buying the note. If a credit event occurs in the underlying2 , this payment will be used to compensate the protection buyer for the default incurred. In return, the protection seller pays a regular premium to the protection buyer and refunds the remaining part of the principal on termination of the contract. In an unfunded credit derivative, typified by a credit default swap (CDS, see Figure 2.2), the protection seller does not make an advance payment to the protection buyer. In this case, the payment is made on termination of the contract, if there is a credit event. See the monthly report of Deutsche Bundesbank (2004a and 2004b) for details of the structure of CLNs and CDSs.

Funds Protection Buyer Reference Asset

Protection Seller Principal & Interest According to the Performance of the Reference Asset

Figure 2.1 Structure of a CLN. Premium Protection Buyer Reference Asset

Protection Seller Compensation in Case of Default

Figure 2.2 Structure of a CDS.

1 A current version of the documentation can be downloaded from the ISDA’s website at http..//www. isda.org. 2 In the following we denote the underlying portfolio as the underlying.

2.2. Credit Risk Transfer Instruments

9

In this chapter we will provide further details concerning CDSs. Then we will turn our attention to the structure and the valuation of collateralized debt obligations (CDOs). We will discuss the most typical kinds of CDOs existing today, and we will explain in detail the Gaussian copula model which is considered the market standard for valuation of CDOs.

2.2 2.2.1

Credit Risk Transfer Instruments Credit Default Swaps

A CDS may be regarded as an insurance against single-name credit risk. The buyer of protection regularly pays a fixed fee or premium to the seller of protection within the term of the swap as long as no credit event occurs. Should a prespecified credit event occur at some point prior to the contract’s maturity, the seller of protection owes a payment to the buyer of protection, thereby insulating the buyer from a financial loss. In the market for credit derivatives, a CDS is by far the most popular and quantitatively the most significant form of credit derivative (see Blanco et al. (2005)). Note that the CDS market is more liquid than the corporate bond market (see Amato and Gyntelberg (2005)). One reason for the higher liquidity is that it is more standardized. Another reason is, that market participants can “go long credit risk” without a cash payment, and they can “go short credit risk” with less difficulty and at lower cost than with corporate bonds.

2.2.2

CDS Indices

While an investor may enter into a CDS transaction to get exposure to the credit risk of one obligor, trading based on indices of CDSs has become more popular in recent years (see Hull and White (2004)). CDS index contracts cover the default risk of the names in an index. As mentioned in Amato and Gyntelberg (2005), the market liquidity of the index contracts is enhanced by different aspects: First of all, the main traded CDS indices, Trac-x and iBoxx, were consolidated in April 2004 into a single index family under the names Dow Jones CDX (for North America and emerging markets) and Dow Jones iTraxx (for Europe and Asia). The Dow Jones CDX and the Dow Jones iTraxx comprise the 125 most actively traded single name CDS contracts in the respective markets. Equal weighting is given to each name. The composition of a given index series remains static until its maturity, except for the defaulted entities which are eliminated from the index. However, the index is scheduled

10


to roll over every six months in order to rebalance the portfolio if necessary. Thus, a new rebalanced index series is launched and associated securities are issued. A second reason for the liquidity of index contracts is the clear geographic focus, the stable composition, and the standardized maturities. There are indices for particular sectors. For instance, the Dow Jones CDX.NA.IG 5yr index gives the average fiveyear credit default swap margin for a portfolio of 125 investment grade US companies. Similarly, the Dow Jones iTraxx EUR 5yr index gives the average five-year credit default swap margin for a portfolio of 125 investment grade European companies. Finally, as mentioned above, there are two different contract formats available to meet the investors’ needs in respect of funding and counterparty risk. An unfunded contract is simply a multi-name CDS. A funded contract can be considered as a bond, where the protection buyer pays a regular premium to the protection seller. In return, the protection buyer receives a pool of collateral securities from the protection seller. If a default occurs in the underlying of the index, the protection buyer sells a corresponding amount of the collateral securities to recover the loss incurred. In an unfunded contract, the protection buyer is exposed to counterparty risk; in a funded contract, the protection buyer is not exposed to counterparty risk but to the risk of credit deterioration in the collateral securities. See Jarrow and Yu (2001) and Chen and Filipović (2004) for the valuation of credit derivatives in the presence of counterparty default risk.

2.2.3

Collateralized Debt Obligations

The fundamental idea behind a CDO is to take a pool of defaultable assets and issue securities, whose cash flows are backed by the payments due on the assets. In a typical CDO, the underlying portfolio is transferred to a special purpose vehicle (SPV), which then issues several tranches of notes. Using a rule for prioritizing the cash flow payments of the underlying assets to the issued tranches, it is possible to redistribute the credit risk of the underlying portfolio. Thus, securities with a variety of risk-return profiles are created. Among others, see Lucas et al. (2006), Schönbucher (2003), and Bluhm et al. (2003) for a detailed overview of the structure and the analysis of CDOs. These publications are the sources for our illustration in Section 2.2.3. We denote the underlying portfolio as reference portfolio or collateral. Note that the same term has been used in Section 2.2.2 in the context of funded credit derivatives for the portfolio of securities used to compensate the protection buyer for defaults in the reference portfolio. We suppose that the actual meaning of the term ‘collateral’ can be derived from the context.


11

There are no predetermined rules as to how many tranches a CDO may contain. Exemplarily, we consider three tranches unless mentioned otherwise: one equity, one mezzanine, one senior tranche. The CDO notes have different levels of seniority, in the sense that the senior tranche has coupon and principal payment priority over the mezzanine tranche, while the mezzanine tranche has coupon and principal payment priority over the equity tranche. Consequently, the income from the collateral assets is paid first to the senior tranche. After the claims of the senior tranche are met, the remaining proceeds flow to the mezzanine tranche. After the mezzanine tranche claims are satisfied, the residual income is paid on the notes of the equity tranche, if any revenues are left. We can also consider the payoff of the different tranches as a function of the portfolio loss. Thus, potential losses in the underlying portfolio are first absorbed by the equity tranche. If losses exceed the value of the equity tranche, they affect the mezzanine tranche. If losses in the underlying portfolio exceed the value of the equity and the mezzanine tranche, they are absorbed by the senior tranche. Therefore, subordinated tranches protect more senior tranches against credit losses. In return, they receive a higher coupon for taking on greater credit risk. While the equity tranche is the most subordinated tranche, it is also the note that possibly pays the highest margin. However, since the equity return is linked to the performance of the underlying portfolio, it has no ex ante upper or lower limit. The senior tranche and the mezzanine tranche are rated tranches but the equity tranche is not rated. Prior to issuance, the originator chooses the size of the CDO tranches such that they achieve the desired ratings. Even if the assets in the reference portfolio are not investment grade, it is possible to create a structure where most tranches obtain high investment grade ratings by concentrating the default risk in a small first-loss layer. Since the structure of the CDO redistributes the risk of the underlying portfolio, assets that individually have a limited appeal to investors can be transformed into securities with a range of different risks matching the risk-return preferences of a larger investor base. A CDO that has a reference portfolio composed of bonds is called a CBO (collateralized bond obligation); likewise, if the collateral consists of loans, it is called a CLO (collateralized loan obligation). CDOs that are backed by asset-backed securities (ABS) or by mortgage-backed securities (MBS) are called structured finance CDOs (SF CDOs). A CDO squared (CDO2 ) is backed by classes of securities from other CDOs. 2.2.3.1

Arbitrage and Balance Sheet CDOs

We differentiate between arbitrage transactions and balance sheet transactions, based on the motivation of the originator.

12


Arbitrage deals are motivated by the opportunity to generate value by repackaging collateral into tranches. Financial theory suggests that all tranches of a CDO structure should together have the same market value as the underlying assets. But in practice this is often not the case. The CDO tranches can sometimes be sold for a cumulative price that exceeds the sum of the market values of the underlying assets. Therefore, a CDO represents a potential arbitrage opportunity. The originator, who often retains the equity tranche, hopes to earn the difference between the yield offered on the collateral assets and the payments made to the tranches. An arbitrage transaction is successful when it offers a competitive return for the equity tranche. Therefore, the purpose of an arbitrage deal is to exploit the difference in margins between risky sub-investment grade securities with a high yield and less risky investment grade securities. In a balance sheet transaction, the motivation of the originator is to remove debt instruments from the balance sheet. Often this is done in order to free up the regulatory or economic capital that a bank would otherwise be obliged to hold. 2.2.3.2

Cash Flow and Market Value CDOs

CDOs can have either cash flow or market value structures. This classification describes how the CDO tranches are protected from potential losses. In a cash flow transaction, the CDO tranches are sized so that interest and principal cash flows from the collateral assets are expected to cover the tranche requirements. The expectation is based on an assessment of the default probability, the recovery rate, and the default correlation between the assets. If the quality of the assets declines, the cash flows are diverted from subordinate notes to senior notes. The asset manager focuses on controlling defaults and recoveries and tries to generate cash flow for the senior and the mezzanine tranche without the active trading of bonds. There are no forced collateral liquidations. In a cash flow transaction, the proceeds to meet the CDO debt obligations come primarily from interest and principal payments from the collateral assets. Tranche ratings are based on collateral defaults and recoveries and on the timely receipt of interest and principal payments from the collateral. The asset manager must meet certain requirements in order to maintain the rating of the CDO. In market value deals, investors are exposed to the market value of the underlying collateral, that must be “marked to market” regularly. Certain market overcollateralization tests ensure that the outstanding debt can be covered. If the tests are not met, collateral sales and liability redemptions may be required. See Goodman and Fabozzi (2002) for details on market overcollateralization tests. Market value deals constitute a minority of CDOs, but they are very useful when the cash flow from the collateral is not predictable.


13

The asset manager is not constrained by a need to match the cash flows of the collateral assets to those of the debt tranches. However, the asset manager has to maintain and improve the market value of the collateral. The funds for interest payments are obtained from collateral interest receipts and from the liquidation process. The funds for principal payments are obtained by liquidating the collateral. The ratings of the CDO tranches are based on the price volatility, liquidity, and market value of the collateral, and on the latter’s diversity. 2.2.3.3

Static Structures and Managed Structures

Furthermore, we distinguish between static deals and managed deals. In static CDO structures, the underlying portfolio remains fixed throughout the life of the CDO. Market participants can make their investment decision in full knowledge of the composition of the underlying portfolio. Investors in static CDOs are primarily exposed to credit risk. In managed CDO structures, the asset manager actively manages the underlying portfolio by reinvesting cash flows from the CDO’s collateral and by buying and selling assets. At the time of the investment decision, market participants do not know the composition of the underlying portfolio. Moreover, the composition will change over time. Only the identity of the asset manager is known as well as certain investment guidelines. Consequently, investors in managed CDOs are exposed to credit risk and the risk of poor management. 2.2.3.4

Cash Structures and Synthetic Structures

Another important distinction is that between the above-mentioned cash structures and synthetic CDOs. See Goodman (2002) for an introduction to synthetic CDOs. Our presentation in Sections 2.2.3.4 and 2.2.3.5 partially follows Cousseran and Rahmouni (2005), Hyder (2002), and O’Kane (2001). In the case of a CDO with cash structure (cf. Figure 2.3), the originator transfers the credit risk of the underlying portfolio via the true sale of assets to a SPV. In return, the SPV issues the CDO securities, which are then sold to investors. Not all CDO tranches must be sold in the form of notes. Credit risk can also be transferred by means of credit derivatives. A synthetic CDO absorbs the economic risk but not the legal ownership of the reference portfolio. Since the underlying assets are not transferred, the client relationship is preserved, and cash flows on the underlying assets do not need to be managed.

14


The most common synthetic CDO type is a partially funded synthetic CDO (cf. Figure 2.4). In a partially funded synthetic CDO, the originator transfers one part of the credit risk by means of a super senior default swap. This credit default swap is referred to as super senior default swap because it corresponds to the part of the structure that is best protected against losses. The super senior default swap transaction represents the unfunded part of the structure. In exchange for protection, the originator pays a premium to the super senior counterparty. The remaining part of the credit risk is transferred to a SPV by means of a CDS. Likewise, the originator pays a premium to the SPV in exchange for protection. In turn, the SPV issues funded CDO tranches. The issuance amount corresponds to the part of the underlying nominal which is not protected by the super senior default swap. The CDO investor pays the nominal of the respective tranche at the beginning of the transaction. The proceeds from the sale of the CDO tranches are put into a collateral account and are invested in risk-free assets. The interest payments from these collateral assets and the CDS premia from the originator are used to pay the tranche margins. In the event of a credit default, a part of the risk free collateral assets is sold to compensate the originator. This sale results in a reduced repayment of the principal of the tranches. Fully funded synthetic CDOs and unfunded synthetic CDOs are also possible. The fully funded structures imply higher costs but they are less exposed to counterparty risk than are unfunded structures. In this study, we focus on the risk and the valuation of unfunded synthetic CDOs. Funds Protection Buyer Reference Portfolio

Funds Protection Seller Tranches

SPV True Sale of Assets

Principal & Interest

Figure 2.3 CDO transaction with cash structure. Credit Default Swap Protection Buyer Reference Portfolio

Credit Default Swap

Credit Linked Notes

SPV

Funds

Super Senior Swap Protection Seller Tranches

Principal & Interest

Collateral Risk-Free Assets

Figure 2.4 Structure of a partially funded synthetic CDO.


2.2.3.5

15

Single-Tranche Deals

Beyond synthetic CDOs there are single-tranche deals. According to Amato and Gyntelberg (2005), single-tranche CDOs constitute one of the main growth areas in the CDO market. A single-tranche CDO is a CDO where the arranger of the deal sells only one tranche from the capital structure of a synthetic CDO. Single-tranche CDOs have several advantages compared to traditional CDO structures. First, single-tranche deals can be designed for a specific investor’s wishes with regard to underlying names, subordination level, and tranche size. This reduces the risk of moral hazard or adverse selection in the choice of the underlying assets and it also reduces the risk of conflicting interests between the investors in different tranches. Second, singletranche CDOs are relatively easy to set up and selling costs are generally low. In traditional CDO deals the arranger does not assume a risk. The risk is completely transferred to the tranche investors by means of a SPV. In traditional CDO deals, the arranger is an investment bank or an asset management firm who is merely responsible for placing the CDO notes. In the case of arbitrage CDOs, the arranger may also act as the originator of the transaction or may even actively manage the underlying assets. However, in single-tranche deals there is no SPV. Thus, the arranger becomes the direct counterpart of the investor and is confronted with different aspects of risk. There are two approaches to carrying out single-tranche transactions (see Cousseran and Rahmouni (2005) for details). In the first approach, the arranger of the single-tranche deal sells the credit risk of only one tranche of the capital structure. The position can be hedged by selling protection in the CDS market or through transactions based on CDS indices. A delta hedging strategy is applied that offsets the arranger’s exposure to credit risk and market risk. Since the risk changes over time, dynamic hedging has to be applied. The amount of protection to be sold is determined by the delta, which describes the sensitivity of the tranche value to changes in the spreads of the underlying assets. A neutral position can be created by selling protection for delta times the tranche nominal. Generally, the tranche delta is higher than unity (often tranches are sold at the mezzanine level), so single-tranche deals are classified as leveraged products. In this setup, the arranger tries to earn a profit by buying protection on only one single CDO tranche and by reselling protection on each of the individual reference entities included in the underlying portfolio. In effect, the arranger buys protection in bulk form and sells it in smaller, individual pieces. Obtaining a satisfying dynamic hedge is a challenging task. Since the hedge positions have to be adjusted regularly, the arranger is exposed to liquidity risk. The arranger

16


is exposed to model risk as well, since the obtained values for the delta are model dependent. Additional complexity is added because the arranger has to hedge changes in credit spreads and defaults on a single-obligor basis and the arranger has to consider the correlation structure on a multi-obligor basis. In the second approach, the arranger purchases the underlying credit portfolio. Here the arranger has to keep the unsold tranches and is left with residual credit risk, which has to be managed by means of dynamic delta hedging.

2.2.3.6

Effect of Correlation

For structured credit derivatives that tranche the credit risk of the collateral assets into products of different seniority, it is essential to know the loss distribution of the underlying portfolio. The shape of the loss distribution is especially affected by the correlation structure of the collateral. Assume that all pairwise default time correlations in the portfolio are equal. Then the level of correlation can produce opposite effects on the risk inherent in different CDO tranches. Belsham et al. (2005), Amato and Gyntelberg (2005), and O’Kane (2001) address the effect of correlation on the risk content of CDO tranches. Senior tranches tend to benefit from low correlation among the assets in the underlying portfolio, since strong diversification makes extreme outcomes less likely. Note that senior tranches suffer only if the extreme outcome of very high losses occurs. If we increase the default correlation, the assets in the collateral pool become more likely to default together and larger losses become more probable. Conversely, junior tranches tend to benefit from high correlation. A subordinate tranche may survive only if the extreme outcome of very low losses occurs, which happens more often when correlation is high. Thus, the senior tranche favors low correlation and the equity tranche favors high correlation. While the equity and the senior tranche are very sensitive to correlation, the mezzanine tranche is relatively insensitive in this regard. Additionally, the mezzanine tranche is not necessarily monotone in correlation. The following Figure 2.5 illustrates how the shape of the portfolio loss distribution depends on default correlation. We consider a homogeneous collateral pool of 125 defaultable assets with equal nominal value. We assume that all assets have a default probability of 5% and a recovery rate of 0%, and we assume that their time to maturity is 5 years. As mentioned above, all pairwise default time correlations are supposed to be the same. We consider the portfolio loss distribution for different levels of correlation.


17 Correlation Parameter 20 % 25

20

20 Probability (%)

Probability (%)

Correlation Parameter 0% 25

15 10 5 0

15 10 5

0

20

40 60 80 100 Number of Defaults

0

120

0

100

80

80

60 40 20 0


120

Correlation Parameter 100%

100

Probability (%)

Probability (%)

Correlation Parameter 80%

20

60 40 20

0

20


120

0

0

20


120

Figure 2.5 Portfolio loss distribution for different levels of correlation.

2.2.4

CDS Index Tranches

By the creation of CDS index tranches a new liquid instrument emerged that allows for the trading of credit risk correlations. A CDS index tranche can be compared to a synthetic CDO tranche based on a CDS index. Especially the rules for determining payoffs ensure that an index tranche is economically equivalent to the corresponding tranche of a synthetic CDO. Index tranches are standardized as far as the underlying portfolio and the structure of the tranches is concerned. The high level of liquidity that constitutes the popularity of these products has mainly been achieved by the above-mentioned standardization and also by the fact that the single-name CDS market and the CDS index market are highly liquid. By investing in CDS index tranches, protection sellers can participate in a specific segment of the CDS index default loss distribution. The tranches are prioritized in order of seniority. In the case of the Dow Jones CDX.NA.IG 5yr index, successive tranches are responsible for 0% to 3%, 3% to 7%, 7% to 10%, 10% to 15%, and 15% to 30% of the losses. In case of the Dow Jones iTraxx EUR 5yr index, successive tranches are responsible for 0% to 3%, 3% to 6%, 6% to 9%, 9% to 12%, and

18


12% to 22% of the losses. As mentioned in Chapter 1, the standard market model for the pricing of CDS index tranches is the one-factor Gaussian copula model, according to e.g. Burtschell et al. (2005a), Duffie (2004), Finger (2004), Friend and Rogge (2004), Hull and White (2004), and Schönbucher (2003). Within this framework, the market generally assumes that the default intensities, the recovery rates, and the pairwise default time correlations are constant and equal across all assets, according to Hull and White (2004).

2.3 2.3.1

Credit Risk Modeling Single-Name Credit Risk: Intensity-Based Models

A variety of different approaches to the pricing of single-name default risk has been explored in the literature and implemented by practitioners. Following the common classification in the literature on default risk, we distinguish between structural models and reduced-form models. In the structural approach, the focus is on modeling the default risk that is specific to a particular obligor (a firm). Often structural models are referred to as firm value models. These models are based on the evolution of the firm’s value relative to some default-triggering barrier. The time of default is specified endogenously within the model as the first moment when the firm’s value passes the given threshold. The firm value approach was first proposed by Black and Scholes (1973) and extended by Merton (1974) and Black and Cox (1976). Further papers that use the structural model in combination with risk-free interest rates are Merton (1977), Geske (1977), Hull and White (1995), and Zhou (1997). Longstaff and Schwartz (1995) extend the approach of Black and Cox (1976) in a model with risky interest rates. Briys and de Varenne (1997) develop a corporate bond valuation model which is rooted in the contribution of Longstaff and Schwartz (1995). Using a discounted default barrier and taking into account early default and interest rate risk, they provide closed-form solutions for defaultable bonds. Varying the Longstaff and Schwartz (1995) setup, Schöbel (1999) presents a valuation model for risky corporate bonds. Triggering default by the firm’s asset to debt ratio, Schöbel derives solutions for the prices of corporate discount and coupon bonds that can be evaluated easily. Closed-form solutions can be derived if the level of leverage is held constant under the risk-neutral measure.

2.3. Credit Risk Modeling

19

In the reduced-form approach the value of the firm is not modeled. The default event is triggered by some exogenously given jump process. Within the reduced-form framework we distinguish between intensity-based models, which concentrate on the modeling of default, and credit migration models, which also consider migrations between credit rating classes. Early papers on the reduced-form approach are Pye (1974), Ramaswamy and Sundaresan (1986), Litterman and Iben (1991), Jarrow and Turnbull (1995), and Madan and Unal (1998). Duffie and Lando (2001) clarify the connection between intensity-based models and the above-mentioned firm value models in a setup with asymmetric information. Throughout this study we concentrate solely on intensity-based models. Notations, definitions, and theorems in Sections 2.3.1 and 2.3.2 are taken partly from Schönbucher (2003). Further sources for the illustration are Duffie and Singleton (2003) and Bielecki and Rutkowski (2002).

2.3.1.1

Stopping Times and the Hazard Rate Function

Consider a filtered probability space (Ω, (Ft )(t≥0) , P ), where the filtration (Ft )(t≥0) describes the information structure of the setup, and P describes the risk-neutral probability measure. We assume that the default of an asset takes place at the random time τ ≥ 0. Given the information in Ft , it is possible to determine whether the default has happened (τ ≤ t) or not (τ > t). Since Ft contains enough information to determine whether the event occurred, the random variable τ ≥ 0 is a stopping time, which means that {τ ≤ t} ∈ Ft for t ≥ 0. Let F (t) = P (τ ≤ t) denote the probability distribution function and f (t) the corresponding density function of τ . The conditional probability distribution function is F (t, T ) = P (τ ≤ T |τ > t) and the corresponding conditional density function is f (t, T ), where t ≤ T . We assume that F (t) < 1 and F (t, T ) < 1 for all t, T with t ≤ T . The hazard rate function is then defined as h(t) =

f (t) 1 − F (t)

(2.1)

and the conditional hazard rate function is defined as h(t, T ) =

f (t, T ) . 1 − F (t, T )

(2.2)

20


Considering the transition from discrete time to continuous time eases the interpretation of the hazard rate. It can be interpreted as the arrival probability of the stopping time per unit of time. We obtain in the unconditional case the following equation: lim

Δt→0

1 F (t, t + Δt) = Δt = = = =

1 P (t < τ ≤ t + Δt) Δt P (τ > t) 1 P (τ ≤ t + Δt) − P (τ < t) lim Δt→0 Δt P (τ > t) 1 F (t + Δt) − F (t) lim Δt→0 Δt 1 − F (t) f (t) 1 − F (t) h(t) . lim

Δt→0

Let F (t, T, T +Δt) denote the conditional default probability over the interval [T, T +Δt] as seen from t, assuming that there was no default until T . Then we obtain in the conditional case the following relation: lim

Δt→0

1 F (t, T, T + Δt) = Δt = = = =

1 P (T < τ ≤ T + Δt|τ > t) Δt P (τ > T |τ > t) 1 P (τ ≤ T + Δt|τ > t) − P (τ ≤ T |τ > t) lim Δt→0 Δt P (τ > T |τ > t) 1 F (t, T + Δt) − F (t, T ) lim Δt→0 Δt 1 − F (t, T ) f (t, T ) 1 − F (t, T ) h(t, T ) . lim

Δt→0

From the definitions in (2.1) and (2.2) we derive the differential equations ∂ log(1 − F (t)) ∂t

(2.3)

∂ log(1 − F (t, T )) ∂T

(2.4)

h(t) = − and h(t, T ) = −

subject to the initial conditions F (0) = 0 and F (t, t) = 0. From the differential equations (2.3) and (2.4) we obtain for all 0 ≤ t ≤ T the unconditional probabilities t h(s)ds F (t) = 1 − exp − 0

(2.5)


and

21

t h(s)ds f (t) = h(t) exp −

(2.6)

0

as well as the conditional probabilities F (t, T ) = 1 − exp −

T

h(t, s)ds

(2.7)

h(t, s)ds .

(2.8)

t

and

f (t, T ) = h(t, T ) exp −

T t

In the following, we outline how single-name default risk is specified in the intensitybased approach. In this framework, we assume that the default of one obligor is triggered by the first jump of a particular counting process. Intensity-based models differ primarily in the specification of the counting process. As starting point we consider Poisson processes with constant intensities. These processes can be extended to allow for time-dependent intensities. It is also possible to model stochastic intensity dynamics with a generalization of the Poisson process, the Cox process. The peculiarities of these processes will be discussed in the following. In order to introduce the counting process already mentioned above, consider the following collection of stopping times {τ1 , . . . , τn } where τi < τj and τi > 0 for i, j = 1, . . . , n, i < j. This collection of times describes a point process. We define the associated counting process as the stochastic process N(t) =

n

1{τi ≤t} .

(2.9)

i=1

N(t) represents the total number of events that have occurred in the time interval [0, t]. Let τ = inf{t ∈ R+ |N(t) > 0}

(2.10)

define the obligor’s default time. The obligor’s survival probability S(t, T ) from time t to time T , given survival until t, can then be derived as follows: S(t, T ) = 1 − F (t, T ) = P (N(T ) = 0|N(t) = 0) .

(2.11) (2.12)

22


We assume that the counting process N(t) is driven by a (possibly stochastic) intensity λ(t). For the counting process N(t) the intensity and the hazard rate are closely linked. Under suitable regularity conditions3 , which we assume to be met in this study, the intensity and the local conditional hazard rate coincide, i.e. λ(t) = h(t, t). Note that hazard rates are also defined for future dates. For instance, h(t, T ) is defined for T > t. However, there is only one λ(t).

