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Table of Contents
Preface; Objectives, Audience and Structure; Description of Contents by Chapter; Acknowledgements; Contents; List of Figures; 1 Introduction and Context; 1.1 Synopsis of Credit Derivative Products; 1.1.1 Credit Default Swaps; 1.1.2 First to Default Swaps; 1.1.3 Collateralized Debt Obligations; 1.2 Motivation for Credit Correlation Models; 1.3 A Timeline of Credit Correlation Modelling; References; Theoretical Tools; 2 Mathematical Fundamentals; 2.1 Credit Pricing Building Blocks; 2.1.1 Cox Process; 2.1.2 Three Building Blocks; 2.2 Point Processes, Filtrations and Intensities
2.2.1 Counting Process2.2.2 Doubly Stochastic Poisson Process; 2.2.3 Watanabe's Characterization; 2.2.4 Stochastic Intensity; 2.2.5 Predictable Intensities; 2.2.6 Change of Filtration; 2.2.7 Random Time Change; 2.3 Copulas; 2.3.1 Sklar's Theorem; 2.3.2 Dependence Concepts; 2.3.3 Elliptical Copulas; 2.3.4 Archimedean Copulas; 2.3.5 Marshall-Olkin Copulas; References; 3 Expectations in the Enlarged Filtration; 3.1 The Dellacherie Formula; 3.2 Generalized Dellacherie Formula; References; 4 Copulas and Conditional Jump Diffusions; 4.1 Introduction; 4.2 The Model
4.3 Interacting Ità ́and Point Processes4.4 The Copula Approach; 4.5 Numerical Examples; 4.6 Conclusion; Note; References; Correlation Models: Practical Implementation; 5 Correlation Demystified: A General Overview; 5.1 Base Correlation; 5.1.1 One-Factor Gaussian Copula; 5.1.2 Pricing CDOs; 5.1.3 Large Homogenous Portfolio; 5.1.4 FFT and Recursion; 5.1.5 Normal, Poisson and Stein Approximations; 5.1.6 Compound Correlation; 5.1.7 Base Correlation Curve; 5.2 Skew Rescaling; 5.2.1 Portfolio Loss Rescaling; 5.2.2 Probability Loss Rescaling; 5.2.3 Tranche Loss Rescaling
5.2.4 Mapping, Blending and Interpolation5.3 CDO2; 5.3.1 Loss Copula; 5.3.2 Conditional Loss Copula; 5.3.3 Bespoke CDO2 Skew; 5.3.4 Summary; 5.4 Expected Tranche Loss Surface; 5.4.1 The Problem; 5.4.2 Equity Tranche Forward Rate Curve; 5.5 Entropy Maximization; 5.5.1 Principle of Maximum Entropy; 5.5.2 Problem Formulation; 5.5.3 Dual Problem; 5.5.4 Regularization; 5.5.5 Minimum Relative Entropy; 5.6 Concluding Remarks; References; 6 Correlation Skew: A Black-Scholes Approach; 6.1 Introduction; 6.2 Building a Black-Scholes Model; 6.3 Stochastic CEV Model; 6.4 Calibration Example
6.5 Skew Dynamics6.6 Risk Management; References; 7 An Introduction to the Marshall-Olkin Copula; 7.1 Introduction; 7.2 Genesis of the Marshall-Olkin Model; 7.2.1 Construction of Correlation; 7.2.2 Copula Function; 7.2.3 Numerical Implementation; 7.3 Calibration; 7.3.1 Background Radiation; 7.3.2 The Expanding Universe; 7.3.3 The Big Bang State; 7.3.4 Correlation Regimes; 7.4 Marshall-Olkin Vs Gaussian Copula; 7.4.1 Multimodal Default Distribution; 7.4.2 Correlation Term Structure; 7.4.3 Correlation Skew; 7.5 Conclusion; Notes; References; 8 Numerical Tools: Basket Expansions
2.2.1 Counting Process2.2.2 Doubly Stochastic Poisson Process; 2.2.3 Watanabe's Characterization; 2.2.4 Stochastic Intensity; 2.2.5 Predictable Intensities; 2.2.6 Change of Filtration; 2.2.7 Random Time Change; 2.3 Copulas; 2.3.1 Sklar's Theorem; 2.3.2 Dependence Concepts; 2.3.3 Elliptical Copulas; 2.3.4 Archimedean Copulas; 2.3.5 Marshall-Olkin Copulas; References; 3 Expectations in the Enlarged Filtration; 3.1 The Dellacherie Formula; 3.2 Generalized Dellacherie Formula; References; 4 Copulas and Conditional Jump Diffusions; 4.1 Introduction; 4.2 The Model
4.3 Interacting Ità ́and Point Processes4.4 The Copula Approach; 4.5 Numerical Examples; 4.6 Conclusion; Note; References; Correlation Models: Practical Implementation; 5 Correlation Demystified: A General Overview; 5.1 Base Correlation; 5.1.1 One-Factor Gaussian Copula; 5.1.2 Pricing CDOs; 5.1.3 Large Homogenous Portfolio; 5.1.4 FFT and Recursion; 5.1.5 Normal, Poisson and Stein Approximations; 5.1.6 Compound Correlation; 5.1.7 Base Correlation Curve; 5.2 Skew Rescaling; 5.2.1 Portfolio Loss Rescaling; 5.2.2 Probability Loss Rescaling; 5.2.3 Tranche Loss Rescaling
5.2.4 Mapping, Blending and Interpolation5.3 CDO2; 5.3.1 Loss Copula; 5.3.2 Conditional Loss Copula; 5.3.3 Bespoke CDO2 Skew; 5.3.4 Summary; 5.4 Expected Tranche Loss Surface; 5.4.1 The Problem; 5.4.2 Equity Tranche Forward Rate Curve; 5.5 Entropy Maximization; 5.5.1 Principle of Maximum Entropy; 5.5.2 Problem Formulation; 5.5.3 Dual Problem; 5.5.4 Regularization; 5.5.5 Minimum Relative Entropy; 5.6 Concluding Remarks; References; 6 Correlation Skew: A Black-Scholes Approach; 6.1 Introduction; 6.2 Building a Black-Scholes Model; 6.3 Stochastic CEV Model; 6.4 Calibration Example
6.5 Skew Dynamics6.6 Risk Management; References; 7 An Introduction to the Marshall-Olkin Copula; 7.1 Introduction; 7.2 Genesis of the Marshall-Olkin Model; 7.2.1 Construction of Correlation; 7.2.2 Copula Function; 7.2.3 Numerical Implementation; 7.3 Calibration; 7.3.1 Background Radiation; 7.3.2 The Expanding Universe; 7.3.3 The Big Bang State; 7.3.4 Correlation Regimes; 7.4 Marshall-Olkin Vs Gaussian Copula; 7.4.1 Multimodal Default Distribution; 7.4.2 Correlation Term Structure; 7.4.3 Correlation Skew; 7.5 Conclusion; Notes; References; 8 Numerical Tools: Basket Expansions