2.3.1.2

Homogeneous Poisson Processes

We will now introduce an important example of counting processes. The Poisson process is the most widely used approach to model arrivals in a system. But first we define independent increments and stationary increments. Definition (Independent Increments) A counting process N(t), t ≥ 0 has independent increments if the number of events occurring in disjoint time intervals are independent. Therefore, the random variables N(t1 ) − N(t0 ), N(t2 ) − N(t1 ), . . . , N(tn ) − N(tn−1 ) are independent for all 0 ≤ t0 < t1 < · · · < tn . Definition (Stationary Increments) A counting process N(t), t ≥ 0 has stationary increments if the number of events occurring in a certain time interval depends only on the length of the time interval and not on the starting point. Therefore, the distribution of the random variable N(t + Δt) − N(t), Δt ≥ 0 is independent of t. Consider now the following definition of a Poisson process with homogeneous intensity parameter λ. Definition (Poisson Process) A non-decreasing integer-valued process N(t), t ≥ 0 is said to be a Poisson process with intensity λ > 0 if the following conditions are satisfied: 3

For details see Schönbucher (2003).


23

1. N(0) = 0, 2. the increments of N(t) are independent, and 3. for all t < T and n = 0, 1, . . . P (N(T ) − N(t) = n) =

1 (T − t)n λn exp (−λ(T − t)) . n!

Note that a Poisson process has stationary increments, i.e. the increments N(T ) − N(t) are independent of Ft . Furthermore, the time interval between two jumps is exponentially distributed and the probability of contemporaneous defaults is zero. The survival probability can be easily derived from equation (2.12): S(t, T ) = exp(−λ(T − t)) . Using homogeneous Poisson processes, we obtain term structures of credit spreads which do not change over time t and which remain flat as T changes. This can be concluded from the specification of the continuous hazard rate: ∂ S(t, T ) =λ. h(t, T ) = − ∂T S(t, T )

2.3.1.3

Inhomogeneous Poisson Processes

In order to obtain more realistic term structures of credit spreads, time dependent intensities can be introduced. Thus, let N(t) be a Poisson process with inhomogeneous intensity λ(t). Definition (Inhomogeneous Poisson Process) A non-decreasing integer-valued process N(t), t ≥ 0 is said to be an inhomogeneous Poisson process with non-negative intensity function λ(t) > 0 if the following conditions are satisfied: 1. N(0) = 0, 2. the increments of N(t) are independent, and 3. for all t < T and n = 0, 1, . . . P (N(T ) − N(t) = n) =

1 n!

T t

n λ(s)ds exp − t

T

λ(s)ds .

24


The intensity function is a function of time only. Again, we can derive the survival probability from equation (2.12): S(t, T ) = exp −

T

λ(s)ds .

t

In the inhomogeneous setup we obtain the continuous hazard rate, which now depends on the future point in time: ∂ S(t, T ) = λ(T ) . h(t, T ) = − ∂T S(t, T )

Using the inhomogeneous setup, non-flat term structures of credit spreads can be modeled, because the hazard rate depends on the future point in time T . 2.3.1.4

Cox Processes

However, the above-mentioned frameworks cannot cope with stochastic dynamics of credit spreads. Scenarios that are regularly observed in the market cannot be reproduced. Therefore we consider Cox processes, i.e. generalized Poisson processes. Cox processes allow to model stochastic dynamics of the default intensity. Consider a stochastic process Xt . Let this process generate a filtration, the so-called background filtration (Gt )t≥0 . The background information is then G = ∪t≥0 Gt . Definition (Cox Process) A counting process N(t), t ≥ 0 with stochastic intensity process λ(t) is said to be a Cox process, if N(t) is an inhomogeneous Poisson process with intensity function λ(t) conditional on the background information G. The associated survival probability is S(t, T ) = E exp −

T

λ(s)ds .

t

In practice, credit risk exposure often depends on a portfolio of defaultable assets. Therefore, we are interested in a model for multi-name credit risk. The above-mentioned


25

intensity-based framework can be used to specify the marginal distribution of the default time for each asset in the underlying of a credit portfolio. Assuming independence among the assets, any pricing problem associated with the portfolio can be managed. Under this assumption, the exposure to multiple defaults will not lead to an increase in model complexity. However, the independence assumption is not realistic. In the following, we allow for arbitrary dependence structures. We outline how the joint distribution of default times can be determined, given the marginal default time distributions.

2.3.2

Multi-Name Credit Risk: Copula Models

Four main approaches have been followed in the literature, extending the intensity-based model to incorporate default correlation. The first approach introduces default correlation into models with stochastic intensities in a very intuitive way: it incorporates dependence into the dynamics of the intensities. In this approach defaults are triggered by the first jump of Cox processes, the intensities of which are correlated. Conditional on the realizations of the intensities, the default indicator processes are independent Poisson processes. However, this approach cannot reproduce arbitrary levels of dependence. The strongest possible correlation can be achieved for perfectly correlated intensities. Schönbucher (2003) shows that the maximal default correlation is of the same order as the default probabilities. In some cases this level of dependence may be sufficient. But in general, correlated intensities cannot reproduce realistic levels of dependence. The second approach, proposed by Duffie and Singleton (1998), extends the abovementioned model by introducing additional default intensities for joint defaults. The default of each subportfolio is triggered by its own jump process. This setup allows for arbitrary levels of dependence up to perfect default dependence. However, some problems remain. It is not intuitively clear how to choose the default intensities for the different credit events. Additionally, the number of possible subsets grows exponentially with the number of defaultable entities. Consequently, the model is very difficult to implement and to calibrate. Another major drawback of this approach is the unrealistic distribution of defaults over time. In reality, defaults often conglomerate but do not occur at exactly the same time. Thus, a realistic period of crisis cannot be modeled. The third model addresses the problem of appropriate time resolution by introducing the aspect of credit risk contagion. After a major default, the credit risk of related obligors

26


is supposed to increase. Davis and Lo (2001a, 2001b) and Jarrow and Yu (2001) model the higher risk for related obligors by increasing their default intensities. After a certain period of time, the default intensities return to their original values. Modeling infectious defaults is very intuitive. Unfortunately, this approach is difficult to calibrate and not very tractable. In order to ensure consistent pricing, valuation models for multiple underlying assets should use multivariate default distribution functions conformant with the marginal default distribution functions used in corresponding models for a single underlying asset. The fourth pricing approach, which incorporates dependence by means of copula functions, allows the modeling of the dependence structure to be separated from the modeling of individual defaults. Li (2000) introduced the copula approach to credit risk. To model single-name credit risk, he used an intensity-based approach without considering the dynamics of default intensities. Schönbucher and Schubert (2001) extended this approach and incorporated the dynamics of default intensities. According to Schönbucher (2003), copula models dominate the above-mentioned frameworks, because they generate realistic dependence structures and are easy to implement and calibrate. In the remainder of this study, we concentrate on the copula function approach associated with an intensity-based model with constant intensities. This framework is frequently used in practical applications. In the following, vectors x and vector-valued functions f(x) are denoted in boldface. 1 is the vector (1, . . . , 1) and 0 is the vector (0, . . . , 0). Note that F (x) denotes the joint distribution of x = (x1 , . . . , xn ), and F(x) denotes the vector of marginal probabilities (F1 (x1 ), . . . , Fn (xn )). Comparisons between vectors are meant component-wise. We present an elementary fact about continuous distribution functions, the definition of the copula function, as well as some basic properties of copula functions. Proposition Let X denote a random variable and let F denote the associated continuous distribution function. Then Y = F (X) is again a random variable and it is uniformly distributed on [0, 1]. Conversely, let U denote a random variable with uniform distribution on [0, 1]. Then Z = F [−1] (U) has the distribution function F .


27

Definition 1 (Copula) An n-dimensional copula is a function C : [0, 1]n → [0, 1] that has the following properties: 1. There are random variables U1 , . . . , Un with range [0, 1] such that C is their joint distribution function. 2. The marginal distribution functions of C are uniformly distributed, i.e. for all i = 1, . . . , n, ui ∈ [0, 1] C(1, . . . , 1, ui, 1, . . . , 1) = ui .

A copula function can be used to link arbitrary marginal distribution functions to a multivariate distribution function. Sklar (1959) focused on the converse. He outlined that any multivariate distribution function can be written in the form of a copula. The marginal distribution functions and the dependence structure can be separated. The dependence structure is completely characterized by the copula function. Theorem (Sklar) Let F denote the joint distribution function of the random variables X1 , . . . , Xn with marginal distribution functions F1 , . . . , Fn . Then there exists an n-dimensional copula C such that for all x ∈ Rn F (x) = C(F1 (x1 ), . . . , Fn (xn )) = C(F(x)) .

(2.13)

Consequently, C is the distribution function of F(X) = (F1 (X1 ), . . . , Fn (Xn )). If F1 , . . . , Fn are continuous distribution functions, then C is unique. Otherwise C is uniquely determined on range(F1 ) × · · · × range(Fn ). We give an additional definition of the copula function. This formal definition is also frequently used. Like the previous definition, it clarifies that the copula can be interpreted as an n-dimensional distribution function on [0, 1]n with uniform margins. Definition 2 (Copula) An n-dimensional copula is a function C : [0, 1]n → [0, 1] that has the following properties:

28


1. The marginal distribution functions Ci (u) = C(1, . . . , 1,

u

, 1, . . . , 1) are

ith component

uniformly distributed, i.e. Ci (u) = u for all i = 1, . . . , n, u ∈ [0, 1].

2. For every u ∈ [0, 1]n , C(u) = 0 if at least one component of the vector u is 0. 3. For all a, b ∈ [0, 1]n with a ≤ b the measure that C assigns to the hypercube with corners a and b is non-negative, i.e. we have 2 2 i1 =1 i2 =1

···

2

(−1)i1 +i2 +···+in C(u1,i1 , u2,i2 , . . . , un,in ) ≥ 0

in =1

where uk,1 = ak and uk,2 = bk for all k = 1, . . . , n. Suppose that the random variables X1 , . . . , Xn have continuous distribution functions F1 , . . . , Fn . Then we can derive the copula density c(F1 (x1 ), . . . , Fn (xn )) from the function C(F1 (x1 ), . . . , Fn (xn )) as outlined below. Knowing the copula density can be particularly useful when it comes to calibrating the model to market data. Note that we apply the following relation between the density function fi (xi ) of a random variable Xi and the cumulative distribution function Fi (xi ): f (xi ) =

∂F (xi ) . ∂xi

By considering

∂ n F (x1 , . . . , xn ) ∂x1 . . . ∂xn n ∂ n [C(F1 (x1 ), . . . , Fn (xn ))] · = fi (xi ) ∂F1 (x1 ) . . . ∂Fn (xn ) i=1

f (x1 , . . . , xn ) =

= c(F1 (x1 ), . . . , Fn (xn )) ·

n

fi (xi )

i=1

we obtain the result f (x1 , . . . , xn ) . c(F1 (x1 ), . . . , Fn (xn )) = n i=1 fi (xi )

(2.14)

Copula functions are the most general framework for describing the dependence structure of random variables. Compared to the linear correlation concept, which fails to capture important aspects of risk, copulas describe comprehensively all components of dependence. Needless to say, there are numerous different specifications of copula functions. In the following we concentrate on the Gaussian copula, which plays an important role in many applications. One reason for its popularity is that the dependence structure is well


29

understood. The Gaussian copula contains the dependence structure of the multivariate standard normal distribution. The free parameters specify the pairwise dependence between the different assets. Additionally, it is easy to draw random variables from the Gaussian copula. Definition (Gaussian Copula) Let X1 , . . . , Xn denote standard normally distributed random variables with correlation matrix Σ and let Φ denote the univariate standard normal distribution function. Then the joint distribution function CΣGa (u1 , . . . , un ) of the random variables Ui := Φ(Xi ), i = 1, . . . , n is a copula and it is called the Gaussian copula to the correlation matrix Σ. Let ΦnΣ denote the multivariate standard normal distribution function and Φ−1 the inverse of the univariate standard normal distribution function. Then the Gaussian copula is according to (2.13) CΣGa (u1, . . . , un ) = ΦnΣ (Φ−1 (u1 ), . . . , Φ−1 (un )) . The corresponding copula density can be obtained using the result (2.14): cGa Σ (Φ1 (x1 ), . . . , Φn (xn )) =

1 1 T −1 x) n 1 exp(− 2 x Σ (2π) 2 |Σ| 2

n 1 1 2 √ i=1 2π exp(− 2 xi )

,

where xT denotes the transposed vector of x. Defining ui = Φ(xi ) and the column vector ψ = (Φ−1 (u1 ), . . . , Φ−1 (un ))T , we obtain cGa Σ (u1 , . . . , un ) =

1 1

|Σ| 2

1 exp(− ψ T (Σ−1 − En )ψ) , 2

where En denotes the n-dimensional unity matrix.

2.3.3

Valuation of Synthetic CDOs

In the following, the valuation of synthetic CDO tranches is introduced. The premium that is paid by the protection buyer to the protection seller is often called spread. It is quoted in basis points (bps) per annum of the contract’s nominal value. Note that this spread is not the same type of concept as the yield spread of a corporate bond to

30


a government bond. Rather, it is the annual price of protection quoted in bps of the nominal value and not based on any risk-free bond or any benchmark interest rate. In this study we use the term spread to describe both a premium for protection and a yield spread. Which is meant will be obvious from the context. In order to calculate the CDO tranche spread, the protection leg and the premium leg have to be evaluated separately. The premium leg reflects the expected value of the payments by the protection buyer, while the protection leg indicates the expected value of the compensation payments. The CDO tranche spread is chosen so that the values of the two legs are equal. Our presentation in Section 2.3.3 is mainly based on the ideas of Laurent and Gregory (2005). In the remainder of this study, valuation takes place in the framework set by the Gaussian copula model in combination with an intensity-based approach with constant intensity parameter.

2.3.3.1

Joint Distribution of Default Times in the Gaussian Copula Approach

We consider one asset with default time τ . Let N(t), t ≥ 0 be a Poisson process with constant intensity λ. According to the intensity-based approach, τ denotes a random variable with continuous distribution function F (t) = P (τ ≤ t) = 1 − exp (−λt). Consequently, the random variables F (τ ) and 1 − F (τ ) have uniform distributions on [0, 1]. This setup allows us to model default times as follows. Let U be a uniform random variable on [0, 1]. The random variable τ is related to the random variable U by τ =−

log(U) . λ

(2.15)

If we use this setup to simulate two random default times τi and τj , independence of the respective uniform random variables Ui and Uj will imply independence of the default times τi and τj . In the following, we consider n assets with stochastic default times τ1 , ..., τn and intensities λ1 , ..., λn . The cumulative distribution function of the default time τi , i = 1, . . . , n is Fi (t) = P (τi ≤ t) = 1 − exp (−λi t). The joint distribution function of τ1 , ..., τn is F (t1 , . . . , tn ). Correlation between default times can be modeled as follows. Let X1 , ..., Xn be standard Gaussian random variables with distribution function Φ and correlation matrix Σ.


31

The uniformly distributed random variable that triggers the default of asset i = 1, . . . , n can be defined as Ui = 1 − Φ(Xi ) .

(2.16)

Then we obtain correlated default times by τi = Fi−1 (Ui ) .

(2.17)

Using this approach, dependence between the assets is modeled by a Gaussian copula function CΣGa . According to (2.15) and (2.16), the joint distribution of τ1 , ..., τn is therefore F (t1 , ..., tn ) = P (τ1 ≤ t1 , ..., τn ≤ tn ) log(Un ) log(U1 ) ≤ t1 , ..., − ≤ tn ) = P (− λ1 λn = P (U1 ≥ exp(−λ1 t1 ), ..., Un ≥ exp(−λn tn )) = P (1 − Φ(X1 ) ≥ exp(−λ1 t1 ), ..., 1 − Φ(Xn ) ≥ exp(−λn tn )) = P (Φ(X1 ) ≤ F1 (t1 ), ..., Φ(Xn ) ≤ Fn (tn )) = CΣGa (F1 (t1 ), ..., Fn (tn )) . Within the general Gaussian copula setup as described above, arbitrary dependence structures between the random variables X1 , ..., Xn can be modeled. Thus, the general Gaussian copula model is a very powerful tool able to reflect many real-world scenarios. However, high levels of generalization are often associated with high levels of computational complexity. It is not surprising that this rule holds too for the general Gaussian copula setup. First, for arbitrary correlation matrices, the model leads to portfolio loss distributions that cannot be computed analytically making it often necessary to rely on Monte-Carlo techniques. Second, a full correlation matrix Σ contains many parameters that have to be specified. The number of free correlation parameters increases quadratically with the number of assets. The curse of dimensionality can be reduced by assuming that only one latent factor drives the dependence between the default times. This so-called one-factor Gaussian copula model leads to a restriction of the correlation structure, but in general the one-factor model is considered to provide sufficient degrees of freedom to replicate real dependence structures. The number of free correlation parameters coincides with the number of assets.

32


Moreover, the one-factor approach allows for semi-analytical portfolio loss distributions which accelerates the pricing process considerably (see for example Laurent and Gregory (2005)). The Gaussian one-factor copula model, to be discussed in the following, is generally used in financial applications.

2.3.3.2

Joint Distribution of Default Times in the Gaussian One-Factor Copula Approach

In the one-factor approach, dependence between the standard Gaussian random variables X1 , ..., Xn is introduced by the following construction: ˆ+ Xi = ρi X

1 − ρ2i X i , i = 1, . . . , n ,

(2.18)

ˆ X 1 , . . . , X n are independent standard normally distributed where ρi ∈ [−1, 1] and X, ˆ can be interpreted as the common risk factor and random variables. The component X X 1 , . . . , X n can be interpreted as idiosyncratic risk factors. Conditional on the common ˆ = ˆ the random variables X1 , . . . , Xn are independent. We obtain Corr(Xi , X) factor X ρi and Corr(Xi , Xj ) = ρi ρj . Note that in the one-factor approach the range of the correlation matrix is restricted, i.e. only a subspace of the general correlation matrix is covered. Using this setup, the joint distribution of default times can be derived by exploiting ˆ

the conditional independence property in (2.18). Let FiX (t) describe the distribution ˆ and F Xˆ (t1 , . . . , tn ) the joint distribution function of the default time τi conditional on X ˆ For the marginal conditional default probability we of default times conditional on X. obtain: ˆ

ˆ FiX (t) = P (τi ≤ t|X) log(Ui ) ˆ ≤ t|X = P − λi log(1 − Φ(Xi )) ˆ = P − ≤ t|X λi −1 ˆ = P Xi ≤ Φ (1 − exp(−λi t))|X ˆ = P Xi ≤ Φ−1 (Fi (t))|X ˆ Φ−1 (Fi (t)) − ρi X ˆ = P Xi ≤ |X 1 − ρ2i ˆ Φ−1 (Fi (t)) − ρi X . = Φ 2 1 − ρi


33

This result can be used in combination with the conditional independence of the default times to compute the joint distribution function:

∞

1 −ˆ x2 )dˆ x F xˆ (t1 , . . . , tn ) √ exp( 2 2π −∞ ∞ n −ˆ x2 1 )dˆ x. = Fixˆ (ti ) √ exp( 2 2π −∞ i=1

F (t1 , . . . , tn ) =

The integral in the previous formula can be easily derived using a certain quadrature technique. Similarly, the one-factor approach will ease computation of the distribution of the portfolio loss to be described below. 2.3.3.3

Pricing the Default Leg and the Margin Leg of a CDO

We consider a synthetic CDO whose underlying consists of n reference assets and we assume that asset j has exposure Ej and fractional loss given default Rj . Ij (t) = 1{τj ≤t} denotes the default indicator process. The cumulative portfolio loss at time t is therefore n

L(t) =

Σ R E I (t) . j

j j

(2.19)

j=1

As mentioned above, a CDO is a structured product that can be divided into various tranches. The cumulative default of tranche (A, B), 0 ≤ A < B ≤ Σ nj=1 Ej at time t is the non-decreasing function ωA,B (L) = (L − A)1[A,B] (L) + (B − A)1]B,Σ nj=1 Rj Ej ] (L) .

(2.20)

The tranche spread sA,B depends on the lower attachment level A and the upper attachment level B. Let D(t) be the discount factor for maturity t. T stands for the maturity of the CDO and let t1 , . . . , tI denote the regular payment dates for the CDO margins. 2.3.3.3.1

The Default Leg

Whenever a default occurs that affects the tranche

(A, B), the protection seller has to compensate the protection buyer for the loss incurred. In order to derive the value of the protection leg we consider each default separately. At default of asset j, the protection seller makes a payment that corresponds to the sudden change in the tranche loss, the payment amounting to ωA,B (L(τj )) − ωA,B (L(τj− )). Of course, the protection seller compensates the protection buyer only if asset j defaults before maturity T . This is verified by Ij (T ). Then we summarize the discounted compensation payments for all assets. The default leg is therefore determined by the expected

34


discounted changes of the cumulative default of the tranche during the term of the CDO. Thus, the protection payments can be written as E

n

D(τj )Ij (T )(ωA,B (L(τj )) − ωA,B (L(τj− )))

.

(2.21)

j=1

2.3.3.3.2

The Margin Leg

The protection buyer pays the tranche spread sA,B

based on the outstanding nominal of the tranche. The outstanding nominal of tranche (A, B) at a certain time t is given by the initial tranche value B − A subtracting the cumulative default of the tranche ωA,B (L(t)), i.e. the outstanding nominal is B − A − ωA,B (L(t)). However, defaults often take place between regular payment dates. Paying the tranche spreads based on the outstanding nominal of the tranche at the payment dates does not take the timing of the defaults into account. Thus, we have to take the outstanding nominal of the tranche during the time interval from the previous payment date to the current payment date as the basis for margin payments. In order to deal with this difficulty, we distinguish between regular margin payments and accrued margin payments. In order to derive the value of the regular margin payments, we consider the payments at the dates ti , i = 1, . . . , I separately. The regular margins are paid for the time from the last regular payment date ti−1 to the current regular payment date ti . Thus, we take the factor (ti − ti−1 ) into account. Regular margins are determined based on the outstanding tranche nominal at the current regular payment date, which amounts to B − A − ωA,B (L(ti )). Then we summarize the expected discounted payments for all regular dates. Thus, the regular margin payments are given by sA,B

I

D(ti )(ti − ti−1 )E [B − A − ωA,B (L(ti ))] .

(2.22)

i=1

In order to derive the value of the accrued margin payments, we consider each default separately. Accrued margins are determined based on the increment of the tranche loss at time τj which amounts to ωA,B (L(τj ))−ωA,B (L(τj− )) if asset j defaults before maturity T . Whether the default of asset j occurs before time T is verified by Ij (T ). Accrued margins are paid for the time from the last regular payment date before the default τj (this date is called tp(τj ) ) until τj . Thus, we take the factor (τj − tp(τj ) ) into account. Then we summarize the expected discounted accrued margin payments for all default dates. The accrued margin payments are then given by sA,B E

n j=1

D(τj )Ij (T )(τj − tp(τj ) )(ωA,B (L(τj )) −

ωA,B (L(τj− )))

.

(2.23)


35

A synthetic CDO can be valued analogously to a default swap transaction because the margin payments are exchanged with the default payments on the tranche. The tranche spread sA,B is derived by setting the default payments in (2.21) and the margin payments in (2.22) and (2.23) as equal. 2.3.3.4

Distribution of the Portfolio Loss in the One-Factor Approach

In the remainder of this study we make the following simplifying assumption, unless mentioned otherwise: we assume that the underlying of the CDO is a homogeneous portfolio of assets, i.e. all obligors have the same nominal Ej = n1 , the same loss given default Rj = 1, and the same default intensity λj = λ. Using these assumptions, we can simplify the portfolio loss at time t that we discussed in (2.19) to 1 Ij (t) . n j=1 n

L(t) =

(2.24)

The one-factor approach eases computation of the distribution of the portfolio loss. The conditional independence property of the one-factor approach gives rise to effective pricing techniques. Conditionally, the portfolio loss is the sum of independent individual losses. As a result, the conditional portfolio loss distribution is the convolution of the individual loss distributions. The unconditional portfolio loss distribution can be obtained by evaluating a numerical integral over the single factor. For details see Laurent and Gregory (2005). Other efficient pricing techniques have been discussed in the literature, see Section 2.4. 2.3.3.5

Monte-Carlo Simulation of CDO Tranche Spreads

Note that the setup illustrated in Section 2.3 (where single-name credit risk is described by an intensity-based model and multi-name credit risk is described by a Gaussian copula model) does not allow to derive the portfolio loss distribution in closed form for arbitrary dependence structures. Whenever we decide to use this approach to price portfolio credit derivatives (as it is market practice using the constraining assumption that all pairwise correlations are equal) we have to rely on Monte Carlo simulations as soon as we want to model arbitrary correlation structures. As illustrated above, there are specific versions of the setup (e.g. the one-factor approach) where the portfolio loss distribution can be computed in a semi-analytical way. However, we can still apply Monte-Carlo simulations when useful or necessary.

36


Consider the Gaussian copula CΣGa with correlation matrix Σ. To simulate the portfolio loss at time t in the Gaussian copula approach, we follow the algorithm mentioned below (see Laurent and Gregory (2005) and Schönbucher (2003) for further details).

1. Find the Cholesky decomposition A of Σ. 2. Simulate n independent standard normally distributed random variates z= (z1 , ..., zn ). 3. Set x= Az. Then x is multivariate normal with mean zero and correlation matrix Σ. 4. Set ui = 1 − Φ(xi ), i = 1, . . . , n where Φ denotes the cumulative univariate standard normal distribution. Then the joint distribution function of (1−u) for u= (u1 , ..., un ) is the Gaussian copula CΣGa . 5. The default dates can be derived from the uniform random variates. They are i) given by τi = − log(u , i = 1, ..., n. λ

Using the simulated correlated default dates we can derive the portfolio loss at time t. The k th simulation gives Lk (t). Repeat steps 1 to 5 exactly m times. The estimator for k the cumulative loss is then L(t) = m1 m k=1 L (t).

2.4

Valuation of CDOs: Literature

We discussed in Sections 2.3.1 and 2.3.2 various approaches to the modeling of singlename and multi-name credit risk. In this context we mentioned some pioneering research articles. Many of these models have been applied to the valuation of CDO tranches. In this study we focus on the Gaussian copula model, because it is considered to be the market standard according to e.g. Burtschell et al. (2005a), Duffie (2004), Finger (2004), Friend and Rogge (2004), Hull and White (2004), and Schönbucher (2003). In Section 2.3.3, we outlined in detail the Gaussian copula approach associated with an intensitybased model following the illustration of Laurent and Gregory (2005). However, in addition to the standard market model, there are various other approaches that have been suggested for valuation of CDO tranches. In the following, we will provide a short and by no means comprehensive overview of CDO pricing models.

2.4. Valuation of CDOs: Literature

37

The approach of Hull et al. (2005) is an example of a structural model that is used for valuation of CDO tranches. Hull et al. extend the structural setup of Black and Cox (1976) to allow for an arbitrary number of different obligors. Multivariate hitting times determine the defaults in the underlying portfolio. The firm value processes are correlated Brownian motions. Each firm value process contains a common component that represents the risk common to all obligors in the portfolio and an idiosyncratic component specific to each obligor. The approach of Duffie and Gârleanu (2001) is an example of a reduced form model that is used for valuation of CDO tranches. Duffie and Gârleanu consider a jump diffusion setup with correlated intensities. They capture default dependence by assuming that each intensity process is the sum of two affine processes. One component is common to all obligors and one component is an idiosyncratic process. In the following however, we concentrate on setups that use copula functions to model dependence between the default times of the underlying assets. According to e.g. Burtschell et al. (2005a), these models are predominantly used in the market to price CDOs. In a credit risk context, the default dependence has been modeled by Gaussian copulas, Student t copulas, double t copulas, Clayton copulas, and Marshall-Olkin copulas. Many of the non-Gaussian copula models have primarily been proposed to account for tail dependence among the default times. Burtschell et al. (2005a) provide a comparative overview of these copula models. The Gaussian copula setup is discussed by many researchers, for example Finger (2004), Hull and White (2004), and Schönbucher (2003). Likewise, many active derivatives dealers have written articles on this model, e.g. Ahluwalia et al. (2004), Watkinson and Roosevelt (2004), St Pierre et al. (2004), and O’Kane et al. (2003). The Student t approach is an extension of the Gaussian copula approach. This setup has been used in the area of credit risk by a number of authors, including Demarta and McNeil (2005), Greenberg et al. (2004), Andersen et al. (2003), Embrechts et al. (2003), Frey and McNeil (2003), Mashal and Zeevi (2003), Mashal et al. (2003), and Schlögl and O’Kane (2005). The double t approach is an extension of the Gaussian one-factor copula approach. This setup has been used for the pricing of CDO tranches by Hull and White (2004). The Clayton copula has been considered for credit risk issues by a number of authors, including Madan et al. (2004), Friend and Rogge (2004), Gregory and Laurent (2003),

38


Laurent and Gregory (2005), Schlögl and O’Kane (2005), Rogge and Schönbucher (2003), Schönbucher (2002), and Schönbucher and Schubert (2001). Elouerkhaoui (2003a, 2003b), Giesecke (2003), Lindskog and McNeil (2003), Li (2000), Wong (2000), and Duffie and Singleton (1998) have been considering multivariate exponential models and the Marshall-Olkin copula in a credit risk context. Finger (2004) points out that most market participants rely on the Gaussian copula model. However, there are many points where specific implementations may differ. These issues may produce discrepancies between seemingly similar pricing models. Finger has surveyed different implementations and specifications. First, there may be variations in the specification of the correlation structure. Second, instead of an average spread individual spreads may be taken into account. Third, instead of a finite number of obligors a large pool approximation may be considered (see Schönbucher (2003) for details of the large pool model). Fourth, there may be different approaches to account for the timing of defaults occurring between payment dates. In general the copula correlation is a deterministic parameter, but there are also extensions to the Gaussian copula model involving stochastic correlation. Examples are Burtschell et al. (2005b) and Andersen and Sidenius (2004). For example, Andersen and Sidenius allow default correlation to be higher in bear markets than in bull markets. Many credit risk models are based on a factor approach. Important papers are Frey and McNeil (2003), Gordy (2003), Pykhtin and Dev (2002), Merino and Nyfeler (2002), and Crouhy et al. (2000). Burtschell et al. (2005b) point out that for CDO pricing the factor approach has been predominantly coupled with a copula approach, while it may also be used within the framework of structural models or within the framework of models with correlated intensities. Especially for the pricing of CDOs, the one-factor copula approach has proved to be a powerful tool because of the conditional independence property. In Section 2.3.3, we outlined the one-factor approach of Laurent and Gregory (2005) for the valuation of CDO tranches. Likewise, Hull and White (2004), Andersen et al. (2003), and Gordy and Jones (2003) use the one-factor setup for CDO pricing. A factor approach can be handled using the Fast Fourier transform approach of Laurent and Gregory (2005). An alternative way to exploit the conditional independence property of the factor approach is a recursive algorithm as described by Andersen et

2.4. Valuation of CDOs: Literature

39

al. (2003). It does not carry the burden of calculating Fourier transforms. Likewise, Hull and White (2004) have developed procedures to value tranches of CDOs using a recurrence relationship and an iterative numerical procedure. The model is based on a factor copula model and is an alternative to the Fast Fourier transform approach of Laurent and Gregory (2005). Recently, other factor models based on different distributions have been proposed, so as to bring more flexibility into the dependence structure and to allow for tail dependence. Kalemanova et al. (2005) have proposed a factor copula approach based on Normal Inverse Gaussian distributions in order to price CDO tranches. Under certain conditions, the Normal Inverse Gaussian distribution is stable under convolution and allows for tail dependence. Moosbrucker (2006) points out that default correlation increases in times of recession. In order to account for this effect a stochastic business time is introduced, see also Joshi and Stacey (2005). The idea of a stochastic business time leads to an increase of default correlation in economic recessions. In analogy to the one-factor Gaussian copula, he defines a one-factor Variance Gamma copula. Finally, Baxter (2006) uses a dynamic model of spread evolution within a factor approach to price CDO tranches. In his model the firm value process is the sum of a Brownian motion and a Variance Gamma process.

Chapter 3 Explaining the Implied Correlation Smile 3.1

Introduction

In the preceding Chapter 2 we outlined how a liquid and transparent market for tranched credit risk has evolved in recent years. Especially tranches linked to the reference portfolios CDX and iTraxx have become actively quoted with relatively narrow bid-ask spreads as mentioned in Schönbucher (2006). We also introduced the Gaussian one-factor copula model which has become the market standard for the valuation of index tranches (according to e.g. Burtschell et al. (2005a), Finger (2004), Friend and Rogge (2004), Duffie (2004), Hull and White (2004), and Schönbucher (2003)). The standard market model generally assumes that default intensities, recovery rates, and pairwise default time correlations are constant and equal across all assets in the portfolio (see Hull and White (2004)). Consequently, it is the market standard to rely on only one scalar parameter ρ to describe all pairwise correlations among the default times of the underlying n assets. Since all pairwise correlations are the same, the corresponding correlation matrix Σ = (Σij )i,j=1,...,n with Σij = ρ, i = j and Σij = 1, i = j is denoted as flat. An obvious advantage of the standard market model is its simplicity and its tractability. This may be the main reason why it is used extensively in the market, despite several well-known shortcomings. One major drawback of the standard market model is that it fails to fit the market prices of different index tranches accurately (see Schönbucher (2006)). This deficiency is illustrated by a phenomenon that is regularly observed in the market for credit derivatives: the implied correlation smile. The implied correlation smile became evident when liquidity in the market led increasingly to the quotation of tranched products in terms

42

Chapter 3. Explaining the Implied Correlation Smile

of implied correlation parameters. Implied tranche correlations and implied base correlations are used as quotation devices. In this chapter we will discuss the implied tranche correlation, the implied base correlation, and the emerging correlation smile. We will give an overview of recent publications that attempt to explain and model the correlation smile, and we will provide an additional explanation for the appearance of the smile. We will outline how the use of heterogeneous dependence structures can lead to an allocation of risk to the different tranches that is similar to the risk allocation observed in the market. We will exemplify how non-flat dependence structures can lead to tranche spreads that generate characteristic implied correlation smiles. Thus, we will conclude that the use of heterogeneous dependence structures can improve the fit with market data.1

3.2

Sensitivity of the Tranche Price to the Level of Correlation

It is intuitively clear that the level of correlation has different effects on the risk content of tranches of different seniority (we already discussed this subject in Section 2.2.3.6). The risk content of a tranche can be exemplified by the expected tranche loss and is accordingly reflected in the tranche price. When we modify the correlation structure in the underlying credit portfolio, we change the risk content of the tranches. Some tranches get riskier, some get less risky, some are more sensitive than others. The extent to which the tranches react depends on their attachment levels. Whereas the expected loss of the different CDO tranches depends on correlation in the underlying portfolio, the expected portfolio loss is independent of the correlation structure among the underlying assets. The sum of the expected tranche losses weighted by the respective tranche nominal always yields the portfolio loss. Consider a CDO that is split into three tranches. The equity tranche covers the first losses that occur in the underlying portfolio. If the portfolio loss exceeds the upper attachment level of the equity tranche, the mezzanine tranche will absorb the losses not covered by the equity tranche. Only if the portfolio loss exceeds the upper attachment level of the mezzanine tranche will the senior tranche be affected.

1

See also Hager and Schöbel (2005).

3.2. Sensitivity of the Tranche Price to the Level of Correlation

43

If correlation among the underlying assets is low, the defaults will occur almost independently of each other. The realized portfolio loss will generally be close to its expected value. If correlation is high, assets will tend to default or survive together. Extreme outcomes will be more likely. In the case of low correlation, the number of defaults generally lies around a certain average. Even if only an average number of assets default in the underlying, the equity tranche is regularly heavily affected by this average number of defaults, because the first-loss tranche is generally very small. Low levels of correlation are therefore highly risky for equity tranche investors. In this case, equity tranche spreads are high to compensate investors for taking on high levels of risk. In the case of high correlation, there is a real chance of a scenario occurring with only very few defaults. This scenario would be positive for equity tranche investors. It is also possible that many assets in the underlying portfolio will default. This possibility does not increase the riskiness of the equity tranche to any great extent, since equity tranches are already heavily affected when only an average number of assets default. Equity tranche investments are therefore less risky in high correlation environments and so spreads are low. Thus, expected tranche losses and tranche spreads decrease monotonely with increasing correlation. Contrariwise, senior tranches are less risky when correlation is low. In this case, the realized portfolio loss tends to lie around its expected value, which is generally below the lower attachment level of the senior tranche. When correlation is high, senior tranches become more risky. In this case, it is more likely that assets default together and the portfolio loss may exceed the lower attachment point of the senior tranche. Thus, expected tranche losses and tranche spreads increase monotonely with increasing correlation. As for the mezzanine tranche, there is no monotone relationship. Generally we observe that for low correlation levels mezzanine tranches act like senior tranches, and for high correlation levels mezzanine tranches act like equity tranches. The sensitivity of the tranche price to correlation is illustrated in the following. We consider flat correlation matrices as illustrated in Figure 3.1 for a correlation level of 0.5. In Figure 3.2 we exemplify the sensitivity of the tranche spread to the level of correlation using the following setup. We consider a homogeneous portfolio of 100 defaultable assets. We assume that all assets have equal CDS spreads of 100 bps, equal recovery rates of 40%, and equal nominal value. The time to maturity is 5 years. As mentioned

44


above, all pairwise default time correlations are supposed to be the same. We compute the tranche spreads for the equity tranche (0%-3%), the mezzanine tranche (3%-10%), and the senior tranche (10%-100%).

Figure 3.1 Flat correlation structure.

3.3

The Implied Tranche Correlation

The implied tranche correlation is the uniform correlation level that makes the tranche spread computed by the standard market model equal to its observed market spread. The implied tranche correlation and the implied base correlation to be described below are applied when comparing alternative investments in synthetic CDO tranches. Moreover, implied correlations of traded CDO tranches are used to value off-market portfolio credit derivatives based on the same underlying portfolio as the traded CDO. However, relative value considerations or valuations of an off-market tranche by means of the derived implied correlation parameter of a related traded tranche can be misleading, as we will show in brief. The illustrated price sensitivity of different tranches to correlation and observations in the market show that implied tranche correlations can suffer from both uniqueness and existence problems. Tranche spreads are not necessarily monotone in correlation, thus we may observe market prices that are attainable by two levels of correlation. We may also observe market prices that are not attainable by any choice of flat correlation.

3.4. The Implied Correlation Smile

45

The Equity Tranche

The Mezzanine Tranche 700

5000

Tranche Spread (in bps)


6000

4000 3000 2000 1000 0 0

0.2

0.4 0.6 Correlation

0.8

1

0.8

1

600

500

400

300

200

0

0.2

0.4 0.6 Correlation

0.8

1

The Senior Tranche


100

80

60

40

20

0

0

0.2

0.4 0.6 Correlation

Figure 3.2 Sensitivity of the tranche spread to the level of correlation.

3.4

The Implied Correlation Smile

Quotations available in the market indicate that different CDO tranches on the same underlying portfolio trade at different implied correlations. Generally, mezzanine tranches trade at lower correlation levels than equity and senior tranches. This observation, known as the implied correlation smile, illustrates that the standard market model fails to accurately reproduce the market prices of different tranches. We can say that the standard market model underprices the equity and senior tranches and overprices the mezzanine tranche. In other words, the market charges more for equity and senior tranches and less for mezzanine tranches than the model implies. If the standard market model was the correct pricing model, the implied default correlations would trivially be constant over tranches. Rather we observe a smile-shaped curve when we plot the implied correlations with respect to the different tranches.

46


Figure 3.3 displays the correlation smile derived from the market prices2 of the CDX.NA.IG 5yr index tranches on April, 4 2005.

Implied Tranche Correlation

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0%−3% 3%−7% 7%−10% 10%−15%15%−30% Tranche

Figure 3.3 The implied correlation smile on April, 4 2005. There are numerous reasons for the market prices not being consistent with the standard market model, or for the implied correlation smile appearing in the market. Apparently the standard market model relies on some highly controversial assumptions. On the one hand, the standard market model makes the simplifying assumption that recovery rates and CDS spreads are constant and equal for all obligors. This entails independence of recovery rates and default intensities, even though market data show the converse. Therefore the correlation smile may be a consequence of these simplifying assumptions. On the other hand, the correlation smile may also be due to an incorrect specification of the dependence structure in general. It is well known that the Gaussian copula overestimates the chance of observing just a few defaults, and underestimates the chance of observing a very high or a very low number of defaults. Moreover, the Gaussian copula does not allow for tail dependence (see Schönbucher (2003) for further details of the tail dependence property). Thus, the Gaussian copula model may not accurately reflect the joint distribution of default times. Apart from that, the standard market model makes the simplifying assumption that all pairwise default time correlations are the same. This may be an explanation for the smile as mentioned shortly in Andersen and Sidenius (2004), Duffie (2004), Friend and Rogge (2004), and Gregory and Laurent (2004). Furthermore, the standard market model does not reflect the well-known fact that default correlation increases in economic recessions. 2

Throughout this thesis, we use data that have been provided by JPMorgan Chase.

3.5. The Implied Base Correlation

47

We pointed out that the market charges more for equity and senior tranches, and less for mezzanine tranches, than the standard market model implies. Equity tranches may be underpriced by the Gaussian copula model because the Gaussian copula underestimates the chance of observing only a few defaults, as mentioned above. Supply and demand imbalances may also play a role: mezzanine tranches are extremely popular with investors. Senior tranches may be underpriced by the Gaussian copula model because sellers of protection on senior tranches require a minimum coupon, even though the risk is relatively low. Since the standard market model is based on many assumptions and simplifications, there is a lot of room for researchers to introduce extensions or to discuss alternative pricing approaches. Recently, an increasing number of publications have been dedicated to modeling the correlation smile. We will address the most essential of these research papers in Section 3.7. Even if the pricing of CDO tranches is a very active field of research, no light has yet been shone on whether whether technical, informational, or liquidity factors cause the standard market model’s failure to fit market prices. The correlation smile makes clear that the implied tranche correlation is a quotation device that has to be interpreted cautiously. Obviously it is not appropriate to use the flat implied correlation parameters of traded tranches to price off-market portfolio credit derivatives with the same underlying; nor is it appropriate to use implied correlation parameters for relative value considerations (when comparing alternative investments in CDO tranches).

3.5

The Implied Base Correlation

To cope with some of the difficulties linked to implied tranche correlations, McGinty et al. (2004) have introduced the implied base correlation approach. Sometimes base correlations are quoted in the market instead of tranche correlations. The base correlation for a certain tranche is the flat correlation level that makes the total value of all tranches up to and including the current tranche equal to zero. For instance, in case of the iTraxx Europe 5yr index, the 0% to 9% base correlation is the correlation that causes the sum of the values of the 0% to 3%, the 3% to 6%, and the 6% to 9% tranches to be zero. Market participants apply the base correlation approach within the framework of the standard market model.

48


Figure 3.4 displays the base correlations that are derived from the market prices of the CDX.NA.IG 5yr tranches on April, 4 2005. Obviously what we observe is a skew.

0.8

Implied Base Correlation

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0%−3%

3%−7% 7%−10% 10%−15%15%−30% Tranche

Figure 3.4 The implied base correlation skew on April, 4 2005. The idea behind the base correlation approach is to derive the implied tranche correlations c1 , . . . , cm for a sequence of tranches with lower attachment level 0 and with given upper attachment levels b1 < · · · < bm . These upper attachment points b1 , . . . , bm are the upper attachment levels of standard tranches (a1 , b1 ), . . . , (am , bm ) that are liquidly traded in the market. The implied base correlations are increasing with the upper attachment level, i.e. c1 < · · · < cm for b1 < · · · < bm . Having derived the implied base correlations for all traded CDO tranches, it is possible to derive by interpolation the implied base correlation ca and cb for arbitrary upper attachment levels a and b with b1 ≤ a ≤ b ≤ bm . From ca and cb any tranche (a, b) can be valued. For details see McGinty et al. (2004). McGinty et al. (2004) have explained how base correlations may facilitate the valuation of off-market tranches that have the same collateral pool as actively traded tranches. They point out that base correlations are unique, since equity spreads are monotone. That is, there is only one base correlation for a special upper attachment point. Furthermore, the base correlation curve is a much smoother line than the implied tranche correlation curve. The approximate linearity of the base correlation curve is considered to be the major advantage of the base correlation approach. However, the existence of the implied base correlation parameter is still not guaranteed. Moreover, choosing the right position on the base correlation curve for a certain upper attachment point may be difficult. This is a weakness of the base correlation approach, because the tranche

3.6. Evolution of the Implied Correlation Smile

49

CDX.NA.IG

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80 70 60 50 40 Jan. 04

May 04

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May 05

Sep. 05

Jan. 06

Figure 3.5 CDX.NA.IG index spread. prices are highly sensitive to the correlation parameter. Willemann (2005a) has shown that even if the base correlation setup offers some benefits over the traditional implied tranche correlation approach, it should be interpreted and used carefully. He notes that even if the true default correlation in the model increases, base correlations for some tranches may actually decrease. Furthermore, base correlations are only unique given the set of attachment points. Therefore, the base correlation for a certain attachment point depends on the position of all prior attachment points that are considered. This means that across the North American and European markets base correlations may be different, because the tranche structures are not the same. Further shortcomings of the base correlation approach are outlined in Willemann (2005a). Within the base correlation approach, expected tranche losses have to be derived from tranche spreads and vice versa. This work step requires both, the knowledge and the use of the implied tranche correlation. Thus, any weakness that is inherent to implied tranche correlations is transferred to implied base correlations.

3.6

Evolution of the Implied Correlation Smile

In the following, we provide a short overview of the credit markets in general and the CDO market in particular. We consider the spread of the CDX.NA.IG 5yr and the associated tranche spreads during a two-year interval from January 2004 until December 2005. Figure 3.5 shows the evolution of the CDX in the interval considered. We shortly comment on the course of the CDX index, following the review of Nomura Securities (see Nomura (2004a, 2004b, 2005a, 2005b)).

50


Even if the general economic outlook improved in the first half of 2004, CDS spreads widened (increased). The bombings in Spain led to increased geopolitical concerns in mid-March 2004. As a result, equity volatility increased. Strong spread widening could be observed in the high volatility sector and the telecom/media sector alike. Strong spread widening could also be observed in the financial sector, as higher interest rates became more and more likely. Approaching mid-year, CDS spreads stabilized and tightened (decreased) with lower equity volatility. In the second half of 2004, CDS spreads tightened significantly after widening in the first half of 2004. One possible explanation for this general tightening of spreads is the increase in CDO-related protection selling. The CDX stayed relatively stable in July, but climbed above 65 bps in early August as the stock market declined because of weak payroll figures and weak corporate earnings. As the equity market recovered at the end of August, CDS spreads declined. In late September CDS spreads widened again as well as in mid-October, when CDS spreads of several insurance companies moved wider after a governmental probe. The CDS spreads tightened shortly afterwards until the end of the year 2004. Uncertainties about economic fundamentals, e.g. oil prices, could be counterbalanced by CDO-related protection selling and by decreasing equity volatility. In the first half of 2005, CDS spreads moved wider again after having tightened in the second half of 2004. The slight widening in January and early February may be explained by a decline in synthetic CDO activity. The spreads tightened again in late February and early March as trading volume increased. After General Motors published bad figures in mid-March, the credit market became volatile. On May 5 the credit ratings of General Motors and Ford were downgraded. Following these downgrades, the CDX market and the index tranche market experienced significant volatility levels. The index spreads turned wider from 41 bps in early March to 78 bps in mid-May. Approaching mid-year, the credit derivatives market became relatively stable. However, the overall spreads remained about 20% wider compared to the beginning of the year. In the second half of 2005, CDS spreads tightened and recovered some of the widening experienced in the first half of 2005. The index traded in summer at around 60 bps and moved tighter by early fall to 50 bps. Approaching the end of the year 2005 the index moved around that level. As volatile automobile names (Ford and General Motors) were removed from the on-the-run index, spread volatility declined. The whole period was characterized by increased idiosyncratic risk and by several high-profile credit events. In the subsequent Figure 3.6, we display the evolution of the associated index tranche spreads.


51

To illustrate how the spreads of the traded index tranches can be used to interpret the market’s view of the credit standing of the underlying portfolio, consider exemplarily the following time period. In May 2005 the CDX spread widened sharply, which can be interpreted as a deterioration in the credit quality of the underlying assets. Considering the spreads of the different traded tranches of the CDX, it becomes obvious that the sharp widening in tranche spreads mainly affected the 0%-3% tranche. This fact can be an indicator for a changing rating quality in only a single sector of the economy. In this case, it was caused by the downgrading of Ford and General Motors to sub-investment status. Since the other tranches remained relatively unaffected, these downgrades obviously were not considered to influence credit quality beyond the automobile industry. Given the index spread and the tranche spreads we can derive the implied tranche correlations using the standard market model. The implied tranche correlations are displayed in Figure 3.7. Figure 3.8 displays the evolution of the implied correlation smile over the time interval considered. Note that implied tranche correlations of mezzanine tranches may suffer from existence or uniqueness problems. If tranche spreads can be reproduced by two levels of correlation, we will only consider implied tranche correlation levels that are below 0.75, so as to exclude unrealistic levels of dependence. Note further that the index spread varies over time and influences the implied tranche correlation. Figure 3.9 shows the tranche spreads for different index spreads and different levels of correlation. The above-mentioned increase in the risk of the equity tranche in May 2005 is reflected in the implied default correlation sharply falling. However, we also observe a fall in the implied tranche correlation of the 7%-10% tranche and the 10%-15% tranche, which formerly seemed to be unaffected by the downgrading of Ford and General Motors. To understand this evolution, note that decreasing tranche spreads reflect a deceasing risk exposure of the respective tranches. Note further that an increasing index spread leads trivially to an increase in the risk exposure of all tranches and to increasing tranche spreads. However, the spreads of the 7%-10% tranche and the 10%-15% tranche exhibited a downward slide in spring and summer 2005 in spite of increasing index spreads. This fact can be explained by a correlation level reducing the risk for the respective tranche.



Upfront Payment (in Percent of the Face Value of the Tranche)

52

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Figure 3.6 CDX.NA.IG tranche spreads.

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Figure 3.7 CDX.NA.IG implied tranche correlations.

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Figure 3.8 Evolution of the CDX.NA.IG implied correlation smile.

Jan. 04


55

Spread of the 0%−3% Tranche


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Figure 3.9 Sensitivity of the tranche spread to the level of correlation for different index spreads.

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3.7


Modeling the Correlation Smile: Literature

Within the last two years, a growing number of researchers have examined different approaches to reproducing the prices of traded CDO tranches and by the same token to explaining and modeling the implied tranche correlation smile or the implied base correlation skew. Burtschell et al. (2005a) systematically compare different CDO pricing models and analyze their ability to model the correlation smile. Likewise, Andersen (2005) gives an overview of the state of affairs in CDO pricing. There has been much interest in simple extensions of the Gaussian copula model in order to reproduce CDO tranche prices. In the following we will list some examples. Gregory and Laurent (2004) present an extension of the Gaussian one-factor copula model that allows for a clustered correlation structure by specifying intra and inter sector correlations. The intra sector correlation describes the uniform default time correlation level within a sector, and the inter sector correlation describes the uniform correlation level between the default times of assets that belong to different sectors. Additionally, Gregory and Laurent introduce dependence between recovery rates and default events, which leads to an improved modeling of the smile. Andersen and Sidenius (2004) also extend the Gaussian one-factor model by randomizing recovery rates. In addition, they allow for correlation between recovery rates and defaults. In a second extension, they introduce in formula (2.18) random correlation parameters (characterized by a two point distribution) to permit higher correlation in economic depressions. Both approaches are able to model a smile. There are some approaches that rely on non-Gaussian distributions in order to reproduce the observed correlation smile. We mentioned these models already in Section 2.4. Hull and White (2004) consider a double t copula setup in analogy to the one-factor Gaussian copula setup. They use t distributions to equip the common factor and the idiosyncratic factor with heavy tails. Kalemanova et al. (2005) extend the large homogeneous portfolio setup. They use a factor copula approach based on Normal Inverse Gaussian distributions to reproduce market prices. Joshi and Stacey (2005) suggest a model for default correlation based on the concept of a stochastic business time in order to generate a correlation smile. Joshi and Stacey attempt to model good and bad time intervals from an economic point of view. In good time intervals there are only few shocks to the market, in bad time intervals there are

3.7. Modeling the Correlation Smile: Literature

57

a lot of shocks. They use Gamma processes to calibrate an intensity model to market prices of liquid CDO tranches. Moosbrucker (2006) defines a one-factor Variance Gamma copula on the basis of a structural model. The setup leads to a dependence structure that produces a smile. In order to reproduce market prices, Baxter (2006) models the firm value processes in a structural setup as the sum of Brownian motions and Variance Gamma processes. Hull et al. (2005) present a structural model for multiple obligors. They extend the model to incorporate positive correlation between default correlations and default rates and negative correlation between recovery rates and default rates. They compare the model prices with quoted market prices. Willemann (2005b) values CDO tranches in a structural setup (cf. Hull et al. (2005)). He allows for correlation by the use of diffusive asset value correlations. Furthermore, he allows for correlation through common jumps in asset values. Using this setup he is able to generate a smile. Hull and White (2006) show that a one-factor copula model can be implied from market prices of CDO tranches. Consequently, this approach fits the market quotes. In the simplest implementation of their implied copula model, Hull and White randomize default probabilities. They assume that defaults occur according to a Poisson process whose hazard rate is drawn from a discrete distribution. Thus, the calibration problem is to choose the values of the hazard rate and their respective probabilities. Tavares et al. (2004) propose a composite basket model to deal with the Gaussian copula smile. They combine the copula model (to model the default risk that is driven by the economy) with independent Poisson processes (to model the default risk that is driven by a particular sector and by the company in question). Mashal et al. (2004) show that using the implied correlation approach for relative value assessments may be misleading because the real dependence structure of the underlying is neglected. They suggest an alternative correlation measure for relative value analysis that considers all historical pairwise correlations. Van der Voort (2005) points out that in some cases the default event of one name may have external causes and does not increase the default risk of other names. By

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introducing an extra idiosyncratic term for external default risk, his model allows for totally external defaults. Van der Voort shows that he is able to model a correlation smile. Schönbucher (2006) proposes a model for the joint stochastic evolution of the portfolio loss process and of its probability distribution. As opposed to most other approaches that we described so far, which start from modeling the individual obligors’ default processes (a bottom-up approach), Schönbucher adopts a top-down approach which focuses on the cumulative loss process of the whole portfolio.

3.8

Heterogeneous Dependence Structures

It is important to note that neither the implied tranche correlation approach nor the implied base correlation approach allows for consistent pricing, because these approaches apply different correlation levels for the valuation of different tranches, which contradicts the fact that the underlying portfolio is the same. It is important to note that the correlation parameters of the copula are asset-specific quantities and not tranche-specific quantities. Therefore they ought to be the same, irrespective of the tranche being priced (cf. Tavares et al. (2004)). Furthermore, the observed smile and skew clarify that the implied tranche correlation and the implied base correlation provide only aggregated information about a certain segment of the portfolio loss distribution. Thus, relative value considerations or valuations of off-market products by means of an implied correlation parameter of a related traded tranche can lead to serious mispricing, especially when the performance of the off-market product depends on another segment of the portfolio loss distribution. In the remainder of this chapter, we outline a concept that has the potential to explain the implied correlation smile and to reproduce the market prices of traded CDO tranches. We adhere to the standard market model for the pricing of CDO tranches, excluding the assumption that the underlying correlation structure is flat. Real-world dependence between the default times of different assets is generally characterized by high levels of complexity. It is intuitively clear that the complex relationship between the default times of different assets usually cannot be compressed into one single number. Therefore we allow for arbitrary heterogeneous, i.e. non-flat correlation structures.

3.8. Heterogeneous Dependence Structures

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One purpose of this study is to assess the importance of the dependence structure regarding the valuation of CDO tranches. We illustrate the impact of inhomogeneous correlation structures on the risk content of different CDO tranches. We discuss different dependence structures and analyze the resulting implied tranche correlations. We show that heterogeneous correlation structures often lead to correlation smiles. To the best of our knowledge, there is no publication that analyzes the connection between the correlation structure and the correlation smile in detail. We transfer our findings to real-world markets. Our examples suggest that the correlation smile that is observed in the CDO market is likely to emerge because the actual underlying correlation structure is inhomogeneous. We point out that neglecting the actual correlation structure may lead to serious mispricing, and we show that the use of an appropriate heterogeneous correlation matrix may be capable of reproducing the observed market spreads. We consider a setup that we denote as the extended standard market model. We assume that the standard market model constitutes the true pricing model but allow for the above-mentioned variation: heterogeneous dependence structures are admitted, i.e. the parameters within the Gaussian copula approach can be chosen arbitrarily. All other assumptions of the standard market model apply (e.g. the intensity based approach describes the default time of one obligor; the Gaussian copula describes the dependence between the obligors’ default times; all assets have equal default intensities, equal recovery rates, and an equal nominal). Within this extended standard market model we choose different correlation matrices, compute the associated tranche spreads, and derive the implied tranche correlations. Depending on the chosen correlation structure, we observe different forms of smiles. We choose the extended standard market setup to make sure that the emerging correlation smiles are solely induced by the structure of the underlying correlation matrix. In our setup, there are no pricing discrepancies caused, for instance, by neglecting stochastic recovery rates, stochastic default intensities, correlated recovery rates and default times, or any other aspects that are not reflected appropriately in the pricing model. In our setup, a smile makes clear that a single flat correlation matrix is not able to distribute the shares of risk on the equity, the mezzanine, and the senior tranche in the same way as the actual underlying matrix does. We observe that generally a flat correlation matrix cannot suitably reflect the risk for all tranches simultaneously when the actual underlying correlation structure is not flat.

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Again, we use the setup of Section 3.2. We consider a homogeneous portfolio of 100 defaultable assets, whereby all assets have equal CDS spreads of 100 bps, equal recovery rates of 40%, and equal nominal value. The time to maturity is 5 years. We consider various dependence structures between the default times of the assets. Using these dependence structures we compute the corresponding tranche spreads for the equity (0%-3%), the mezzanine (3%-10%), and the senior tranche (10%-100%). Then we use the obtained tranche spreads to derive the implied tranche correlations.

3.8.1

Heterogeneous Dependence Structures Can Cause Implied Correlation Smiles

In the following, we outline how the credit risk is allocated to the different tranches of a CDO and what kind of correlation smile we obtain when the underlying dependence structure is not flat. First, we give examples that exemplify the existence and the uniqueness problem of the implied tranche correlation of mezzanine tranches.

3.8.1.1

The Existence Problem

Consider a simplified setting where the assets in the underlying portfolio are divided into 10 different sectors of equal size. The correlation matrix in Figure 3.10 illustrates the corresponding correlation structure with 10 sectors of perfect intra sector correlation. The inter sector correlation is zero. Using this dependence structure we obtain the following tranche spreads for the equity, the mezzanine, and the senior tranche: 1741 bps, 886 bps, and 14.5 bps. We obtain a risk allocation to the CDO tranches that differs widely from any risk allocation that could be obtained by a flat correlation matrix. The equity tranche spread of 1741 bps can be reproduced by a flat correlation matrix with a correlation level of 0.45. Likewise, the senior tranche spread of 14.5 bps can be reproduced by a flat correlation matrix with a correlation level of 0.23. However, the high level of risk that is allocated to the mezzanine tranche by the correlation structure in Figure 3.10 cannot be reproduced by any flat correlation structure. This can be seen in Figure 3.2, where all mezzanine tranche spreads are far below 886 bps for flat correlation structures. Thus, when the mezzanine tranche spread is 886 bps we cannot derive an implied correlation parameter. Figure 3.10 displays the existing implied tranche correlations for the equity and the senior tranche, while the implied tranche correlation for the mezzanine tranche is absent.


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Figure 3.10 The existence problem.

3.8.1.2

The Uniqueness Problem

As mentioned above, implied correlations for mezzanine tranches can suffer from uniqueness problems. Consider a correlation matrix with one sector of high correlation consisting of 25 assets. The intra sector correlation is 0.8 and the inter sector correlation is 0.3. The correlation matrix in Figure 3.11 describes the corresponding correlation structure. Using this dependence structure we obtain the following spreads for the equity, the mezzanine, and the senior tranche: 2190 bps, 607 bps, and 25.0 bps. The equity tranche spread of 2190 bps can be reproduced by a flat correlation matrix with a correlation level of 0.34. Likewise, the senior tranche spread of 25.0 bps can be reproduced by a flat correlation matrix with a correlation level of 0.35. However, the mezzanine tranche spread of 607 bps can be reproduced by two different flat correlation matrices with correlation levels 0.02 and 0.39. In this study we proceed in the following way: whenever we have the choice between two correlation levels that reproduce a given tranche spread, we choose the correlation level that is closer to the implied tranche correlation of the equity tranche. Figure 3.11 illustrates the uniqueness problem of the implied tranche correlation of the mezzanine tranche. In this chapter, we do not turn our attention to the direction of the smile. We solely analyze the implied correlation curve to figure out whether the risk allocation inherent to a given dependence structure can be reproduced by a homogeneous dependence structure (this would be illustrated by a flat implied correlation curve). In this respect, the direction of the smile does not influence our line of argument.

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Figure 3.11 The uniqueness problem.

3.8.1.3

Exemplary Heterogeneous Matrices

Consider again a portfolio that consists of 10 non-overlapping commensurate sectors of high correlation against a background of low correlation. Each sector comprises 10 assets. We analyze the influence of varying intra and inter sector correlations on the shape of the smile (see Figure 3.12). First, we keep the intra sector correlation constant at 0.7 and vary the inter sector correlation between 0.3 and 0.5. We find that these correlation structures lead to correlation smiles. In Figure 3.13, we illustrate the impact of keeping the intra sector correlation constant and varying the inter sector correlation on the shape of the smile. Likewise, keeping the inter sector correlation constant at 0.3 and varying the intra sector correlation between 0.5 and 0.7 results in correlation smiles, as can be seen in Figure 3.14. The more the matrices differ from a flat correlation matrix, the more pronounced is the smile. Now we consider a portfolio where the assets are divided into non-overlapping sectors of equal size. We vary the size and consequently the number of these sectors. We analyze 20, 10, 4, and 2 sectors of 5, 10, 25, and 50 assets each. The intra sector correlation is 0.7, the inter sector correlation is 0.3. We observe that an expanding cluster size influences the smile (see Figure 3.15). When the underlying contains 4 clusters of 25 assets each, the observed smile is more pronounced than the smile that is caused by 20 clusters of 5 assets each. However, the smile effect is only intensified up to a certain cluster size. Finally, we consider a portfolio that contains only one single sector of high correlation. All other assets have low pairwise correlations. The size of the cluster varies between 10


63

Pairwise Linear Correlation

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Figure 3.12 Varying the intra sector correlation and the inter sector correlation influences the implied correlation smile. 0.55

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Figure 3.13 Correlation smiles produced by correlation matrices with 10 clusters of equal size. The intra sector correlation is 0.7, the inter sector correlation varies between 0.3 and 0.5. and 75 assets. We assume that the intra sector correlation is 0.8 and the inter sector correlation is 0.2. Positive and negative correlation smiles emerge depending on the cluster size (see Figure 3.16). Up to a certain cluster size, the smile is aggravated by increasing the size of the cluster.

3.8.2

Different Dependence Structures Can Lead to Identical Implied Tranche Correlations

It is intuitively clear that different correlation structures can lead to identical tranche spreads and therefore to identical implied tranche correlations. For instance, consider the following three dependence structures shown in Figure 3.17. These dependence

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Implied Correlation

0.38 0.36 0.34 0.32 0.3

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3%−10% Tranche

10%−100%

Figure 3.14 Correlation smiles produced by correlation matrices with 10 clusters of equal size. The inter sector correlation is 0.3, the intra sector correlation varies between 0.5 and 0.7. structures cannot be transformed into each other by reordering3 the underlying assets. They all lead to identical tranche spreads and consequently to the same implied tranche correlations. Moreover, this example clarifies that it is not readily apparent how a certain correlation structure allocates the risk to the CDO tranches.

3.8.3

Heterogeneous Dependence Structures Do Not Necessarily Lead to Implied Correlation Smiles

Consider a flat correlation matrix with uniform default time correlation 0.5. We compute the CDO spreads for the equity, the mezzanine, and the senior tranche using this flat correlation matrix. From the tranche spreads we derive the implied correlations. Trivially, we get the same implied correlation parameter of 0.5 for all three tranches. Obviously, this example does not lead to a correlation smile. However, we would have obtained the same tranche spreads and the same implied correlations for the equity, the mezzanine, and the senior tranche if we had used the following correlation matrix (see Figure 3.18): one consisting of five commensurate clusters of 20 assets each. The intra sector correlations are (0.9754, 0.8994, 0.6069, 0.4700, 0.4281) the inter sector correlation is 0.3911. In this special case, a flat correlation matrix and a clustered correlation matrix produce identical tranche spreads. If we wanted to price a CDO whose underlying had the clustered correlation structure mentioned above, we could alternatively use the standard market model with correlation parameter 0.5. This would lead to correct CDO margins. 3

In Section 5.5, we illustrate how to represent correlation matrices in a systematic way.

3.9. Conclusion

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However, the last-mentioned example is an exception. Most realistic correlation matrices lead to correlation smiles, i.e. most realistic correlation matrices lead to an allocation of credit risk to the CDO tranches that cannot be reproduced by flat correlation structures. In general, using the standard market model with only one single correlation that describes all pairwise default time correlations leads to severe mispricing, when the actual underlying correlation structure is inhomogeneous.

3.8.4

Heterogeneous Dependence Structures Allow for Flexible Portfolio Loss Distributions

The risk content of the CDO tranches is determined by the probability distribution of the portfolio loss. In Chapter 2.2.3.6, we already discussed the shape of the portfolio loss distribution for flat correlation matrices with different levels of correlation. Using heterogeneous correlation matrices, we can generate a wide variety of different risk profiles. This can be illustrated by displaying the portfolio loss distribution for different dependence structures. Using heterogeneous correlation matrices, we are able to generate portfolio loss distributions that could never be reproduced by flat correlation matrices. The following Figure 3.19 illustrates exemplarily how the shape of the portfolio loss distribution differs from the typical shape generated by flat correlation matrices as we allow for heterogeneous correlation structures. Again we use the setup of Section 2.2.3.6, but, as mentioned above, we use heterogeneous dependence structures. We consider a homogeneous collateral pool of 125 defaultable assets with equal nominal value. We assume that all assets have a default probability of 5%, a recovery rate of 0%, and a time to maturity of 5 years. We display different correlation matrices and the corresponding portfolio loss distributions.

3.9

Conclusion

In this chapter, we outlined that the explanatory power of the implied tranche correlation and the implied base correlation is rather questionable, because these parameters merely provide aggregated information about a certain segment of the portfolio loss distribution. Relative value considerations or valuations of off-market tranches by means of an implied correlation parameter can lead to serious mispricing, because different correlation levels are applied for the valuation of different tranches on the same underlying portfolio. The implied tranche correlation smile and the implied base correlation skew illustrate the

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deficiency of the standard market model, which uses flat correlation matrices, when it comes to pricing CDO tranches consistently. We outlined that the use of heterogeneous dependence structures can lead to an allocation of risk to the different tranches that is similar to the risk allocation observed in the market. We exemplified that heterogeneous dependence structures can lead to tranche spreads that generate characteristic implied correlation smiles. We considered different correlation matrices and examined the resulting correlation smiles. We chose the extended standard market setup to ensure that the implied correlation smiles are solely due to the correlation matrix. Finally, we exemplified that heterogeneous correlation structures offer a vast variety of different portfolio loss distribution structures. Our study indicates that the implied correlation smile that is observed in the CDO market may be caused by an inhomogeneous correlation structure. We suggest taking the implied correlation approach one step further and deriving a dependence structure that reproduces all tranche spreads of a traded CDO simultaneously. If the implied correlation smile emerges because the actual dependence structure is not described adequately, using the appropriate heterogeneous correlation structure will allow for consistent pricing.

3.9. Conclusion

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Figure 3.15 Varying the size and the number of sectors influences the correlation smile. The intra sector correlation is 0.7, the inter sector correlation is 0.3.

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Figure 3.16 When there is only one sector of high correlation, the cluster size influences the correlation smile. The intra sector correlation is 0.8, the inter sector correlation is 0.2.

3.9. Conclusion

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Figure 3.17 Correlation matrices differing in shape but leading to identical implied tranche correlations.

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Figure 3.18 The flat correlation matrix and the correlation matrix with 5 clusters of 20 assets each lead to identical tranche spreads and consequently to identical implied correlation parameters. These correlation matrices do not lead to correlation smiles.

3.9. Conclusion

Figure 3.19 Portfolio loss distribution for heterogeneous correlation matrices.

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Chapter 4 Optimization by Means of Evolutionary Algorithms 4.1

Introduction

In the preceding Chapter 3, we presented a possible explanation for the inability of the standard market approach to fit quoted CDO tranche prices and to model the correlation smile. We suggested overcoming the deficiency of the standard market model by means of non-flat dependence structures. In the subsequent Chapter 5, we will explain how a correlation matrix can be derived from observed tranche spreads such that all tranche spreads of the CDO structure are reproduced simultaneously. This idea can be represented in the form of an optimization problem. This Chapter 4 addresses optimization algorithms. Life in general and the domain of finance in particular confront us with many opportunities for optimization. Optimization is the process of searching for the optimal solution in a set of candidate solutions, i.e. the search space. Optimization theory is a branch of mathematics which encompasses many different methodologies of minimization and maximization. In this chapter we represent optimization problems as maximization problems, unless mentioned otherwise. The function to be maximized is denoted as objective function. Optimization methods are similar to approaches to root finding, but generally they are more intricate. The idea behind root finding is to search for the zeros of a function, while the idea behind optimization is to search for the zeros of the objective function’s derivative. However, often the derivative does not exist or is hard to find. Another difficulty with optimization is to determine whether a given optimum is the global or only a local optimum.

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Chapter 4. Optimization by Means of Evolutionary Algorithms

There are many different types of optimization problems: they can be one- or multidimensional, static or dynamic, discrete or continuous, constrained or unconstrained. Sometimes even the objective function is unknown. In line with the high number of different optimization problems, many different standard approaches have been developed to finding an optimal solution. Standard approaches are methods that are developed for a certain class of problems (though not specifically designed for an actual problem) and that do not use domain-specific knowledge in the search procedure. In case of a discrete search space, the most simple optimization method is the total enumeration of all possible solutions. Needless to say, this approach finds the global optimum but is very inefficient especially when the problem size increases. Other approaches like linear or quadratic programming utilize special properties of the objective function. Possible solution techniques for nonlinear programming problems are local search procedures like the gradient-ascent method, provided that the objective function is real-valued and differentiable. Most local search methods take the approach of heading uphill from a certain starting point. They differ in deciding in what direction to go and how far to move. If the search space is multi-modal (i.e. it contains several local extrema), the local search methods will all run the risk of being stuck in a local optimum. But even if the objective function is not differentiable or if the search space is multi-modal, there will still be some standard approaches that deal with these kinds of problems. For example, global search procedures evaluate a larger part of the search space to take multi-modality into account. However, when the optimization problem is very complex, the number of available algorithms becomes noticeably smaller. Due to their complexity, many realistic optimization problems cannot be solved using standard techniques. In these cases often Evolutionary Algorithms (EAs) are used. In this chapter we will illustrate the idea behind EAs. We will discuss the general scheme of EAs, and we will comment on several well-established evolutionary operators. Some of the information summarized here is taken from Bäck (1996) and Haupt and Haupt (2004). Further sources for this illustration are Eiben (2003), Man et al. (1999), and Mitchell (1996).

4.2

Evolutionary Algorithms

EAs are stochastic search methods that are inspired by organic evolution. They are used to find solutions to complex optimization problems by applying the principles of evolutionary biology to computer science. EAs are typically implemented as computer

4.2. Evolutionary Algorithms

75

simulations in which a population of abstract representations of possible solutions evolves toward better solutions. EAs model the collective learning process within a population. Each individual is defined as a point in the search space. The starting population is initialized by an algorithm-dependent method, and often it is initialized at random. This population is supposed to evolve toward successively better regions of the search space by applying randomized operators of selection, recombination and mutation. The application of selection, recombination, and mutation leads to a new population which replaces the current population in the next iteration of the algorithm. A predefined objective function (often denoted as fitness function) indicates the quality of a certain point in the search space. Inspired by the Darwinian theory of evolution, individuals with higher fitness levels have a greater chance to reproduce than individuals of lower quality. Recombination allows for mixing of parental information, while mutation introduces new characteristics. The combined application of variation and selection generally leads to improving fitness values in consecutive populations. EAs include Evolution Strategies, Genetic Algorithms, Genetic Programming and Evolutionary Programming. Evolution Strategies were developed in Germany by Rechenberg (1973) and Schwefel (1975) in the area of combinatorial optimization (see also Rechenberg (1994) and Schwefel (1995)). Evolution Strategies work with n-dimensional vectors of real numbers as representations of the different individuals. Selection, mutation, and adaption of mutation rates are the most important working mechanisms. The basic principles of Genetic Algorithms were proposed by Holland (1975) in his study on adaptive systems and were further advanced by Goldberg (1989). In Genetic Algorithms solutions are traditionally represented in binary strings, but real-valued encodings are also possible. Selection and recombination are the most significant operators. In the field of Genetic Programming, candidate solutions are represented by trees. In the field of Evolutionary Programming, candidate solutions are represented by finite state machines. The theory of EAs assumes that, at a very general level of description, EAs find optimal solutions by detecting, emphasizing, recombining, and mutating good buildingblocks of candidate solutions. Hence, the idea behind Evolution Strategies is that individuals with high fitness levels can be found in the vicinity of other individuals with high fitness levels. In turn, the idea behind Genetic Algorithms is that a good solution is composed of the buildingblocks of other good solutions.

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Generally, EAs are likely to be competitive with or to exceed standard methods if the optimization problem is very complex. They can handle optimization problems that are intricate due to high dimensionality, non-linearity, non-differentiability, and discontinuity of the objective function, noise in the measurements, multiple kinds of constraints, multimodality, and largeness of the search space or other unusual features of the search space. They are applied if the search space is not well understood and if finding a sufficiently good solution is enough. The fitness function can be based on numerically generated data, experimental data, or analytical functions. EAs are a very comfortable optimization technique when it comes to constraints. Both hard constraints and soft constraints can be incorporated. Hard constraints may not be violated by candidate solutions. Soft constraints may be violated, but if it happens, the individual conflicts with some of the requirements. There are several approaches to how EAs can handle these constraints. One possibility is to generate only legal individuals. Another option is to lower the fitness value of an individual that violates the rules (this works only for soft constraints). Alternatively, such an individual can be eliminated. Sometimes a repair mechanism can be applied to transform an illegal individual into a legal individual. Often optimization problems with constraints are formulated as multi-objective optimization problems. This is a class of optimization problems where EAs have been very effective. Many standard techniques have difficulties in optimizing several objectives at the same time. An EA searches the generations until a termination criterion is met. The termination criterion can be arbitrarily complicated (e.g. taking the diversity of the population into account) or it can be very simple (e.g. termination after a certain number of function evaluations). The most common termination criteria are to stop the algorithm after a certain number of function evaluations or after a certain time, after reaching a sufficiently good result, or after failing to improve the currently known best solution for a certain number of generations. EAs only require the output of the objective function to guide the selection process and a suitable representation of the potential solutions to apply the evolutionary operators to. They often start with a broad sampling of the search space, which reduces the risk of being trapped in a local optimum. However, similar to other heuristic search methods, there is no guarantee that EAs will find the global optimum, though they generally find good solutions in a reasonable amount of time. Additionally, EAs can provide a list of optimum numbers, not just one single solution. There is a large number of problems where EAs are successfully applied, but there are also many traditional problems where EAs are outperformed by other optimization methods. Especially when the search space is well understood, there are often algorithms using domain-specific heuristics that outperform general approaches like EAs.

4.3. Notation

77

The above-mentioned indications are merely loose guidelines that may help to predict whether an EA will be an effective search procedure for a given problem. However, before applying sophisticated EAs, it is reasonable to apply the basic implementations to get an idea of the characteristics of the search space (see Streichert and Ulmer (2005)). Assume that a Monte-Carlo Search performs as well as an Evolution Strategy or a Genetic Algorithm. Then the search space is likely to be either very flat or very craggy and non-causal. If a Hill-Climber or another local-search strategy outperforms a populationbased EA, then the search space is likely to be very simple with one global or only a few local extrema. One drawback of EAs is the high number of function evaluations required. Sometimes these evaluations are computationally expensive. To save computational time, in some cases it is sufficient to evaluate only a part of the objective function or to evaluate a substitute function to guess the quality of a candidate solution. Additionally, if a parallel computer is available, the different processors can evaluate the objective function of different individuals at the same time, since the function evaluations are independent. Therefore, EAs are optimally suited for parallel computations. Further non-trivial variants of parallelization are discussed in 4.6. Compared to a parallel implementation, a serial computation slows down the problem-solving process, especially when a large population size is considered.

4.3

Notation

In the following, EAs will be presented in a language obtained by mixing pseudocode and mathematical notations. We concentrate on the main components of an EA like the population of individuals and the selection, recombination, and mutation operators as well as the evaluation process. We keep the notation as general as possible to cover different specifications of EAs, especially Evolution Strategies and Genetic Algorithms. The notation used in Section 4.3 is partly taken from Bäck (1996). Definition: General Evolutionary Algorithm An Evolutionary Algorithm (EA) is defined as an 8-tuple EA = (I, Φ, Ω, Ψ, s, ι, μ, λ) where I is the search space containing all individuals. Φ : I → R is the objective function that assigns a certain fitness value to the individuals. Ω is a set of probabilistic genetic operators like the recombination operator r and the mutation operator m.

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The mutation operator is generally a Gaussian perturbation that delivers a slightly modified mutant. The selection operator s may change the number of individuals from λ or μ + λ to μ, where μ, λ ∈ N and μ ≤ λ. μ is the number of parent individuals, λ is the number of offspring individuals. Whenever μ = λ, the operator set Ω contains an operator that changes the population size from μ parent individuals to λ offspring individuals. This change is inverted by the selection operator in consideration of the fitness values of the candidates. Note that the genetic operators are always probabilistic, whereas the selection operator can be probabilistic or deterministic. Ψ : I μ → I μ describes the complete process of transforming a population P into a subsequent one by applying genetic operators and selection. ι : I μ → {true, false} denotes the termination criterion of the algorithm. The generation transition function Ψ, which transforms a population into the subsequent one, creates a population sequence. Definition: Population Sequence Let Ψ : I μ → I μ describe the generation transition function of an EA, and let P (0) ∈ I μ describe the initial population. Then the sequence P (0), P (1), P (2), ... is called a population sequence or evolution of P (0) if and only if P (t + 1) = Ψ(P (t)), for all t ≥ 0, i.e. the sequence results from a successive application of the random operator Ψ. We now present a general outline of an EA. Let t denote the respective generation, and let P (t) = {p1 (t), ..., pμ (t)} denote the population at t whereby pi (t), i = 1, ..., μ are the individuals and μ is the population size. Only the individuals in the set P (t) have the chance to reproduce. Each individual in P (t) has the same probability

1 μ

of becoming

a parent. Parents are chosen with replacement from the set P (t). Recombination and mutation is applied until λ individuals are created. Let P ∗ (t) = {p∗1 (t), ..., p∗λ (t)} denote the offspring population that is generated in generation t. When it comes to selecting the μ parents for the next generation P (t + 1), sometimes not only the current offspring individuals P ∗ (t) but also the parents from the current generation P (t) are taken into account. Q ∈ {∅, P (t)} denotes this set that might additionally be considered by the selection operator.

4.3. Notation

79

Objects forming possible solutions within the original problem context are referred to as phenotypes. Some algorithms use an additional solution representation, the so-called genotype. Often the evolutionary operators act only on the genotype. As mentioned above, Genetic Algorithms traditionally use a binary solution representation as genotype. However, real-valued representations are also possible. In our study we only consider real-valued encodings. Generally, the result of an EA is the individual that yields the best fitness value over the complete evolutionary cycle - it does not necessarily agree with the best individual in the final generation. General Scheme of an Evolutionary Algorithm t := 0; Initialization of P (0) : P (0) := {p1 (0), ..., pμ (0)}; Evaluation of P (0) : Φ(p1 (0)), ..., Φ(pμ (0)); while ι(P (t)) = true do Recombination: P ∗ (t) := r(P (t)); Mutation: P ∗ (t) := m(P ∗ (t)); Evaluation of P ∗ (t) : Φ(p∗1 (t)), ..., Φ(p∗λ (t)); Selection: P (t + 1) := s(P ∗ (t) ∪ Q); t := t + 1; end There are numerous different versions of EAs. The common idea behind all these techniques is the same: given a population of individuals, the environmental pressure causes natural selection, which results in rising fitness in the population. Most alternative EA implementations differ primarily in technical details. In the following, we give a short introduction to the algorithms used in this study. We mainly concentrate on Evolution Strategies and Genetic Algorithms, and we also discuss a simple Monte-Carlo Search. But first of all, we comment on some established types of selection, recombination, and mutation. The classification follows to some extent Streichert and Ulmer (2005) and Eiben (2003). Needless to say, there are numerous other variants that will be omitted here. Note that not all of the evolutionary operators mentioned below can be applied in every type of EA.

80

4.4


Evolutionary Operators

We distinguish between object variables and strategy variables. Object variables specify the point in the search space, while strategy variables specify the evolutionary parameters. For instance, the mutation parameters that are used in Evolution Strategies to control the mutation step size and the correlations between the mutations are evolutionary parameters. Whenever strategy variables are used, the solution representation containing the object variables is extended by the strategy variables. The evolutionary operators act on the object variables and the strategy variables. Thus, not only the object variables but also the strategy variables are part of the search process. This feature is called self-adaptation. The issue of changing parameter values during the evolution of an EA is an active area of research.

4.4.1

Selection

Let P = { p1 , ..., pk } denote the set from which μ individuals are chosen. Then we have s : I k → I μ , μ ≤ k. In the following, we consider different ways to define the selection operator s. 4.4.1.1

Elite Selection

The μ best individuals are selected (without replacement) from P to form the parent population in the next generation. 4.4.1.2

Tournament Selection

Choose the best individual of a randomly generated subset of P. Let the subset have a fixed size q ≤ k. Repeat the procedure (generating the subset and choosing with replacement) until μ individuals have been selected. The size of the subset determines the selection pressure. 4.4.1.3

Proportional Selection

Every individual has the chance to be selected with a probability that is proportional to its fitness value. Let P = { p1 , ..., pk } be the set of individuals where it is chosen from. Let {f1 , ..., fk } denote the respective fitness values. Then every individual pi is selected with

fi . This formulation has to be adjusted when a minimization problem Σki=1 fi is considered. Choose the individuals (with replacement) until μ individuals have been

probability selected.

4.4. Evolutionary Operators

4.4.2

81

Recombination

The recombination operator can act on an arbitrary number of individuals from the parent population, but in this study we assume that it acts on only two individuals, i.e. r : I 2 → I. Additionally, in Section 4.4.2 we assume that the individuals are ndimensional vectors of real numbers. Let pα = (xα1 , ..., xαn ) and pβ = (xβ1 , ..., xβn ) be the parent individuals, α, β ∈ {1, ..., μ} and let p∗ = (x∗1 , ..., x∗n ) be the offspring individual. Then we have r(pα , pβ ) = p∗ . In the following, we consider different ways to define the recombination operator r. 4.4.2.1

Flat Crossover

p∗ is a random variable that is uniformly distributed over the range given by the parents, i.e. x∗i ∼ U(min(xαi , xβi ), max(xαi , xβi )), for all i = 1, ..., n. 4.4.2.2

Discrete N-Point Crossover

N ∈ {1, ..., n − 1} points are chosen where the vector components are swapped. For instance, for 1-point crossover and a swapping point i we have p∗ = (xα1 , ..., xαi , xβi+1 , ..., xβn ). 4.4.2.3

Discrete Uniform Crossover

Every single component of the offspring genotype is randomly chosen from the respective components of the parent genotypes, i.e. x∗i ∈ {xαi , xβi }, for all i = 1, ..., n. 4.4.2.4

Intermediate Crossover

The offspring genotype is the mean of the parent genotypes, i.e. p∗ = 4.4.2.5

pα +pβ . 2

Arithmetical Crossover

The offspring genotype is a linear combination of the parent genotypes, e.g. x∗i = γxαi + δxβi , where γ > 0 and δ > 0 are random variables such that γ + δ = 1.

4.4.3

Mutation

Mutation is an asexual operator m : I → I. In this study we assume that it modifies an individual by a Gaussian perturbation. Let z denote a realization of a standard normally distributed random variable, and let zi indicate that the variable is sampled anew for each value of i. Again, we assume that the individuals are n-dimensional vectors of real numbers. Let p = (x1 , ..., xn ) be the parent individual, and let p∗ = (x∗1 , ..., x∗n ) be the

82


offspring individual. Then we have m(p) = p∗ . In the following, we consider different ways to define the mutation operator m. 4.4.3.1

Standard Mutation

One randomly selected element xi is mutated. We obtain x∗i = xi + zi . 4.4.3.2

Global Mutation Without Strategy Parameter

Each element xi is mutated for all i = 1, ..., n. We obtain x∗i = xi + zi . 4.4.3.3

Global Mutation With Strategy Parameter

The genotype is p = (x1 , ..., xn , σ). One strategy parameter σ is used. It controls the mutation step size. All objective variables (x1 , ..., xn ) are mutated, so too is the strategy variable σ. Mutation is formulated as follows: m(p) = p∗ with p∗ = (x∗1 , ..., x∗n , σ ∗ ), whereas σ ∗ = σeτ z and x∗i = xi + σ ∗ zi for all i = 1, ..., n and τ ∝

√1 . n

The parameter σ

is bounded below by a minimum value. 4.4.3.4

Local Mutation

The genotype is p = (x1 , ..., xn , σ1 , ..., σn ). There is one strategy variable for each object variable. Therefore, the mutation step size of each dimension can be adjusted separately. Mutation is formulated as follows: m(p) = p∗ with p∗ = (x∗1 , ..., x∗n , σ1∗ , ..., σn∗ ), whereas σi∗ = σi eτz+τ zi and x∗i = xi + σi∗ zi for all i = 1, ..., n and τ ∝ √1 , τ ∝ √ 1√ . Again, the 2n

2 n

parameters σi are bounded below by given minimum values. 4.4.3.5

1/5-Rule

For the (1 + 1)-ES that will be explained below, Rechenberg (1973) developed a convergence rate theory for n 1 for two characteristic model functions. The so-called 1/5-rule constitutes a classical adaptive method for setting the mutation rate. Rechenberg points out that the ratio of mutations in which the offspring is fitter than the parent to all mutations should be 1/5. Whenever the ratio is greater than 1/5, the standard deviation of mutation is increased; whenever the ratio is less than 1/5, the standard deviation of mutation is decreased. Depending on the ratio, the standard deviation is multiplied or divided by an adjustment factor which should be in [0.817, 1]. The adjustment is performed every n mutations.

4.5. Basic Algorithms

4.5 4.5.1

83

Basic Algorithms Evolution Strategies

As far as selection mechanisms are concerned, Schwefel (1975) introduced an elegant notation characterizing the number of parent and offspring individuals in Evolution Strategies (ESs). He distinguishes between a (μ + λ)- and a (μ, λ)-strategy. The former selects the μ best individuals out of the union of μ parents and λ offspring to form the next parent generation, while the latter selects the μ best individuals out of the offspring only. The first applications of ESs were experimental and dealt with hydrodynamic problems. At that time a two-membered (1 + 1)-strategy (Hill-Climber) was used.

4.5.1.1

Hill-Climber

The Hill-Climber starts with an individual randomly chosen from the search space. In each generation mutation is applied to the individual. Naturally, the crossover procedure is omitted. Mutation is the only variation operator doing the whole search work. The objective function is evaluated and the better individual is kept to breed the next generation. This search procedure is repeated until we get a sufficiently good result or until a given termination criterion is met. The Hill-Climber can be very efficient in simple unimodal search spaces. In multi-modal search spaces, it often converges too fast to a local optimum. Multi-start Hill-Climbers where several local Hill-Climbers start simultaneously from randomly chosen points in the search space can reduce the risk of getting stuck in a local optimum. Another alternative that reduces the risk of premature convergence is Simulated Annealing (see Streichert and Ulmer (2005)). This method can be compared to a Hill-Climber, only the selection scheme is less strict, since it allows short-term degradation of the solution quality. General Scheme of a Hill-Climber t := 0; Initialization of P (0): P (0) := {p1 (0)}; Evaluation of P (0): Φ(p1 (0)); while ι(P (t)) = true do Cloning: p∗1 (t) := p1 (t); Mutation: p∗1 (t) := m(p∗1 (t));

84


Evaluation of p∗1 (t): Φ(p∗1 (t)); Selection: p1 (t + 1) := s(p1 (t) ∪ p∗1 (t)); (Elite Selection) t := t + 1; end 4.5.1.2

The (μ, λ)-Strategy and the (μ + λ)-Strategy

The (μ, λ)-strategy and the (μ + λ)-strategy incorporate the population principle. These strategies start with μ randomly chosen candidates from the search space. Executing recombination and mutation leads to λ new candidates. As mentioned above, ESs focus on the application of very sophisticated mutation operators. The mutation probability is relatively high and the recombination probability relatively low. Therefore mutation eclipses recombination, the recombination operator being sometimes entirely omitted. Whenever the recombination probability is low, the ES has the character of a localized search. In case recombination is applied, the traditional recombination operators of ESs are discrete crossover and intermediate crossover. After creating the offspring individuals and calculating their fitness, the best μ are chosen deterministically (elite selection). In case of the (μ, λ)-strategy, the individuals are chosen from the offspring only, in case of the (μ + λ)-strategy, the individuals are chosen from the offspring and the parents. Often a (μ, λ)-ES is used instead of a (μ + λ)-ES to reduce the risk of premature convergence. Experiments show that both the optimal mutation rate and the convergence velocity increase as λ increases. The generations are searched until an individual with a sufficiently high fitness level is obtained or until a given termination condition is met. Often the population size μ of an ES is small and the selection pressure relatively high due to the deterministic selection operator. Note that ESs are not as closely connected to the natural archetype, mainly because they only use elite selection. The Hill-Climber as described above corresponds to a (1 + 1)-ES. General Scheme of an Evolution Strategy t := 0; Initialization of P (0): P (0) := {p1 (0), ..., pμ (0)} ∈ I μ ; Evaluation of P (0): Φ(p1 (0)), ..., Φ(pμ (0)); while ι(P (t)) = true do Recombination: P ∗ (t) := r(P (t)) ∈ I λ ; Mutation: P ∗ (t) := m(P ∗ (t)); Evaluation of P ∗ (t): Φ(p∗1 (t)), ..., Φ(p∗λ (t)); Selection: (Elite Selection)

4.5. Basic Algorithms

85

if (plus-Strategy) then P (t + 1) := s(P ∗ (t) ∪ P (t)); else P (t + 1) := s(P ∗ (t)); end t := t + 1; end

4.5.2

Genetic Algorithms

Genetic Algorithms (GAs) are likely to be the most popular types of EAs. They are applied by biologists who want to simulate biological systems but are also applied by researchers from other fields. Holland (1975) pointed out that the natural search procedure can be applied to many real-world problems. Holland developed a theory of adaptive systems that communicate with their environments. In general, the initial population of a GA(μ) is created at random. Let μ denote the population size. The idea behind GAs is to create high quality individuals by recombining useful segments of their parents. For that reason, sophisticated crossover operators are applied. All μ parent individuals have the same probability to reproduce. Recombination and mutation lead to μ new individuals, i.e. λ = μ. The recombination operator is the most important evolutionary operator of a GA. The recombination probability is relatively high. In analogy to the natural model, the mutation probability is usually very small. Mutation serves only as an operator introducing new characteristics into the population. From these μ individuals we again choose μ individuals with replacement based on their fitness levels by means of elaborate selection operators. These individuals finally form the offspring population. In this study we focus on generational GAs. In case of a generational GA, the parent population of the next generation is given by the offspring of the current generation. Often GAs use binary representations, but in this study we use real-valued encodings. General Scheme of a Genetic Algorithm t := 0; Initialization of P (0): P (0) := {p1 (0), ..., pμ (0)}; Evaluation of P (0): Φ(p1 (0)), ..., Φ(pμ (0)); while ι(P (t)) = true do Recombination: P ∗ (t) := r(P (t)) ∈ I μ ; Mutation: P ∗ (t) := m(P ∗ (t));

86


Evaluation of P ∗ (t): Φ(p∗1 (t)), ..., Φ(p∗μ (t)); Selection: P (t + 1) := s(P ∗ (t)); t := t + 1; end

4.5.3

Monte-Carlo Search

A Monte-Carlo Search is a random optimization strategy without any feedback. In each generation it randomly generates and evaluates individuals. The best solution that has been found so far is memorized. A Monte-Carlo Search can work with an arbitrary population size. General Scheme of a Monte-Carlo Search t := 0; Initialization of P (0): P (0) := {p1 (0), ..., pμ (0)}; Evaluation of P (0): Φ(p1 (0)), ..., Φ(pμ (0)); Save the Best Solution: pbest = s(P (0)); (Elite Selection) while ι(P (t)) = true do t := t + 1; Initialization of P (t): P (t) := {p1 (t), ..., pμ (t)}; Evaluation of P (t): Φ(p1 (t)), ..., Φ(pμ (t)); Save the Best Solution: pbest = s(P (t) ∪ pbest ); (Elite Selection) end

4.6

Parallel Algorithms

An intuitive but trivial approach to the parallelization of EAs consists in computing the fitness evaluations of different individuals on different computers. This idea makes sense actually, because the fitness evaluations constitute the main part of the computational burden. Apart from this intuitive parallelization, there are further non-trivial variants of parallel implementation. Haupt and Haupt (2004), Bäck (1996), and Goldberg (1989) discuss advantages of parallel EAs and strategies for the parallelization, as well as the implementation of EAs on parallel architectures. Haupt and Haupt emphasize that the nature of the problem and the architecture of the parallel machine decide which parallel EA may work best. Alga and Tomassini (2002), Folino et al. (2001), and Cantú-Paz (2000) are recent publications that deal with the parallelization of EAs.

4.6. Parallel Algorithms

87

In the following, we place emphasis on population models and outline why they are particularly useful for our optimization problem. Population models structure the individuals within a population. We distinguish between global, local, and regional population models1 , following the depiction of Eiben and Smith (2003) and Weicker (2002). Population models differ in the definition of the selection pool. The selection pool describes the part of the population that contains all individuals constituting mutual reproduction partners. Note that we differentiate between population models and population-based models. The term population-based models merely indicates that the model incorporates the population principle.

4.6.1

Global Population Models

The global population model corresponds to the classical EA. In global population models, the population is not structured or divided into subpopulations. All individuals can be combined to produce offspring individuals. The selection pool includes the whole population.

4.6.2

Local Population Model

In local population models (so-called fine-grained models) for each individual a neighborhood is defined. Only individuals from one neighborhood can mate and produce offspring individuals. The selection pool is composed of all individuals sharing the same neighborhood. The information slowly diffuses through the population. Local population models maintain the population’s diversity for a relatively long period of time as a result of niching.

4.6.3

Regional Population Models

In regional population models (so-called coarse-grained models) the population is divided into several disjoint subpopulations. Only individuals from one subpopulation can be combined to produce offspring individuals. The selection pool is composed of all individuals of a subpopulation. For a certain number of generations, the subpopulations evolve independently of each other. From time to time, selected individuals are exchanged between the subpopulations (migration). The genetic diversity in the subpopulations is determined by the number of exchanged individuals and the number of generations between two migrations. Local and regional population models often keep the algorithm 1 Termination is not standardized, e.g. in Bäck (1996) local population models and regional population models are denoted as diffusion models and migration models, respectively.

88


from premature convergence. In the following, we concentrate on regional population models as opposed to global population models. We already pointed out that EAs are optimally suited for parallel computing platforms since the fitness evaluations of different individuals are independent and can be carried out on different processors. When the number of available processors divides the number of individuals, the upcoming tasks can be allocated to the processors in equal shares. However, it is hard to use an arbitrary number of processors (which may even change over time) to full capacity because not all processors are concurrently occupied with the same number of tasks. Thus, some processors may have to wait until the other processors finished their computations. Furthermore, for the purpose of communication and coordination, a control unit is required to schedule the tasks and collect the results in every generation. This transaction additionally decelerates the optimization process. To deal with both above mentioned problems, we can apply the concept of regional population models. Each processor that participates in the evolutionary search manages a subpopulation. The number of subpopulations corresponds to the number of available processors and can be adjusted dynamically without any problem. Thus, the workload is equally balanced between the processors. The subpopulations evolve independently of each other, except for the exchange of selected individuals which takes place after a certain number of generations. Thus, we reduce the overhead of communication.

4.7

Evolutionary Algorithms in Finance: Literature

EAs have been applied successfully to many areas of economics. We can find publications outlining the general relevance of EAs to economic modeling. Other publications can be allocated to game theory, macroeconomics, econometrics, and finance. Apart from these areas, EAs have been used to model the formation process of risk preferences, and EAs also found their way into more uncommon fields of economic activity. In this section, we provide an insight into the application of EAs to financial economics. In parts we follow the survey of Chen (2002). Especially the area of financial economics has attracted a lot of intensive attention from EA researchers. One major field of research is agent-based artificial financial markets, which includes stock markets, foreign exchange markets, future markets, and doubleauction markets. Only a few studies have focused on commodity futures and artificial

4.7. Evolutionary Algorithms in Finance: Literature

89

derivative markets. Another major field is financial engineering. In financial engineering there are publications on a variety of topics, all of them undergoing extensive study: financial forecasting, portfolio optimization, volatility modeling, option pricing, etc. EAs are very popular in financial engineering because they can handle very complex problems. In the following, we list several publications from the above-mentioned fields. LeBaron (2001), Joshi, Parker, and Bedau (1999), Joshi and Bedau (1998), and Palmer et al. (1994) deal with artificial stock markets and EAs. Lux and Schornstein (2005), Arifovic and Gençay (2000), Izumi and Ueda (2000), Izumi and Ueda (1998), and Arifovic (1996) apply EAs in the area of foreign exchange markets. Chen and Kuo (1999) and Chen and Yeh (1997a) deal with EAs in the context of future markets. Chen (2000) and Dawid (1999) connect EAs with double-auction markets. For EAs, financial forecasting is an area of application in financial engineering that is intensively researched. Liu and Yao (2001), Tsang et al. (2000), Kaboudan (1999), and Mahfoud and Mani (1996) use EAs in the context of financial forecasting. O’Neill et al. (2001) and Baglioni et al. (2000) use EAs for portfolio optimization. Chen and Yeh (1997b) deal with EAs in the context of volatility modeling. In the field of option pricing, EAs and in particular Genetic Programming can be applied to approximate the true option pricing formula. It is assumed that the true option pricing formula is made up of a selection of equations describing the contract terms and of other analytical properties describing the behavior of the stock price. In order to start from a locally optimal solution, the model of Black and Scholes (1973) is often introduced in the initial gene pool. It can be shown that the resulting formulas approximate the true solution better than the Black-Scholes model. Researchers proceeding as described above are Chidambaran et al. (1998). Further studies that connect the area of option pricing with EAs are Jay White (1998), Chidambaran et al. (1998), Keber (1999), and Hamida and Cont (2005).

Chapter 5 Evolutionary Algorithms in Finance: Deriving the Dependence Structure 5.1

Introduction

Correct modeling of default time dependence is essential for the valuation of multi-name credit derivatives, because structured products are extremely sensitive to the level of correlation and the shape of the correlation structure (see Sections 2.2.3.6, 3.2, and 3.8.4). Although it is still of interest to find empirical sources of correlation data, people increasingly use the CDO market to derive information about the dependence structure among the underlying assets of a CDO. In the CDO market, an observed tranche spread is considered to be a direct indicator of asset correlation. Increasingly often, a single implied correlation parameter is derived from the tranche spread of a traded CDO. The standard market model forms the basis for computation of the implied correlation parameter, assuming that all pairwise default time correlations are equal. The implied correlation parameter is supposed to reflect the level of dependence in the portfolio. However, we observe that different tranches of a CDO trade at different implied correlation levels, even though the underlying portfolio is the same for all tranches. This is illustrated by the emerging implied correlation smile, a topic we discussed in Chapter 3. The implied correlation of a traded CDO tranche is often used to price off-market products with the same underlying as the traded CDO. The correlation smile shows clearly that it is not appropriate to rely on the implied correlation of a traded CDO tranche for valuing non-standard tranches on the same collateral pool. Potential explanations for the smile have been discussed in Chapter 3. We outlined that an implied correlation smile can emerge when the underlying dependence structure is heterogeneous. Note that dependence structures of real-world portfolios are often

92

Chapter 5. Evolutionary Algorithms in Finance: Deriving the Dependence Structure

characterized by high levels of heterogeneity. This can be the reason for the implied correlation smile that is observed in the market. A flat correlation structure is generally not able to reproduce the portfolio loss distribution generated by a heterogeneous correlation structure. This explains why pricing models with flat dependence structures are often not able to reproduce the risk allocation to CDO tranches that is generated by non-flat dependence structures. If the implied correlation smile observed in the market emerges because the actual correlation structure is not described adequately, use of an appropriate heterogeneous correlation matrix will allow for consistent pricing. In this case, only one correlation matrix is used to price different CDO tranches, which complies with the fact that the underlying portfolio is the same. We point out that heterogeneous dependence structures have the potential to reproduce observed market prices. We suggest taking the implied correlation approach one step further. If the tranches of a CDO structure are quoted in terms of tranche spreads, we suggest deriving a correlation matrix that reproduces all quoted tranche spreads simultaneously. Analogously, if the CDO tranches are quoted in terms of implied tranche correlations, the correlation matrix is chosen such that the corresponding tranche prices lead to implied tranche correlations that are concordant with the quoted implied tranche correlations. Andersen and Sidenius (2004) mention this idea in passing. We do not primarily focus on finding a correlation matrix that coincides with the actual underlying correlation matrix. Our goal is rather to derive a correlation matrix that is consistent with all available market information. In this chapter, we will outline the suggested approach in detail. We will explain systematically how to derive a suitable dependence structure from observed market data by means of EAs. Furthermore, we will illustrate why EAs are appropriate for this kind of application. Chapter 5 is an extended version of Hager and Schöbel (2006) dealing with the setup of our suggested approach.

5.2

The Implied Correlation Structure

As outlined above, we opt for a dependence structure such that all resulting tranche prices are concordant with observed market prices. Analogously, the dependence structure is chosen such that the corresponding tranche prices lead to implied tranche correlations

5.3. The Optimization Problem

93

concordant with an observed correlation smile. After deriving a suitable correlation structure that is able to reproduce all observed CDO tranche spreads or, as the case may be, all observed implied tranche correlations, we can use this dependence to price off-market products with the same underlying. Using this implied correlation structure we can significantly reduce the risk of mispricing off-market derivatives. As the number of obligors in the portfolio increases, the number of free matrix parameters increases quadratically. Thus, for a large number of obligors, it may be disproportionate to imply a full correlation matrix. To reduce the complexity, simplified matrix representations can be considered. For example, we can calibrate the vector components in a one-factor approach. Alternatively, we can assume that the underlying assets belong to different highly correlated sectors, and then we can calibrate intra and inter sector correlations in a cluster approach. Further representations are possible. These simplified approaches are easier to handle because a smaller number of parameters has to be derived. However, the reduction of parameters inevitably leads to a reduction in the set of correlation matrices that can be represented by this approach. Thus, the reduction of parameters bears the risk that the actual correlation matrix cannot be represented using the simplified representation.

5.3

The Optimization Problem

As mentioned in Chapter 3, we assume that the extended standard market model constitutes the true pricing model. All assumptions of the standard market model apply (e.g. the intensity based approach describes the default time of one obligor; the Gaussian copula describes the dependence between the obligors’ default times; all assets have equal default intensities, equal recovery rates, and an equal nominal). Only the assumption that all pairwise default time correlations are the same is abandoned. All model parameters are known except for the pairwise correlations. Suppose that we know the tranche spreads of an actively traded CDO. Given the set of tranche spreads, the parameters defining the correlation matrix are chosen such that all tranche spreads are reproduced simultaneously. In this chapter we assume that the CDO consists of an equity tranche, a mezzanine tranche, and a senior tranche. Our goal is to derive a correlation matrix that replicates the given tranche spreads of the equity tranche, the mezzanine tranche, and the senior tranche simultaneously. Let these target values be denoted se , sm , and ss . It is intuitively clear that, in general, there can be more than one correlation matrix leading to identical tranche spreads se , sm , and ss . There may also be combinations of

94


tranche spreads that cannot be reproduced precisely by any correlation matrix. Unfortunately, it is not possible in our setup to derive the correlation matrix in closed form. Even the portfolio loss distribution and so too the tranche spreads cannot be computed in closed form for arbitrary dependence structures. Moreover, for certain correlation matrices the corresponding tranche spreads have to be obtained via Monte-Carlo simulation. In this case we have to deal with noise. Obviously, the optimization problem discussed in this study is rather complex because the search space is high dimensional and can be multi-modal. The objective function is non-linear, non-differentiable, and may be discontinuous. Often we have to deal with noise, and often constraints have to be considered. Since our optimization problem is characterized by these properties, the number of applicable optimization techniques is restricted. Therefore, we choose EAs to address this challenging problem.

5.4

Description of the Genotypes

In an EA, the first work step is to create a random initial population of candidate solutions in order to get a broad sampling of the search space. Therefore, we have to find a procedure that randomly generates valid dependence structures. Let Σ = (Σij )i,j=1,...,n denote the correlation matrix and let n denote the number of obligors in the portfolio. Correlation matrices are symmetric positive semi-definite matrices whose matrix elements are in [−1, 1]. That is, we have Σ = ΣT (symmetric), where ΣT denotes the transposed matrix of Σ, and xT Σx ≥ 0 for all x ∈ Rn×1 (positive semi-definite). The diagonal elements of a correlation matrix are ones. In the following, we introduce three approaches to describe the dependence, these being the Cholesky approach, the onefactor approach, and the cluster approach. In the Cholesky approach, we use a Cholesky decomposition C of the correlation matrix Σ as genotype. Without loss of generality, the correlation matrix Σ can then be represented as Σ = C T C. In the one-factor approach we use a row vector ρ = (ρi )i=1,...,n as genotype. The pairwise linear correlation between the default time of asset i and the default time of asset j can then be computed as Σij = ρi ρj , i, j = 1, . . . , n, i = j and Σij = 1, i = j. Using the one-factor approach, we can avoid Monte-Carlo simulations and provide semi-explicit expressions for CDO tranche spreads (see Laurent and Gregory (2005)). In the cluster approach, we assume that the correlation matrix Σ consists of several sectors of high correlation against a background of low correlation. Σ is then defined by the intra and inter sector correlations. After determining the correlation matrix Σ from the respective genotype, Σ can be used to derive the corresponding tranche spreads in the extended standard market

5.4. Description of the Genotypes

95

approach. Within the EA, the set of tranche spreads (or, as the case may be, the set of implied tranche correlations) obtained from a certain genotype represents the phenotype. Suppose we observe tranche spreads derived within the framework of the extended standard market model using an arbitrary correlation matrix that is known in advance. And suppose we want to reproduce these tranche spreads using the above-mentioned matrix representations. We expect to obtain a correlation matrix that either coincides with the original matrix or leads to the same risk allocation as the original matrix. However, it is possible that the one-factor approach or the cluster approach are not able to reproduce the given combination of tranche spreads. Since we know there is at least one matrix that reproduces the tranche spreads (the original matrix), in our setup, this indicates that the original matrix cannot be described by a one-factor representation or a cluster representation as the case may be, and that there is no other matrix leading to the same risk allocation as the original matrix. However, using the Cholesky approach we will always be able to reproduce the tranche spreads, since any symmetric matrix has a Cholesky factorization, as will be outlined below. Thus, in the Cholesky approach, the actual underlying correlation matrix can always be represented by a corresponding genotype. In the following, we will outline the peculiarities of the three different approaches that have to be considered as we perform the EA.

5.4.1

The Cholesky Approach

Correlation matrices are symmetric and positive semi-definite. Since any correlation matrix Σ is symmetric, Σ can be represented in the form Σ = C T C, where C is an upper triangular matrix C ∈ Rn×n with cij ∈ R for i = 1, ..., n, j = i, ..., n and cij = 0 for i = 2, ..., n, j = 1, ..., i − 1. This can be illustrated as follows (for symmetric matrices we only display the upper triangular): ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ T C C = ⎜ ⎜ ⎜ ⎜ ⎝

0

0

···

0

c12

c22

0

···

c13

c23

c33

0 .. .

c14 .. .

c24 .. .

c34 .. .

..

c1n

c2n

c3n

···

.

⎞

⎛ ⎟ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ 0 ⎠

c11

cnn

c11

c12

c13

c14

···

0

c22

c23

c24

···

0 .. .

0 .. .

c33

c34 .. .

···

0

0

···

0

c1n

⎞

c2n ⎟ ⎟ c3n .. .

cnn

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

96


⎛ ⎜ ⎜ ⎜ = ⎜ ⎜ ⎜ ⎝

c2 11

c11 c12

c11 c13

c11 c14

2 (c2 12 + c22 )

(c12 c13 + c22 c23 )

(c12 c14 + c22 c24 )

···

(c12 c1n + c22 c2n )

2 2 (c2 13 + c23 + c33 )

(c13 c14 + c23 c24 + c33 c34 )

···

(c13 c1n + c23 c2n + c33 c3n )

.

.

···

c11 c1n

. . .

.

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

2 2 (c2 1n + c2n + · · · + cnn )

⎛ ⎜ ⎜ ! ⎜ = ⎜ ⎜ ⎝

1 Σ12 1

Σ13

Σ14

···

Σ23

Σ24

···

1

Σ34 .. .

···

Σ1n

⎞

Σ2n ⎟ ⎟ Σ3n ...

⎟ ⎟=Σ. ⎟ ⎠

1

From the above-mentioned equation it is obvious that the matrix elements of C can be easily derived from the matrix elements of Σ using a recurrence formula. Especially, every column of C is normalized to unity. Consequently, any correlation matrix can be represented by a (not necessarily unique) Cholesky factorization. The following theorem states under which condition the Cholesky factorization is unique. Theorem (Cholesky Factorization) Any symmetric and positive definite matrix A ∈ Rn×n has a unique Cholesky factorization A = LT L, where L is an upper triangular matrix L ∈ Rn×n , i.e. lij ∈ R for i = 1, ..., n, j = i, ..., n and lij = 0 for i = 2, ..., n, j = 1, ..., i − 1. 5.4.1.1

Initialization

In the Cholesky approach, the first step in the evolutionary search process is to generate a population of upper triangular matrices. In this approach we have to calibrate

n(n+1) 2

free parameters. Note that a randomly generated upper triangular matrix C does not necessarily lead to a valid correlation matrix Σ = C T C. Therefore, we have to apply the following repair mechanism. 5.4.1.2

Repair Mechanism

A randomly generated upper triangular matrix C automatically leads to a symmetric positive semi-definite matrix Σ = C T C, since ΣT = (C T C)T = C T C = Σ and xT Σx = xT C T Cx = (Cx)T Cx ≥ 0 for all x ∈ Rn×1 . To make sure that the off-diagonal matrix elements Σij , i, j = 1, ..., n, i = j are in [−1, 1] and that the diagonal elements Σij , i = j are 1, we have to normalize the matrix C column-wisely. Define for that reason a matrix


C ∗ with matrix elements c∗ij = √

97 cij c21j +c22j +...+c2jj

. The resulting matrix C ∗ T C ∗ meets the

demands of a correlation matrix. We use the normalized upper triangular matrix not only to derive a valid correlation matrix but to maintain the modified representation as genotype. Thus, we redefine C such that C = C ∗ . The normalization procedure is carried out in each generation after the recombination and mutation operators have been applied. 5.4.1.3

Evaluation and Program Termination

Various functions can be taken to indicate the quality of an arbitrary genotype and of the corresponding correlation matrix Σ. To measure the quality of Σ, we can compute the corresponding equity, mezzanine, and senior tranche spreads se (Σ), sm (Σ), and ss (Σ) and compare them to the given target spreads se , sm , and ss . Consider e.g. the following function f1 , which measures the maximum relative deviation of the obtained tranche spreads from the target spreads: f1 (Σ) = max(

|se (Σ) − se | |sm (Σ) − sm | |ss (Σ) − ss | , , ). se sm ss

Alternatively, we can define the function f2 that measures the sum of the relative deviations of the obtained tranche spreads from the target spreads: f2 (Σ) =

|se (Σ) − se | |sm (Σ) − sm | |ss (Σ) − ss | + + . se sm ss

In order to assess the quality of a genotype with corresponding correlation matrix Σ, we can also compare implied tranche correlations. Assume that σe (Σ), σm (Σ), and σs (Σ) denote the implied tranche correlations that are derived from the equity, mezzanine, and senior tranche spreads se (Σ), sm (Σ), and ss (Σ), where Σ describes the correlation structure. We compare these implied tranche correlations with the implied tranche correlations σe , σm , and σs that are derived from the given target spreads se , sm , and ss . Consider exemplarily the function f3 , which measures the sum of the deviations of the obtained implied tranche correlations from the target implied tranche correlations: f3 (Σ) = |σe (Σ) − σe | + |σm (Σ) − σm | + |σs (Σ) − σs | .

98


Alternatively, squared deviations can be considered which is common practice in many optimization problems. Note that when the above-mentioned function definitions are used these functions have to be minimized, with low values standing for high quality. In a population-based optimization strategy with λ individuals, we neglect the overall performance of a certain generation and just take the best individual in that generation. In this study the objective function registers the quality of the best individual that has been generated so far. Let z(t) denote the objective function at time t, let Σi,τ denote the ith individual in generation τ , i ∈ {1, ..., λ}, τ ∈ {1, ..., t}, and let f (Σi,τ ) indicate the quality of the matrix Σi,τ . Consequently, in a population-based strategy, the value of the objective function in generation t that has to be minimized in the course of the generations is z(t) =

min

(f (Σi,τ )) .

i∈{1,...,λ}, τ ∈{1,...,t}

Note that often both se (Σi,τ ), sm (Σi,τ ), ss (Σi,τ ) and se , sm , ss are obtained by MonteCarlo simulation. In this case, we have to deal with approximations. We consider two cases of suitable termination conditions. Naturally, reaching the optimum of the objective function (0 in our case) with a certain precision should be used as a stopping condition. Therefore, we stop our algorithm as soon as the objective function falls below a predefined value. Furthermore, we terminate the EA as soon as the total number of function evaluations reaches a given limit. The same evaluation procedures and termination criteria are applied in the one-factor and in the cluster approach. 5.4.1.4

Recombination

In this study, we assume that the recombination operator acts on only two individuals. We select two candidates from the current parent population and use these individuals to create a new candidate. In case of the 1-point-crossover procedure, one random point p ∈ {1, ..., n−1} is selected where the columns of the parent matrices C 1 , C 2 are swapped to produce the offspring C.


⎛

C

1

⎜ ⎜ ⎜ = ⎜ ⎜ ⎜ ⎝ ⎛

C

2

⎜ ⎜ ⎜ = ⎜ ⎜ ⎜ ⎝ ⎛

⎜ ⎜ ⎜ C = ⎜ ⎜ ⎜ ⎝

5.4.1.5

99

c111

c112

···

c11p

0

c122

c12p

0 .. .

0 .. .

··· .. .

0

0

c211 0

c212 c222

0 .. .

0 .. .

0

0

c111 0

c112 c122

0 .. .

0 .. .

0

0

c11(p+1) c12(p+1)

···

c11n

···

c12n .. .

0

c1nn

···

c21n

···

c22n .. .

0

c2nn

···

c21n

···

c22n .. .

0

c2nn

··· ··· ··· .. .

c21p c22p

c21(p+1) c22(p+1)

c11p c12p

c21(p+1) c22(p+1)

··· ··· ··· .. . ···

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Mutation

In case of global mutation without a strategy parameter, we simply add realizations of a normally distributed random variable Z with expected value 0 to each matrix element. zij indicates that the random variable is sampled anew for each i = 1, ..., n, j = i, ..., n. Thereby we transform an individual C into the mutant C. ⎛ ⎜ ⎜ ⎜ C = ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ C = ⎜ ⎜ ⎝

c11

c12

c13

···

0

c22

c23

···

0 .. .

0 .. .

c33

··· .. .

0

0

···

0

c1n

⎞

c2n ⎟ ⎟ c3n .. .

⎟ ⎟ ⎟ ⎠

cnn

c11 + z11

c12 + z12

c13 + z13

···

0

c22 + z22

c23 + z23

···

0 .. .

0 .. .

c33 + z33

··· .. .

0

0

···

0

c1n + z1n

⎞

c2n + z2n ⎟ ⎟ c3n + z3n .. .

⎟ ⎟ ⎟ ⎠

cnn + znn

Alternative recombination or mutation schemes are carried out analogously.

100 Chapter 5. Evolutionary Algorithms in Finance: Deriving the Dependence Structure

5.4.2

The One-Factor Approach

5.4.2.1

Initialization

Assume now that we want to derive a correlation matrix created by a one-factor approach. In the one-factor approach, the genotype is a row vector. Thus, we start generating a population of vectors ρ ∈ [−1, 1]1×n . Using the one-factor approach, we have to calibrate n free parameters. 5.4.2.2

Repair Mechanism

A randomly generated vector ρ ∈ R1×n automatically leads to a symmetric positive = xT ρT ρx = T = (ρT ρ)T = ρT ρ = Σ and xT Σx = ρT ρ, since Σ semi-definite matrix Σ T n×1 (ρx) ρx ≥ 0 for all x ∈ R . But Σ is not a correlation matrix because the ith diagonal where D denotes entry is ρ2i and not 1. Thus, define a correlation matrix Σ as Σ = Σ+D, a diagonal matrix with 1 − ρ2i as the ith diagonal entry. Trivially, Σ is symmetric and semi-definite, because D is semi-definite. Note that mutation can breed vector elements with |ρi | > 1. To make sure that the pairwise correlations ρi ρj are in [−1, 1], define a censored vector ρ∗ = (ρ∗i )i=1,...,n with ρ∗i = min(max(ρi , −1), 1). This representation leads to a valid correlation matrix. Thus, redefine ρ as ρ = ρ∗ . Consequently, we maintain the modified representation of the genotype. Recombination and mutation are carried out as is customary (cf. Sections 4.4.2 and 4.4.3). Selection, evaluation, and program termination are carried out in analogy to the Cholesky approach. The same applies to the following cluster approach.

5.4.3

The Cluster Approach

5.4.3.1

Initialization

Assume we want to imply a correlation matrix exhibiting a cluster structure. We assume there are m sectors of high correlation against a background of low correlation. We suppose we know the different sizes of the different sectors and merely want to imply the intra and inter sector correlations. α1 , ..., αm denote the levels of the intra sector correlations, while β denotes the level of the inter sector correlation. Consider the following structure of the correlation matrix:

5.5. A Systematic Approach to Describe the Dependence Structure

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ Σ=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1

α1

α1 .. .

1

α1

···

···

1

α1

α1

1

···

.. .. . ···

β .. .

···

⎞

.. .

α1 .. .

. 1

αm

αm .. .

1

αm

···

β .. .

101

⎟

··· ⎟ ⎟

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . ⎟ · · · αm ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ 1 αm ⎠ αm

1

In the cluster approach, we use the vector γ ∈ [0, 1]m+1 as genotype of our algorithm. γ = (γi)i=1,...,m+1 represents the intra and inter sector correlations. To guarantee that the matrix is positive semi-definite, we define the inter sector correlation β as min(γ1 , ..., γm+1 ). Using the cluster approach, we have to calibrate m+1 free parameters. 5.4.3.2

Repair Mechanism

/ [0, 1]. Therefore define a censored vecMutation can lead to vector elements with γi ∈ tor γ ∗ = (γi∗ )i=1,...,m+1 with γi∗ = min(max(γi , 0), 1). Again, the inter sector correlation ∗ ). Then redefine γ as γ = γ ∗ . Consequently, we maintain is defined as min(γ1∗ , ..., γm+1

the modified representation of the genotype and use it to derive the correlation matrix Σ. The idea behind the above-mentioned repair mechanisms is to alter the genotype to obtain a valid representation of the correlation matrix. The modification of the genotype does not necessarily have to be maintained in the next generation. After the correlation matrix is derived, the genotype can be set back to its original value. However, in this study Lamarckism is applied. Lamarckism is the theory which states that acquired characteristics of an individual can be passed to its offspring. In the experiments without Lamarckism, the actual genotypes are not affected by the repair mechanism. Whenever Lamarckism is applied, the genotype is ultimately altered.

5.5

A Systematic Approach to Describe the

Dependence Structure We showed in Section 3.8.2 that different correlation matrices can lead to an identical allocation of credit risk to the different CDO tranches and consequently to identical CDO


tranche spreads. In Figure 5.1, we display exemplarily three correlation matrices that yield identical tranche spreads. Since the tranche spreads are computed by Monte-Carlo simulations we admit deviations of up to 3% of the relevant target value. The first matrix is generated by a Cholesky approach, the second by a one-factor approach, and the third by a cluster approach.

1

0.5

0 1

2

Cluster Approach Pairwise Linear Correlation


Cholesky Approach

3

4

5

6

7

8

Assets

9 10

9 10 78 56 4 3 12 Assets

1

0.5

0 1

2

3

4

5

6

Assets

7

8

9 10

9 10 78 56 4 3 12 Assets


One−Factor Approach

1

0.5

0 1

2

3

4

5

6

Assets

7

8

9 10

9 10 78 56 4 3 12 Assets

Figure 5.1 Different dependence structures can lead to identical tranche spreads. Note that for large n it can be hard to figure out whether two arbitrarily ordered correlation matrices that reproduce identical tranche spreads have the same or different underlying correlation structures. Since n assets can be ordered in n! different ways, we can obtain many different representations of the same dependence structure. The different matrix representations result from a simultaneous exchange of rows and columns. For that reason, we compare eigenvalues and eigenvectors to distinguish whether two correlation matrices represent the same or different dependence structures.


103

Any symmetric matrix A ∈ Rn×n has exactly n real orthogonal eigenvectors. Furthermore, there is an orthogonal matrix XA and a diagonal matrix Λ such that A = XA ΛXAT , where Λ is the diagonal matrix of the eigenvalues and XA is the matrix of the normalized = GAG−1 with a regular eigenvectors. If we transform the matrix A into the matrix A has the same eigenvalues and the same symmetry quadratic matrix G, then the matrix A as A. Consider now a permutation π : {1, ..., n} −→ {1, ..., n} and the ⎛ of n elements ⎞ eπ(1) ⎜ . ⎟ . ⎟ appropriate permutation matrix P , which is defined as P = ⎜ ⎝ . ⎠ with ei being eπ(n) the ith row vector in the identity matrix En . As permutation matrices are orthogonal matrices, we have P T P = P P T = En . The product of a matrix A with a permutation matrix P on the left (P A) permutes the rows of A; likewise, multiplication on the right (AP ) permutes the columns of A. Define B = P AP T and consider the eigenvalue decomposition of A and B, which is A = XA ΛXAT and B = XB ΛXBT , where XB is the matrix of normalized eigenvectors of B. Since A and B have the same eigenvalues, we can write B = P AP T = P (XA ΛXAT )P T = (P XA )Λ(XAT P T ) = (P XA )Λ(P XA )T = XB ΛXBT . Therefore, we have XB = P XA . We conclude that different representations of one correlation structure can be identified up to permutation by comparing eigenvalues and eigenvectors. A permutation can be defined by the following algorithm.

5.5.1

The Nearest Neighbor Algorithm

To represent correlation matrices in a systematic way, we choose a nearest neighbor algorithm. The nearest neighbor algorithm orders the assets such that highly correlated assets are grouped together. The input is a correlation matrix that indicates the pairwise correlations Σij between asset i and asset j, i, j = 1, ..., n. The initial ordering of the reference numbers of the assets is (1, 2, 3, . . . , n). The output is a permutation of (1, 2, 3, . . . , n) which describes a new and systematic ordering of the assets according to the nearest neighbor algorithm. The resulting permutation is unique, if all off-diagonal elements of the correlation matrix are different. First, we choose the pair of assets exhibiting the highest pairwise correlation. Denote the reference numbers of these assets as a1 and a2 . a1 and a2 are positioned next to each other, forming the first group. The pairwise correlation between asset a1 and asset a2 does not have to be considered again. Therefore, the respective matrix element is set to −∞. Thereafter, we select again the pair of assets with the largest matrix entry. It is the pair with the second largest pairwise correlation. Denote the reference numbers of


these assets as a3 and a4 . Assume that {a1 , a2 } and {a3 , a4 } are disjoint sets. Then a3 and a4 form the second group. Otherwise, assume that a3 is identical with a1 . Then a4 is attached to one side of the group containing a3 . To find out which side of the group in consideration a4 should be attached to, we have to compare the pairwise correlations between a4 and each of the borderline elements. In our case, the borderline elements are a1 and a2 . We already know that the correlation between the assets a1 and a4 is higher than the correlation between the assets a2 and a4 . Therefore, place a4 next to its “nearest neighbor,” which is a1 , keeping the order in the first group unchanged. Now, the first group contains the sequence {a4 , a1 , a2 } or vice versa. The pairwise correlation between the assets a3 and a4 does not have to be considered again, and the relevant matrix element is set to −∞. Likewise, correlations between two arbitrary members of the same group do not have to be taken into account any more (they are set to −∞), because these assets (and their respective group members) have already been joined. That means, in our case, that the pairwise correlation between a2 and a4 can be neglected in the next iteration, and is therefore set to −∞. Continue choosing the assets with the largest matrix entry. It is possible that both of these assets belong to existing groups. Groups are merged such that those two borderline elements with the highest pairwise correlation are placed next to each other. The algorithm terminates as soon as there is one group containing the complete ordering of all n assets. 5.5.1.1

General Scheme of the Nearest Neighbor Algorithm

function order = NearestNeighborAlgorithm (Σ) for i = 1 : n for j = 1 : i Σ(i,j) = -∞; end end groups = zeros(n,n); % In groups, we order the individuals according to their pairwise correlations. % Disjoint groups of highly correlated assets are registered in different lines. while Σ = −∞ · ones(n,n) % Iteration is stopped as soon as the complete ordering of all n assets % is displayed in one row of groups and all elements of Σ are set to −∞. maximum = max(Σ(i,j)), i,j = 1,...,n; [r,c] = find(Σ==maximum);


% r and c denote the row and column index of the maximal entry. if groups == zeros(n,n) g = 1; % g denotes the group number. groups(g,1:2) = [r,c]; z = [r,c]; % The members of the currently considered group are stored in z. else if (neither r nor c belong to an existing group) g = g + 1; groups(g,1:2) = [r,c]; z = [r,c]; elseif (r belongs to an existing group) & (c does not belong to an existing group) add c to one side of the group of r such that the correlation between c and its new neighbor is as high as possible; z contains the members of the joint group of c and r; elseif (c belongs to an existing group) & (r does not belong to an existing group) add r to one side of the group of c such that the correlation between r and its new neighbor is as high as possible; z contains the members of the joint group of c and r; else (r and c belong to existing groups) add the group of c to one side of the group of r such that the correlation at the crossing point is as high as possible; the former group of c is dissolved; z contains the members of the joint group of c and r; end end gs = max(size(z));

% gs indicates the size of group z;

for i = 1 : gs for j = 1 : gs Σ(z(i),z(j)) = -∞; % The pairwise correlations between the elements of one % existing group do not have to be considered again. end

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end end order = z; % In the final cycle z contains the reference numbers of all n assets % in the desired order. The following Figure 5.2 shows correlation maps and the corresponding correlation matrices before and after we applied the nearest neighbor algorithm1 . Note that the matrices in Figure 5.1 are also ordered by means of the nearest neighbor algorithm.

Figure 5.2 Dependence structures can be ordered in a systematic way.

1 The algorithm is based on code published by Barry M. Wise, Eigenvector Research, Inc., available at http://www.eigenvector.com/MATLABarea.html.

5.6. Conclusion

5.6

107

Conclusion

In this chapter, we outlined how to derive a correlation matrix that reproduces all given implied tranche correlations of a CDO structure simultaneously. If the implied correlation smile emerges, because the actual correlation structure is heterogeneous, use of an appropriate non-flat correlation structure will allow for consistent pricing. We outlined how to derive the dependence structure by means of EAs and we illustrated why EAs are appropriate for this kind of application. We discussed three different matrix representations: the Cholesky approach, the onefactor approach, and the cluster approach. We considered simplified solution representations to reduce the complexity of the optimization problem. Simplified approaches are easier to handle because the number of parameters that have to be calibrated is relatively small. Furthermore, we discussed different ways to describe the fitness value of potential solution matrices. Finally, we outlined how correlation matrices can be identified by their eigenvectors and eigenvalues. This analysis enables us to decide whether two arbitrarily ordered correlation matrices represent the same or different dependence structures. We also discussed a nearest neighbor algorithm for ordering correlation matrices systematically.

Chapter 6 Experimental Results 6.1

Introduction

In this chapter, we will deal with the implementation of the optimization problem that we illustrated in the previous Chapter 5. To assess potential and tractability of our suggested approach, the analysis in Section 6.3 and Section 6.4 is based on simulated data. We choose a particular correlation matrix as starting point. This correlation matrix has to be compatible with the representation used for the genotypes in the optimization process. Using the correlation matrix in conjunction with the extended standard market model, we derive a set of tranche spreads. Then we apply our algorithm to find a correlation matrix that reproduces these tranche spreads precisely. Of course, we already know that there is at least one global optimum for this particular problem setup. Our goal is to find either the correlation matrix we used as starting point or any other matrix leading to the given tranche spreads. Working with simulated data simplifies the assessment of the performance, and it simplifies the assessment of the quality of the obtained solution. Working with simulated data excludes pricing discrepancies caused for instance by neglecting stochastic recovery rates, stochastic default intensities, correlated recovery rates and default times or any other aspects that are not reflected appropriately in our pricing model. We compare different evolutionary search procedures in order to get some idea of the characteristics of the search space. In Section 6.6 and Section 6.7 we analyze our suggested approach based on market data. In Section 6.3 we compare the performance of two basic algorithms: we consider a Hill-Climber and a population-based ES. In Section 6.4 we compare the performance of more advanced algorithms, considering different implementations of ESs and GAs. The

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Chapter 6. Experimental Results

comparison is based on the number of function evaluations as well as on the number of generations needed to obtain a satisfying solution. In Section 6.6 we consider different versions of parallel algorithms and parallel implementations. We show that parallel algorithms can significantly accelerate the optimization process. Using market data we quantify the speedups that can be obtained. In Section 6.7 we derive a correlation matrix that reproduces a set of tranche spreads which is actually observed in the market.1

6.2

Solution Evaluation

In general, the fitness value of an individual is obtained by simply evaluating the target function. Depending on the optimization problem, the evaluation of the target function can be time-consuming. Since EAs require a high number of target function evaluations, the speed of the optimization process can be unacceptably slow. If a partial evaluation of the target function provides enough information to assess the fitness value of an individual, the “lazy evaluation”-technique can be applied to save computational time. If the partial evaluation indicates a very bad fitness value, the remaining part of the evaluation is skipped. Relying solely on the guess is generally sufficient to guide the evolutionary search while the individuals are still far away from the optimum. An alternative approach dealing with the curse of time-consuming function evaluations uses model surrogates to approximate the true target function. A model is trained on the basis of the fitness values of individuals that have been evaluated using the true target function. The trained model is used to guide the evolutionary search by preselecting the most promising individuals. The approach assumes that the trained model approximately reflects the fitness ratings of the true model. The function surrogate is chosen such that it saves computational time compared to the true fitness function. By means of this strategy, time-consuming fitness evaluations can be largely avoided. This approach is often used in combination with ESs and is therefore referred to as the Model Assisted Evolution Strategy.

6.2.1

Using the Expected Tranche Loss as Proxy for the Tranche Spread

In this study, we want to find a dependence structure that reproduces a set of observed tranche spreads using EAs. However, computing the tranche spreads for an arbitrary correlation matrix can be very time-consuming. To save computational time, we can use 1

See also our conference report Hager and Schöbel (2006).

6.2. Solution Evaluation

111

one of the above-mentioned methods. Numerous other approaches are also possible. In Sections 6.3 and 6.4, we use the expected tranche loss instead of the tranche spread. We search for a dependence structure that reproduces a given set of expected tranche losses instead of a given set of tranche spreads to avoid time-consuming evaluations. Computing the expected tranche loss requires much less computational time than computing the tranche premium. It is intuitively clear that there is a close connection between the expected tranche loss and the tranche premium, because the protection buyer pays the tranche premium to compensate the protection seller for the default that might affect the tranche. This connection is illustrated in Figure 6.1 for flat dependence structures. We use the following setup: a homogeneous portfolio of 100 assets, with all assets having equal CDS spreads of 100 bps, equal recovery rates of 40%, and an equal nominal. The time to maturity is 5 years. We compute the tranche spreads and the expected tranche losses for the equity (0%-3%), the mezzanine (3%-10%), and the senior tranche (10%100%). Since expected tranche losses are not readily available in the market, we proceed in the following way: we assume that a set of tranche spreads is given, then we derive the corresponding implied tranche correlations from the tranche spreads. It is also possible that the implied tranche correlations are given as input variables. In any case, we use the implied tranche correlations to compute the expected tranche losses. Then we search for a correlation matrix that reproduces the expected losses of all tranches simultaneously. Note that this procedure can lead to a correlation matrix that is only an approximation of the actual solution. In this case it can be the starting point for another evolutionary search, using the tranche spreads as target values. However, common sense justifies using the expected tranche loss as proxy for the tranche spread. Thus, we proceed as outlined above. To compute the tranche spreads and the expected tranche losses, in this chapter we apply Monte-Carlo simulations. The respective standard errors depend on the attachment and detachment level of the tranche and on the underlying correlation structure. The number of iterations is chosen such that the standard error is less than 2%.

112

Chapter 6. Experimental Results The Equity Tranche

The Mezzanine Tranche

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Figure 6.1 The expected tranche loss vs. the tranche premium, scaling factor k=0.04274. Note that we use a unique scaling factor for all tranches.

6.2.2

Equivalent Information Content: Density of the Portfolio Loss and Expected Tranche Loss

For illustrative purposes, we prove an equivalence relationship between the formulation via the density of the portfolio loss and the formulation via the expected loss of a tranche with lower attachment level zero. We assume that the portfolio contains infinitely many credits. Let L ∈ [0, 1] denote the cumulative portfolio loss in percent and let f (l) denote the probability density function of L. In analogy to (2.20), we define the cumulative default of tranche (0, B) as ω0,B (l) = l · 1[0,B] (l) + B · 1]B,1] (l) .

(6.1)

The expected loss of tranche (0, B) in percent of the tranche value is then defined as EL(B) =

E [ω0,B (l)] . B

(6.2)

6.3. Performance Comparison: Basic Strategies

113

In the following, we show that the information content of the density of the portfolio loss and the information content of the expected loss of a tranche with lower attachment level zero are equivalent. 6.2.2.1

Deriving the Expected Tranche Loss from the Density of the Portfolio Loss

Using the probability density function, we can derive the expected tranche loss. 1 E l · 1[0,B] (l) + B · 1]B,1] (l) B 1 1 B lf (l)dl + f (l)dl . = B 0 B

EL(B) =

6.2.2.2

Deriving the Density of the Portfolio Loss from the Expected Tranche Loss

Using the expected tranche loss, we can derive the probability density function. Consider first the following derivative: 1 d 1 B d EL(B) = lf (l)dl + f (l)dl dB dB B 0 B B 1 lf (l)dl . = − 2 B 0 Applying the appropriate differential operator on the expected tranche loss we obtain f (B) = −

1 d d B2 [EL(B)] . B dB dB

Similar results can be obtained by considering a finite number of credits in the underlying portfolio.

6.3 6.3.1

Performance Comparison: Basic Strategies Setup

In the following, we want to evaluate the performance of two basic algorithms in respect of the Cholesky approach, the one-factor approach, and the cluster approach. We consider a homogeneous portfolio of 10 assets. All assets have equal default intensities

114


of 1%, equal recovery rates of 40%, and an equal unit nominal. We consider a CDO consisting of an equity tranche (0%-5%), a mezzanine tranche (5%-15%), and a senior tranche (15%-100%). The time to maturity is 5 years. For simplicity’s sake, in this setup we just consider non-negative pairwise linear correlations. In the Cholesky approach, the genotype is an upper triangular matrix in R10×10 ; in the one-factor approach, the genotype is a row vector in [0, 1]10 . In the cluster approach, we assume that there are two sectors of equal size. We have to derive the intra sector correlations and the inter sector correlation. Therefore, the genotype is in [0, 1]3 . Since we consider only 10 assets, the number of free parameters is still conveniently sized, even in the Cholesky approach. In Section 6.3, we apply a (1 + 1)-ES and a (3, 12)-ES to guide the evolutionary search. We use elite selection and global mutation without a strategy parameter. We mutate individuals by adding realizations of normally distributed random variables with expected value 0 and standard deviation 0.05. The mutation probability is 1.0. In case of the (3, 12)-ES, we apply 1-point crossover. The crossover probability is 1.0. To indicate the quality of a genotype and the corresponding correlation matrix Σ, we choose the function f1 (Σ) that was introduced in Section 5.4.1. We replace the tranche spreads by the respective expected tranche losses, since we want to reproduce a given set of expected tranche losses. We terminate the algorithm as soon as the quality of an individual falls below 3%. We admit deviations of up to 3%, because we compute the tranche spreads and the expected tranche losses in this chapter using Monte-Carlo simulations. We purposely design the algorithms in a very fundamental way so as to get at the idea that is at the bottom of EAs. Note that there is room for enhanced performance when more sophisticated algorithms are applied. See Section 6.4 and Section 6.6 for a first impression of the performance of more elaborate algorithms. We showed in Section 3.8.2 that different correlation matrices can lead to identical tranche spreads. Likewise, we can show that different correlation matrices can lead to identical expected tranche losses. However, there are expected tranche losses that can be generated by only one correlation matrix. Recall that we consider only non-negative pairwise correlations. When the underlying is perfectly diversified, the expected loss of the equity tranche is maximized and the expected loss of the senior tranche minimized. When the underlying is perfectly correlated, the expected loss of the equity tranche is minimized and the expected loss of the senior tranche maximized (see Section 2.2.3.6).


115

In our setup, the expected loss of the whole portfolio is 2.9262%. Since we consider a homogeneous portfolio, the expected loss equals the probability of default multiplied by the loss given default. This value is not affected by the correlation structure. Different correlation structures do not change the overall risk, they just redistribute the risk on the different CDO tranches according to their seniority. Whenever the expected losses of two of the three tranches and the expected loss of the whole portfolio are known, the expected loss of the third tranche is uniquely determined. Therefore, the expected tranche losses (39.3469%, 9.2067%, 0.0450%) can only be obtained by a portfolio of completely independent assets, while (4.8771%, 4.8771%, 2.5820%) can only be obtained by a portfolio of perfectly correlated assets. When discussing and comparing the (1+1)-ES and the (3, 12)-ES regarding the Cholesky approach, the one-factor approach, and the cluster approach, the performance comparison has to be handled with care. First of all, the number of free parameters differs largely depending on the approach considered. Naturally, the number of free parameters influences the performance of the optimization process fundamentally, this aspect has to be taken into account. Furthermore, if the target values can be reproduced precisely by several correlation structures, the existence of multiple optima will influence the performance of the optimization process. If there are several optima, the evolutionary search will probably find the optimum in a shorter period of time, and different runs will come up with different solutions. To improve comparability we choose a unimodal search space. We choose the extreme case of complete independence as starting point of our consideration. In this case, the correlation matrix is the identity matrix. We use this matrix to compute the expected tranche losses. As mentioned above, when the underlying assets are completely independent, the expected tranche losses are (39.3469%, 9.2067%, 0.0450%) for the equity, the mezzanine, and the senior tranche. These expected tranche losses are the target values. We use EAs to find a correlation matrix that reproduces these target values. First we consider the Cholesky approach, then the one-factor approach, and finally we consider the cluster approach. Note that the identity matrix can be described by the genotypes of all three approaches. This implies that the optimum is attainable in all three approaches. Using the identity matrix as our basis, we can avoid ambiguities in the optimization process, because the target values can only be reproduced when the underlying assets are completely independent. We show how the evolutionary search leads to correlation matrices that are - as expected - to a certain order of precision very close to the identity matrix.

116


6.3.2

Results

We analyze the performance of the (1+1)-strategy and the (3, 12)-strategy for the abovementioned setup. Our focus is on the decline of the objective function in the course of the generations. Low values of the objective function stand for a high quality of the obtained solution. To ensure that we obtain reliable results, we repeat each evolutionary search process 50 times and compute the mean value of the objective functions over the 50 runs. In addition to the mean value, we also consider the 10% and the 90% quantiles. Figures 6.2 and 6.3 illustrate the convergence behavior of the Cholesky, the one-factor, and the cluster approaches. Note that for different approaches with different population sizes the initial values of the objective function do not necessarily coincide because the objective function reflects the fitness of the best individual in the whole population. Furthermore, note that a straight line represents exponentially fast convergence since we consider a logarithmic scale. (1+1)−ES, Cholesky Approach

30 20

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Figure 6.2 Convergence behavior (mean value and quantiles): (1 + 1)-ES.

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(3,12)−ES, Cholesky Approach

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Figure 6.3 Convergence behavior (mean value and quantiles): (3, 12)-ES.

Aside from the course of the objective function, the parameters 1 zk (1) − zk (Tk ) , 50 k=1 zk (1)λ(Tk − 1) 50

ω1e =

50 T −1

ω2e =

k 1{zk (τ +1) =zk (τ )} 1 , 50 k=1 τ =1 λ(Tk − 1)

1 zk (1) − zk (Tk ) , 50 k=1 zk (1)(Tk − 1) 50

ω1g =

50 T −1

ω2g =

k 1{zk (τ +1) =zk (τ )} 1 50 (Tk − 1) τ =1

k=1

are a guideline for assessing and comparing the performance of the different algorithms. In the k th run, the objective function zk (t) indicates the quality of the best individual that has been generated up to generation t (compare Section 5.4.1). Let Tk denote the

118


total number of generations in the k th run and let λ be the size of the offspring generation. Consequently, λTk denotes the total number of function evaluations in the k th run. ω1e indicates the average normalized improvement of the objective function per function evaluation. We consider the normalized improvement of the objective function in order to take the different primary values of the objective function (i.e. the different values in the first generation) into account. ω2e denotes the average number of improvements in the objective function proportionate to the number of function evaluations. Analogously, ω1g and ω2g indicate the performance based on the number of generations. Tables 6.1, 6.2, and 6.3 display the resulting values of ω1e , ω2e , ω1g , and ω2g in the Cholesky, the one-factor, and the cluster approach. Note that the explanatory power of comparisons based on these parameters is very limited when algorithms with different population sizes are considered. Generally, the initial value of the objective function is lower for a population-based strategy than for a (1 + 1)-strategy. Likewise, in a (1 + 1)-strategy each single improved individual is registered, whereas in a population-based strategy we only register the improvement of the best individual in the population. Consequently, these parameters should only be considered as a loose guideline. Table 6.1 The parameters ω1e , ω2e , ω1g and ω2g in the Cholesky approach. ω1e

ω2e

ω1g

ω2g

(1 + 1)-ES 0.00007 0.00731 0.00007 0.00731 (3, 12)-ES 0.00009 0.00589 0.00104 0.07067

Table 6.2 The parameters ω1e , ω2e , ω1g and ω2g in the one-factor approach. ω1e

ω2e

ω1g

ω2g

(1 + 1)-ES 0.00472 0.13801 0.00472 0.13801 (3, 12)-ES 0.00815 0.07224 0.09779 0.86687

Table 6.3 The parameters ω1e , ω2e , ω1g and ω2g in the cluster approach. ω1e

ω2e

ω1g

ω2g

(1 + 1)-ES 0.02098 0.31511 0.02098 0.31511 (3, 12)-ES 0.00913 0.07552 0.10955 0.90626


119

Compare first the convergence behavior of the Cholesky, the one-factor, and the cluster approach. In the Cholesky approach we have 55 free parameters, in the one-factor approach 10 free parameters, and in the cluster approach 3 free parameters. When the number of free parameters is low, it does not take long to search the parameter space and find an optimal solution. Therefore, the cluster approach converges faster than the one-factor approach, and the one-factor approach converges faster than the Cholesky approach. This relation is valid for the (1 + 1)-strategy and the (3, 12)-strategy. In Figure 6.3 this order is not obvious when we compare the cluster approach with the one-factor approach, because the primary value of the objective function in the cluster approach is higher than in the one-factor approach. However, considering the parameters ω1e , ω2e , ω1g , and ω2g , it is obvious that the cluster approach really converges faster than the one-factor approach. Compare now the convergence behavior of the (1 + 1)-strategy and the (3, 12)-strategy. To that end, consider Figure 6.4. We find that in the Cholesky approach the (3, 12)-ES clearly outperforms the (1 + 1)-ES. In the one-factor approach the convergence behavior does not differ a lot, and in the cluster approach the (1 + 1)-ES converges faster than the (3, 12)-ES. The parameters ω1e , ω2e , ω1g , and ω2g are not very expressive of this comparison. On the one hand, the initial value of the objective function is often lower in a population-based strategy than in a (1 + 1)-strategy. Consequently, the parameters ω1e and ω1g cannot illustrate the difference between the performance of the (1 + 1)-strategy and the (3, 12)-strategy. Likewise, in the (1 + 1)-strategy the parameters ω2e and ω2g register the improvement of each single individual, but in the (3, 12)-strategy the parameters ω2e and ω2g only register the improvement of the best individual in the population. Hence, these parameters should at best be considered a loose guideline. For instance, we use these parameters to illustrate why the (1 + 1)-strategy outperforms the (3, 12)-strategy in the cluster approach. Note that in the cluster approach 31.511% of all new individuals show an improved fitness in the (1+1)-strategy. In the (3, 12)-strategy at best one of the 12 offspring individuals (i.e. 8.33%) can cause an improvement, since we focus on the performance of the best individual in the population. This can be the reason why the (1 + 1)-strategy outperforms the (3, 12)-strategy in the cluster approach. The same reasoning can be used to explain why the (1 + 1)-strategy performs well compared to the (3, 12)-strategy in the one-factor approach. In the following Figure 6.5, we display correlation matrices produced by one of the 50 runs in the Cholesky, the one-factor, and the cluster approaches. The corresponding expected tranche losses are in ([38.1665%, 40.5273%], [8.9305%, 9.4829%], [0.0437%, 0.0464%]), i.e. the maximal deviation from (39.3469%, 9.2067%, 0.0450%) is 3%.

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Chapter 6. Experimental Results (1+1)−ES vs. (3,12)−ES, Cholesky Approach

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Figure 6.4 Convergence behavior (mean value): (1 + 1)-ES vs. (3, 12)-ES.

6.4

Performance Comparison: More Advanced Algo rithms

6.4.1

Setup

In the following, we want to evaluate the performance of different algorithms in the Cholesky and the one-factor approaches. Again, we consider a homogeneous portfolio of 10 assets with an equal unit nominal. The default intensity of each obligor is 1%, and the recovery rate is 40%. The CDO consists of three tranches. The respective attachment and detachment levels are 0%5%, 5%-15%, 15%-100%. The time to maturity is 5 years. Again, we only consider non-negative pairwise linear correlations.

6.4. Performance Comparison: More Advanced Algorithms

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Figure 6.5 Dependence structures leading to the desired expected tranche losses. We consider a Monte-Carlo Search, a Hill-Climber, a (4, 20)-ES, and a generational GA(40). In case of the (4, 20)-ES, the mutation probability is 0.95, and the crossover probability is 0.50. We apply elite selection, 1-point crossover, and three different mutation operators: the 1/5-rule, global mutation with a strategy parameter, and global mutation without a strategy parameter. In case of the GA(40), the mutation probability is 0.50, and the crossover probability is 0.95. We apply proportional selection and tournament selection, 1-point, 3-point, and intermediate crossover, and global mutation without a strategy parameter. We mutate individuals by adding realizations of normally distributed random variables with expected value 0 and standard deviation 0.05, unless explicitly mentioned otherwise. To indicate the quality of a genotype and the corresponding correlation matrix Σ, we choose the function f2 (Σ) that was introduced in Section 5.4.1, where we replace the tranche spreads by the respective expected tranche losses. The threshold for program termination is 5%.

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Our goal is to find a correlation matrix that models an observed correlation smile. The given implied tranche correlations are 0.24 for the equity tranche, 0.05 for the mezzanine tranche, and 0.35 for the senior tranche. First, we compute the expected tranche losses of the equity, the mezzanine, and the senior tranche using the respective implied correlations. We get 31.96%, 9.71%, and 0.45%. Then we have to find a correlation matrix reproducing all three expected tranche losses simultaneously. As opposed to the experiment in Section 6.3, there are several distinct matrices that meet our requirements. However, this fictitious correlation smile describes a more realistic dependence structure than the dependence structure in Section 6.3, which assumed perfect diversification.

6.4.2

Results

To make sure we obtain reliable results, we repeat each evolutionary search process 25 times. We compute the mean value of the objective functions over the 25 runs, and we also consider the 10% and the 90% quantiles. In the k th run, the objective function zk (t) indicates the quality of the best individual found up to time t. We consider the decline of the objective function in the course of the generations. Again, we consider the parameters 1 zk (1) − zk (Tk ) , 25 k=1 zk (1)λ(Tk − 1) 25

ω1e =

25 T −1

ω2e =

k 1{zk (τ +1) =zk (τ )} 1 , 25 k=1 τ =1 λ(Tk − 1)

1 zk (1) − zk (Tk ) , 25 k=1 zk (1)(Tk − 1) 25

ω1g =

25 T −1

ω2g =

k 1{zk (τ +1) =zk (τ )} 1 25 k=1 τ =1 (Tk − 1)

as a guideline for assessing and comparing the performance of the different algorithms. Table 6.4 and Table 6.5 display the resulting values in the Cholesky and the one-factor approach. Note the limited explanatory power of these parameters that has been outlined above.


123

Table 6.4 The parameters ω1e , ω2e , ω1g and ω2g in the Cholesky approach. ω1e

ω2e

ω1g

ω2g

Monte-Carlo Search

5.35 ·10−5

87.61 ·10−5

5.35 ·10−5

87.61 ·10−5

(1 + 1)-ES

9.02 ·10−5

533.65 ·10−5

9.02 ·10−5

533.65 ·10−5

(4, 20)-ES Global Mutation 1/5-Rule

14.50 ·10−5

723.17 ·10−5

289.92 ·10−5

14463.47 ·10−5

(4, 20)-ES Global Mutation with Str. Param.

9.19 ·10−5

408.77 ·10−5

183.87 ·10−5

8175.47 ·10−5

(4, 20)-ES Global Mutation without Str. Param.

9.39 ·10−5

422.85 ·10−5

187.78 ·10−5

8456.91 ·10−5

GA(40) Prop. Selection 1-Point Crossover

8.74 ·10−5

306.83 ·10−5

349.69 ·10−5

12273.09 ·10−5


8.87 ·10−5

300.80 ·10−5

354.91 ·10−5

12032.13 ·10−5

GA(40) Prop. Selection Interm. Crossover

8.43 ·10−5

297.82 ·10−5

337.16 ·10−5

11912.93 ·10−5

GA(40) Tourn. Selection 1-Point Crossover

12.42 ·10−5

556.87 ·10−5

496.64 ·10−5

22274.92 ·10−5

124


Table 6.5 The parameters ω1e , ω2e , ω1g and ω2g in the one-factor approach. ω1e

ω2e

ω1g

ω2g

Monte-Carlo Search

39.40 ·10−5

338.17 ·10−5

39.40 ·10−5

338.17 ·10−5

(1 + 1)-ES

175.11 ·10−5

4388.60 ·10−5

175.11 ·10−5

4388.60 ·10−5

(4, 20)-ES Global Mutation 1/5-Rule

126.68 ·10−5

124.62 ·10−5

2533.65 ·10−5

2492.31 ·10−5

(4, 20)-ES Global Mutation with Str. Param.

164.86 ·10−5

126.37 ·10−5

3297.17 ·10−5

2527.47 ·10−5

(4, 20)-ES Global Mutation without Str. Param.

127.28 ·10−5

114.33 ·10−5

2545.64 ·10−5

2286.67 ·10−5


22.45 ·10−5

282.41 ·10−5

897.92 ·10−5

11296.57 ·10−5


23.19 ·10−5

251.53 ·10−5

927.48 ·10−5

10061.22 ·10−5

GA(40) Prop. Selection Interm. Crossover

1.22 ·10−5

8.16 ·10−5

48.76 ·10−5

326.53 ·10−5

GA(40) Tourn. Selection 1-Point Crossover

58.98 ·10−5

955.61 ·10−5

2359.10 ·10−5

38224.28 ·10−5


6.4.2.1

125

Monte-Carlo Search vs. (1+1)-ES

First, we compare a Monte-Carlo search and a (1 + 1)-ES for purposes of analyzing the search space. We consider the Cholesky approach (cf. Figures 6.6 and 6.7) and the one-factor approach (cf. Figures 6.8 and 6.9). We display the mean value of the objective functions over the 25 runs, and we display the 10% and the 90% quantiles. We also summarize the mean values of the objective functions of different algorithms in one plot in order to ease comparison. Generally, the Monte-Carlo search is rather inefficient, especially in high-dimensional search spaces. Whenever the Monte-Carlo search performs as well as a Hill-Climber or a population-based EA, the search space is probably very flat or very craggy or non-causal. However, in our case the Hill-Climbing strategy clearly outperforms the random search in both, the one-factor and the Cholesky approach. Often, (1 + 1)-strategies are very efficient in simple unimodal search spaces. Frequently, however, they cannot handle situations in which there are several local optima. If a Hill-Climber starts up the wrong hill, it has no chance to know that it has found an inferior optimal solution. 6.4.2.2

(1+1)-ES vs. Multistart (1+1)-ES

Next we extend the (1 + 1)-ES to a multistart (1 + 1)-ES. A multistart (1 + 1)-ES reduces the risk of premature convergence. Additionally, if the fitness function is noisy, HillClimbing techniques are often irrecoverably led astray by the noise, whereas populationbased EAs perform well in the presence of small amounts of noise. Using the multistart (1 + 1)-ES, we obtain different solution matrices, which leads to the conclusion that the search space is multimodal. In the following Figure 6.10 (for the Cholesky approach) and Figure 6.11 (for the one-factor approach) we display correlation matrices that have been produced by one of the 25 runs in the Cholesky and the one-factor approach respectively. The corresponding expected tranche losses are sufficiently close to (31.96%, 9.71%, 0.45%), i.e. the sum of the relative deviations is less than 5%.

126

Chapter 6. Experimental Results (1+1)−ES, Cholesky Approach 120 100 80

60

60 Objective Function

Objective Function

Monte−Carlo Search, Cholesky Approach 120 100 80

40

20

10

5

40

20

10


5 0

2000


4000 6000 Generations

8000

10000

0

2000


8000

10000

Figure 6.6 Convergence behavior in the Cholesky approach (mean value and quantiles): (1 + 1)-ES and Monte-Carlo search. Monte−Carlo Search vs. (1+1)−ES, Cholesky 120 100 80 Objective Function

60 40

20

10 Monte−Carlo (1+1)−ES 5 0

2000


8000

10000

Figure 6.7 Convergence behavior in the Cholesky approach (mean value): (1 + 1)-ES vs. MonteCarlo search. 6.4.2.3

(4,20)-ES vs. GA(40)

We now compare different implementations of a (4, 20)-ES and a GA(40). In case of the (4, 20)-ES, we use elite selection and 1-point crossover. Our focus is on the application of different mutation operators. We consider the 1/5-rule, global mutation without a strategy parameter, and global mutation with a strategy parameter that controls the mutation step size. The mutation probability is 0.95, and the crossover probability is 0.50. In case of the GA(40), we focus on the selection and crossover parameters. We use proportional selection and tournament selection with a tournament group size of 10. We use 1-point crossover, 3-point crossover, intermediate crossover, and we use global mutation without a strategy parameter. The crossover probability is 0.95, the

6.4. Performance Comparison: More Advanced Algorithms Monte−Carlo Search, One−Factor Approach

(1+1)−ES, One−Factor Approach

25

25 10% Quantile Mean Value 90% Quantile

15

10

0

500

1000 Generations

1500


20 Objective Function

Objective Function

20

5

127

15

10

2000

5

0

500

1000 Generations

1500

2000

Figure 6.8 Convergence behavior in the one-factor approach (mean value and quantiles): (1 + 1)-ES and Monte-Carlo search.

Objective Function

Monte−Carlo Search vs. (1+1)−ES, One−Factor 25 Monte−Carlo (1+1)−ES 20 15

10

5

0

500

1000 Generations

1500

2000

Figure 6.9 Convergence behavior in the one-factor approach (mean value): (1 + 1)-ES vs. MonteCarlo search. mutation probability is 0.50. To compare the performance of the different algorithms, consider Tables 6.4 and 6.5 as well as Figures 6.13, 6.15, 6.17, and 6.19 where we display the mean values of the objective functions. Mean values and quantiles are displayed in Figures 6.12, 6.14, 6.16, and 6.18. Comparing the course of the objective functions and comparing the parameters ω1e , ω2e , ω1g , ω2g leads to consistent results, as long as we compare algorithms with identical population size. However, when it comes to comparing the performance of algorithms with different population size, considering the parameters ω1e , ω2e , ω1g , ω2g does not lead to reliable results, as has been outlined above. 6.4.2.3.1

Cholesky Approach First, we compare the different ES implementations

in the Cholesky approach. The performance of different algorithms differs considerably.

128


The global mutation strategy with the 1/5-rule outperforms the other approaches. The global mutation strategy with one strategy parameter performs worse than the other implementations. After this, we compare the different GA implementations in the Cholesky approach. The GA with tournament selection combined with 1-point crossover leads to the best result, the GAs with proportional selection and 1-point or 3-point crossover perform moderately, and the GA with proportional selection and intermediate crossover yields the worst result. It is important to know that by means of recombination the hyperbody formed by the parents generally cannot be left by the offspring individuals. Especially the intermediate crossover technique causes volume reduction, since it successively narrows the search space such that eventually the optimal solution cannot be attained anymore after a few generations.


Cholesky Approach

1

0.5

0 1

2

3

4

5

6

7

8

Assets

9 10

9 10 78 56 4 3 12 Assets

Figure 6.10 Exemplary solution matrix in the Cholesky approach.


One−Factor Approach

1

0.5

0 1

2

3

4

5

6

Assets

7

8

9

10

9 10 78 56 4 3 12 Assets

Figure 6.11 Exemplary solution matrix in the one-factor approach.


(4,20)−ES, Global Mutation With Str. Param., Cholesky 120 100 10% Quantile 80 Mean Value 90% Quantile 60 Objective Function

Objective Function

(4,20)−ES, Global Mutation, 1/5−Rule, Cholesky 120 100 10% Quantile 80 Mean Value 90% Quantile 60 40

20

10

5

40

20

10

200 300 400 500 Generations (4,20)−ES, Global Mut. Without Str. Param., Cholesky 120 100 10% Quantile 80 Mean Value 90% Quantile 60 Objective Function

129

0

100

0

100

5

0

100

200 300 Generations

400

500

40

20

10

5

200 300 Generations

400

500

Figure 6.12 Convergence behavior in the Cholesky approach (mean value and quantiles): Evolution Strategies. (4,20)−ES, Cholesky Approach 120 100 80

Global Mutation, 1/5−Rule Global Mutation With Str. Param. Global Mutation Without Str. Param.

Objective Function

60 40

20

10

5

0

100

200 300 Generations

400

500

Figure 6.13 Convergence behavior in the Cholesky approach (mean value): Evolution Strategies in comparison.

130

Chapter 6. Experimental Results (4,20)−ES, Global Mut. With Str. Param., One−Factor 25 10% Quantile Mean Value 20 90% Quantile Objective Function

Objective Function

(4,20)−ES, Global Mutation, 1/5−Rule, One−Factor 25 10% Quantile Mean Value 20 90% Quantile 15

10

5

15

10

5 0

20

40 60 Generations

80

100

0

20

40 60 Generations

80

100

Objective Function

(4,20)−ES, Global Mut. Without Str. Param., One−Factor 25 10% Quantile Mean Value 20 90% Quantile 15

10

5

0

20

40 60 Generations

80

100

Figure 6.14 Convergence behavior in the one-factor approach (mean value and quantiles): Evolution Strategies. (4,20)−ES, One−Factor Approach 25 Global Mutation, 1/5−Rule Global Mutation With Str. Param. Global Mutation Without Str. Param.

Objective Function

20 15

10

5

0

20

40 60 Generations

80

100

Figure 6.15 Convergence behavior in the one-factor approach (mean value): Evolution Strategies in comparison.


GA(40), Prop. Selec., 3−Point Crossover, Cholesky 120 100 10% Quantile 80 Mean Value 90% Quantile 60 Objective Function

Objective Function

GA(40), Prop. Selec., 1−Point Crossover, Cholesky 120 100 10% Quantile 80 Mean Value 90% Quantile 60 40

20

40

20

10

5

10

100 150 200 250 Generations GA(40), Tourn. Selec., 1−Point Crossover, Cholesky 120 100 10% Quantile 80 Mean Value 90% Quantile 60

50

Objective Function

Objective Function

0

5

100 150 200 250 Generations GA(40), Prop. Selec., Interm. Crossover, Cholesky 120 100 10% Quantile 80 Mean Value 90% Quantile 60 40

20

10

5

131

0

50

0

50

40

20

10

0

50

100 150 Generations

200

250

5

100 150 Generations

200

250

Figure 6.16 Convergence behavior in the Cholesky approach (mean value and quantiles): Genetic Algorithms. GA(40), Cholesky Approach 120 100 80

Prop. Selection,, 1-Point Crossover Prop. Selection,, 3-Point Crossover Prop. Selection,, Interm. Crossover Tourm. Selection,, 1-Point Crossover

Objective Function

60 40

20

10

5

0

50

100 150 Generations

200

250

Figure 6.17 Convergence behavior in the Cholesky approach (mean value): Genetic Algorithms in comparison.

132


15

10

5

GA(40), Prop. Selec., 3−Point Crossover, One−Factor 25 10% Quantile Mean Value 20 90% Quantile Objective Function

Objective Function

GA(40), Prop. Selec., 1−Point Crossover, One−Factor 25 10% Quantile Mean Value 20 90% Quantile

0

10

20 30 Generations

40

5

0

10

20 30 Generations

40

0

10

20 30 Generations

40

50

GA(40), Tourn. Selec., 1−Point Crossover, One−Factor 25 10% Quantile Mean Value 20 90% Quantile Objective Function

Objective Function

GA(40), Prop. Selec., Interm. Crossover, One−Factor 25 10% Quantile Mean Value 20 90% Quantile

10

10

5

50

15

15

50

15

10

5

0

10

20 30 Generations

40

50

Figure 6.18 Convergence behavior in the one-factor approach (mean value and quantiles): Genetic Algorithms. GA(40), One-Factor Approach 25

Prop. Selection,, 1-Point Crossover Prop. Selection,, 3-Point Crossover Prop. Selection,, Interm. Crossover Tourm. Selection,, 1-Point Crossover

Objective Function

20 15

10

5

0

10

20 30 Generations

40

50

Figure 6.19 Convergence behavior in the one-factor approach (mean value): Genetic Algorithms in comparison.


133

Then we compare the performance of a Hill-Climber, an ES, and a GA on the basis of function evaluations needed to obtain a sufficiently good result. From each group, we choose the algorithm that performed best in our previous analysis. We consider the (1 + 1)-ES, the (4, 20)-ES with global mutation and 1/5-rule, and the GA(40) with tournament selection and 1-point crossover (cf. Figure 6.20). In the Cholesky approach, the (1 + 1)-ES does not provide sufficiently good results. The (4, 20)-ES with global mutation and 1/5-rule performs best. The GA with tournament selection and 1-point crossover leads to good results too. Finally we compare the performance of the algorithms on the basis of generations (cf. Figure 6.21). In this case, the GA with tournament selection and 1-point crossover outperforms the other strategies. Again, the (1 + 1)-ES is not competitive, yielding the worst results.

Objective Function

Perf. on the Basis of Function Evaluations, Cholesky 120 100 (1+1)−ES (4,20)−ES, Global Mutation, 1/5−Rule 80 GA(40), Tourn. Selec., 1−Point Crossover 60 40

20

10

5

0

2000

4000 6000 8000 Function Evaluations

10000

Figure 6.20 Cholesky approach: performance (mean value) based on evaluations. Perf. on the Basis of Generations, Cholesky 120 100 80

(1+1)−ES (4,20)−ES, Global Mutation, 1/5−Rule GA(40), Tourn. Selec., 1−Point Crossover

Objective Function

60 40

20

10

5

0

50

100 150 Generations

200

250

Figure 6.21 Cholesky approach: performance (mean value) based on generations.

134

6.4.2.3.2


One-Factor Approach

We now compare the different ES implementa-

tions in the one-factor approach. Their performance is nearly identical, with the confidence intervals widely overlapping. There is only a very small difference, but the global mutation strategy with one strategy parameter outperforms the other approaches. Then we compare the different GA implementations in the one-factor approach. The performance of the different GA implementations differs considerably. The GA with tournament selection combined with 1-point crossover leads to the best result. The GAs with proportional selection and 1-point or 3-point crossover perform moderately. The GA with proportional selection and intermediate crossover converges too fast to a unsatisfactory solution. The intermediate crossover technique is likely to cut off that part of the search space that contains the optimal solution. As in the Cholesky approach this implementation yields the worst performance. Now we compare the performance of the different algorithms on the basis of function evaluations needed (cf. Figure 6.22) to obtain a sufficiently good result. We choose algorithms that performed well in our previous analysis. We consider the (1 + 1)-ES, the (4, 20)-ES with global mutation and one strategy parameter, and the GA(40) with tournament selection and 1-point crossover. In the one-factor approach, the Hill-Climbing strategy performs slightly better than the ES implementation, indeed there is only a very small difference. Whereas the GA implementation cannot keep up with the (1 + 1)-ES or the (4, 20)-ES implementations. In a final step, we compare the performance of the algorithms on the basis of generations (cf. Figure 6.23). In this case the (1 + 1)-ES is not competitive. The (4, 20)-ES with global mutation and one strategy parameter and the GA(40) with tournament selection and 1-point crossover are the best algorithms, with the ES performing a little better than the GA.

6.5. Implementation of a Parallel System

135

Objective Function

Perf. on the Basis of Function Evaluations, One−Factor 25 (1+1)−ES (4,20)−ES, Global Mut. With Str. Param. 20 GA(40), Tourn. Selec., 1−Point Crossover 15

10

5

0

500


2000

Figure 6.22 One-factor approach: performance (mean value) based on evaluations.

Perf. on the Basis of Generations, One−Factor 25

Objective Function

20

(1+1)−ES (4,20)−ES, Global Mut. With Str. Param. GA(40), Tourn. Selec., 1−Point Crossover

15

10

5 0

10

20 30 Generations

40

50

Figure 6.23 One-factor approach: performance (mean value) based on generations.

6.5

Implementation of a Parallel System

In the following, we consider different versions of parallel algorithms (in particular, we compare the classical global population model with regional population models), and we consider different versions of parallel implementations (in particular, we compare a serial implementation using one processor with parallel implementations using several processors). We show that parallel algorithms can significantly accelerate the optimization process up to superlinear speedups.2 For realization of the optimization procedure, a software system is required that incorporates the following three components: 2

We thank Clemens Dangelmayr for computational assistance.

136


a distributed system platform to coordinate the parallel3 optimization processes on different processors; a framework for the optimization via EAs which can be parallelized using the system platform; an implementation of the theoretical model to evaluate the objective function. The system platform used in this context is discussed in Dangelmayr (2005) and Blochinger et al. (2006). It is based on state-of-the-art technology, e.g. Java 5.0 4 , AspectJ 1.55 , and Eclipse 3.16 . The advantage of the system platform is that it can be combined with arbitrary parallel architectures. Changing resources like new processors are managed dynamically. The platform provides and integrates advanced functionality required for parallel computing. The system platform is expanded to incorporate a framework for parallel optimization in the form of EAs. For evaluation of the objective function, we use MATLAB 7.07 . Between the system platform and MATLAB a generic interface is realized. Different approaches are possible to realize the interface (e.g. the communication via the Java Native Interface and Shared Libraries with the MATLAB Runtime Engine is one option (see Dangelmayr (2006) for details and further options). The software system is tested on a cluster computer8 with 16 IBM Dual XEON 2.6 GHz processors of 2 GB RAM each. The cluster computer is connected via a 100 MBit/s network, which represents the current industry standard.

6.6 6.6.1

Performance Comparison: Parallel Algorithms Setup

We compare population models that differ in the number of subpopulations. In our study, the number of processors ranges from 1 to 14. When we implement the classical global population model, we use only one processor. When we implement a regional population model, the number of subpopulations coincides with the number of processors used. In regional population models, migration takes place in the following way:

3

For an introduction to parallel computing see Grama (2003). For details of the programming language Java see Flanagan (2005) and Lindholm and Yellin (1997). For details of aspect-oriented programming and AspectJ see Miles (2005). 6 Fairbrother et al. (2004) describe Eclipse as an open source platform for development of Java applications. 7 Visit The MathWorks on http://www.mathworks.com. 8 For details of cluster computers see Tanenbaum (2002). 4

5

6.6. Performance Comparison: Parallel Algorithms

137

any time a new individual is found that exhibits better fitness values than every other individual found so far, it is integrated in every other subpopulation. We illustrate our approach using market data for the on-the-run Dow Jones CDX.NA.IG 5 yr index on June 17, 2004. The Dow Jones CDX.NA.IG 5 yr comprises the 125 most actively traded single name CDS contracts in the North American investment grade segment. An equal weighting is given to each name. The time to maturity is 5 years. We assume that the assets have equal recovery rates of 40%. The CDX index level is 62 bps. The given upfront payment for the 0%-3% tranche is 43.25%, the given tranche spread for the 3%-7% tranche is 342.5 bps, 126.0 bps for the 7%-10% tranche, 50.0 bps for the 10%-15% tranche, and 13.50 bps for the 15%-30% tranche. By convention, the equity tranche is quoted in a different way from the other tranches of the CDO structure. The market quote of 43.25% means that the tranche holder receives 500 bps per year on the outstanding principal plus an initial payment of 43.25% of the tranche principal. The tranche spreads of the mezzanine and senior tranches are paid annually. Our goal is to find a correlation matrix that models the observed tranche spreads. To indicate the quality of an obtained correlation matrix, we choose the fitness function f2 that was introduced in Section 5.4.1. The threshold for program termination is 25%. In our study, we choose the well-established one-factor approach for the optimization process, in line with Section 5.4.2. In a first step, different versions of serial algorithms are compared to analyze which parameter combinations and evolutionary operators perform best in this setup. In a second step, the combination of parameters and operators that performed best are analyzed in a parallel setup. Having compared different versions of serial algorithms, we choose a (5 + 25)-ES. We then apply proportional selection (see Section 4.4.1), i.e. every individual has the chance to be selected with a probability proportional to its fitness value. Individuals are selected with replacement. Individuals are recombined using intermediate crossover (see Section 4.4.2), where the offspring genotype is the mean of the parent genotypes. The crossover probability is 1. We apply global mutation (see Section 4.4.3). We mutate individuals by adding realizations of normally distributed random variables with expected value 0 and randomly varying variance on each element of the genotype. The mutation probability is 1. To compare the performance of the different algorithms with regard to the problem in consideration, the algorithms search the generations until the termination criterion is met (i.e. until an individuum is found that exceeds a predefined fitness level). In order to determine the speedups, the running times are compared. We start with the running time obtained

138


when the maximal number of processors is used. Recursively, we compute the speedups of the different implementations up to the serial implementation. The efficiency is defined as the speedup divided by the number of processors.

6.6.2

Results

To make sure that we obtain reliable results, we repeat each implementation 25 times. We consider the mean value of the objective functions over the 25 runs. We consider the decline of the objective function in the course of the generations. The speedups and the efficiency for different numbers of processors are displayed in Table 6.6. Superlinearity can be realized for our optimization problem: using 14 processors we obtain a speedup of 1700% which corresponds to an efficiency of 125%. Superlinearity is possible, because the concept of a regional population model differs substantially from the concept of a classical global population model. For our optimization problem, the introduction of regionality clearly leads to performance gains.

Table 6.6 Speedup and efficiency obtained by parallel algorithms and parallel implementation. Processors

Speedup

Efficiency in %

1 4 8 14

1.00 6.12 11.70 17.56

100 153 146 125

6.7

Deriving the Dependence Structure From Market Data

In the following we aim to derive a correlation matrix modeling a set of tranche spreads (or analogously a correlation smile) which is actually observed in the market. We choose the same setup as in 6.6. We illustrate our approach using market data for the on-the-run Dow Jones CDX.NA.IG 5 yr index on June 17, 2004. Thus, we consider a homogeneous portfolio of 125 assets with an equal unit nominal. The time to maturity is 5 years. The recovery rate is assumed to be 40%. The CDX index level on June 17, 2004 was 62 bps.

6.7. Deriving the Dependence Structure From Market Data

139

The given upfront payment for the 0%-3% tranche was 43.25%, while the given tranche spread for the 3%-7% tranche was 342.5 bps, 126.0 bps for the 7%-10% tranche, 50.0 bps for the 10%-15% tranche, and 13.50 bps for the 15%-30% tranche. We derive the implied tranche correlations from market spreads using the standard market model. The resulting implied tranche correlations are 0.175 for the 0%-3% tranche, 0.041 for the 3%-7% tranche, 0.165 for the 7%-10% tranche, 0.198 for the 10%-15% tranche, and 0.268 for the 15%-30% tranche. We aim to find a suitable correlation matrix such that the market quotes on June 17, 2004 can be reproduced for all tranches. To indicate the quality of an obtained correlation matrix, we choose the fitness function f2 . In order to limit the number of free parameters and to speed up the evolutionary search, we choose the one-factor approach for the optimization process. Using the one-factor approach, the genotype is a row vector in [−1, 1]125 . Note that we do not know in advance whether the actual dependence structure of the underlying portfolio can be represented by a one-factor setup. However, experience shows that the one-factor approach generally has enough degrees of freedom to obtain a satisfying fit to market data. The implied correlation matrix that is the result of the evolutionary search is pictured in Figure 6.24. Using the implied correlation matrix, we obtain a very good fit to market data (cf. Tables 6.7 and 6.8). Figure 6.25 illustrates the quality of the fit by displaying the implied correlation smiles. We emphasize that our approach is competitive with other models recently been suggested when it comes to modeling the implied correlation smile. Nevertheless, we admit that using the derived correlation matrix does not always yield a perfect fit to market data. There are several potential explanations for this deficiency. First, it is possible that the algorithm is unable to handle 125 free parameters and consequently does not find the optimum. However, this aspect can be neglected. Whenever we work with simulated data and try to reproduce the tranche spreads that are obtained using an arbitrary predefined correlation matrix, these target spreads can generally be reproduced even for 125 free parameters. Second, note that we terminated our algorithm after a relatively short period of time in order to limit the computational burden. This may explain the merely approximate solution. Third, it is possible that the actual correlation structure of the underlying portfolio cannot be encoded in the form of a one-factor setup. Finally, there may be other aspects that influence the characteristic smile structure apart from the heterogeneity of the underlying dependence structure.

140



1

0.8

0.6

0.4 1 0.2

25 50

0

75 1

25

100

50

75

100

125

Assets

Assets

Figure 6.24 The implied correlation matrix.

Table 6.7 Tranche spreads (in bps per year). 0%-3% 3%-7% Market Spreads Model Spreads

4325.0 4492.0

7%-10% 10%-15% 15%-30%

342.5 342.3

126.0 123.8

50.0 58.2

13.5 13.4

Table 6.8 Implied tranche correlations. 0%-3% 3%-7% 7%-10% 10%-15% 15%-30% Market Correlations Model Correlations

0.175 0.157

0.041 0.041

0.165 0.162

0.198 0.218

0.268 0.268

6.8. Conclusion

141


0.4 Market Implied Correlation Smile Model Implied Correlation Smile

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0%−3%

3%−7% 7%−10% 10%−15%15%−30% Tranche

Figure 6.25 Fitting the implied correlation smile.

6.8

Conclusion

In this chapter, we analyzed the performance of several algorithms regarding the optimization problem illustrated in Chapter 5. We presented some basic and some more advanced implementations of EAs and discussed their performance in a Cholesky, a onefactor, and a cluster approach. We showed that population-based algorithms can often significantly speed up convergence, especially when performance is measured based on the number of generations needed to obtain a sufficiently good result. Considering a one-factor setup, we illustrated how the use of a regional population model can speed up the evolutionary search. Comparing a regional population model with a classical global population model, the optimization procedure requires less objective function evaluations to obtain a certain fitness level when the regional population model is applied. The implementation of a regional model on a parallel computing platform reduces computational time for two reasons. First, we get a speedup because function evaluations are carried out simultaneously on different processors. Second, a model that incorporates regionality requires less objective function evaluations than a classical global population model. Using the regional model, we find a sufficiently good solution more often or with less objective function evaluations. We conclude that it is advantageous to implement a regional population model even on a serial/standard computer. Serial implementation of the parallel algorithm (pseudoparallel implementation) is highly recommended.

142


Not least, the analysis of our suggested approach on the basis of market data shows that the use of a heterogeneous correlation matrix allows for a very good fit to market data. This result emphasizes the relevance of our setup for the current academic debate. It is assumed that future developments will give an additional boost to algorithms that can be easily parallelized. On September 26, 2006 Intel presented a prototype of a processor with 80 cores. This processor is supposed to perform a trillion floating-point operations per second - a teraflop - on a single chip. The company hopes to have the 80core processor ready for commercial production around the end of the decade. Presently, dual-core processors constitute the market standard. EAs and other algorithms that are suitable for parallel implementation will benefit from the development of chips with multiple cores, because different threads can run simultaneously on different cores.

Chapter 7 Summary and Outlook In this thesis we discussed the valuation of correlation products, in particular the valuation of CDS index tranches. The one-factor Gaussian copula approach, which is considered to be the market standard, assumes in its simplest version that all pairwise default time correlations in the underlying portfolio are the same. Using this setup, different correlation levels are needed to reproduce the market prices of different tranches on the same underlying portfolio. Generally, mezzanine tranches trade at lower correlation levels than equity and senior tranches. This phenomenon has come to be denoted as the correlation smile. The correlation smile illustrates that the standard market model is not able to consistently reproduce market prices, because different parameter sets are used to reproduce the prices of the different tranches of one CDO structure. In this study, we focused on the use of heterogeneous correlation structures, in order to explain the observed correlation smile. We exemplified that inhomogeneous dependence structures can generate characteristic correlation smiles when used in combination with the Gaussian copula model. We concluded that the observed correlation smile may occur because the actual correlation structure of the underlying portfolio is inhomogeneous. Therefore the use of heterogeneous dependence structures can improve the fit to market data. We suggested deriving a correlation matrix chosen such that all tranche prices of a traded CDO can be reproduced when this correlation matrix is used in conjunction with the Gaussian copula approach. We pointed out that using an appropriate correlation structure is advantageous, because only one dependence structure is used for the valuation of all CDO tranches - which agrees with the fact that the underlying is the same. The obtained correlation matrix can be used to price off-market products with the same underlying as the traded CDO.

144

Chapter 7. Summary and Outlook

The price to pay for a better fit to market data is the increased complexity of model calibration. As there is no standard optimization technique for deriving the correlation structure from market prices, we used EAs. These algorithms were chosen because they can handle optimization problems that are very complex due to high dimensionality, non-linearity, non-differentiability, and discontinuity of the objective function, noise in the measurements, multiple kinds of constraints, multi-modality, and largeness of search space or other unusual features thereof. EAs can be applied when there is little or no a priori knowledge of the problem in consideration. No information or initial estimate of the optimum is required. Similar to other heuristic search methods, there is no guarantee that EAs will find the global optimum, but they generally find good solutions in a reasonable amount of time. To speed up the evolutionary search, we deployed the concept of parallelization. Besides the parallel implementation, which consists in the simultaneous valuation of the fitness of different individuals on different processors, we considered parallel algorithms, in particular population models, where different subpopulations evolve independently of each other. We showed that parallelization can significantly accelerate the optimization process. We emphasized that the parallel implementation of parallel algorithms even allows for superlinear speed-ups. We thus secured a considerable improvement in the usability of our suggested approach. We analyzed the applicability of our approach using market data. We were able to obtain correlation matrices that provide considerably good fits to market data. We emphasized that our approach is competitive with other models that have recently been suggested to reproduce the prices of CDS index tranches. However, there may be sets of tranche prices that cannot be reproduced sufficiently well in a reasonable amount of time. We provided potential explanations for this deficiency.

In conclusion, we summarize our two main results. First, we illustrated that heterogeneous correlation structures are a possible explanation for the observed correlation smile. Second, we applied EAs in the area of portfolio credit derivatives and we outlined how to use this optimization method to derive a correlation structure that allows quoted tranche prices to be reproduced consistently.

Chapter 7. Summary and Outlook

145

Even if a relatively liquid and transparent market for tranched credit risk evolved in recent years, we point out that the credit derivatives market is still not yet fully developed. At the current stage it is supposable that market imperfections contribute biased effects which aggravate the fit to observed data. Irrespective of these misbalances, it is important that pricing models are developed further. In this study we illustrated the tremendous influence of heterogeneous default dependence structures on the risk content of portfolio credit derivatives. In this regard, frameworks have to be enhanced so as to realistically capture credit risk correlations. The development of stochastic correlation models incorporating the dynamics of tranche prices beyond purely spread-driven dynamics will be a major challenge.

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Credit Suisse's Credit Portfolio Modeling Handbook

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Advances in Evolutionary Algorithms

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