Financial Engineering with Copulas Explained (eBook)

Dr. Matthias Scherer is Professor of Mathematical Finance at the Technische Universität München, where he gives lectures in Mathematical Finance and Statistics. His research interests span Mathematical Finance, but focus on credit-risk analysis and the application of copulas. He holds a PhD from the University of Ulm, and a Masters in Mathematics from Syracuse University. Dr. Scherer has co-authored numerous articles on financial topics including dependence modeling and the book Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications. Dr. Jan-Frederik Mai is Quantitative Analyst at XAIA Investment GmbH. He holds a PhD in Financial Mathematics from Technische Universität München and is co-author of numerous research articles in the field of dependence modeling and of the book Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications.

Cover 1
Half-Title 2
Title 4
Copyright 5
Dedication 6
Contents 8
1 What Are Copulas? 18
1.1 Two Motivating Examples 18
1.1.1 Example 1: Analyzing Dependence between Asset Movements 18
1.1.2 Example 2: Modeling the Dependence between Default Times 23
1.2 Copulas and Sklar’s Theorem 23
1.2.1 The Generalized Inverse 26
1.2.2 Sklar’s Theorem for Survival Functions 28
1.2.3 How to Apply Sklar’s Theorem? 29
1.3 General Copula Properties 31
2 Which Rules for Handling Copulas Do I Need? 36
2.1 The Fréchet–Hoeffding Bounds 36
2.2 Switching from Distribution to Survival Functions 38
2.3 Invariance Under Strictly Monotone Transformations 42
2.4 Computing Probabilities from a Distribution Function 45
2.5 Copula Derivatives 47
2.6 Constructing New Copulas from Existing Ones 49
3 How to Measure Dependence? 52
3.1 Pearson’s Correlation Coefficient 52
3.2 Concordance Measures 56
3.2.1 Using Kendall’s t and Spearman’s ?S 61
3.3 Tail Dependence 62
4 What are Popular Families of Copulas? 66
4.1 Gaussian Copulas 66
4.1.1 Important Stylized Facts of the (Bivariate) Gaussian Copula 68
4.1.2 Generalization to Elliptical Copulas 72
4.2 Archimedean Copulas 74
4.2.1 Stylized Facts of Archimedean Copulas 77
4.2.2 Hierarchical Archimedean Copulas 79
4.3 Extreme-value Copulas 81
4.3.1 Marshall–Olkin Copulas 83
4.3.2 Stylized Facts of Extreme-Value Copulas 87
4.4 Archimax Copulas 88
5 How to SimulateMultivariate Distributions? 91
5.1 How to Simulate from a Copula? 95
5.1.1 Simulation Based on Analytical Techniques 95
5.1.2 Simulation Along a Stochastic Model 96
5.1.3 Practical Guide for the Implementation 100
6 How to Estimate Parameters of a Multivariate Model? 102
6.1 The Method ofMoments 103
6.1.1 Some Theoretical Background 104
6.2 Maximum-Likelihood Methods 105
6.2.1 Perfect Information about the Marginal Laws 106
6.2.2 Joint Maximization Over a and ?: Full Maximum-Likelihood 107
6.2.3 Inference Functions forMargins (IFM)Method 107
6.3 Using A Rank Transformation to Obtain (Pseudo-)Samples 108
6.3.1 Visualization of theMethods 111
6.4 Estimation of Specific Copula Families 113
6.4.1 Taylor-made Estimation Strategies for Extreme-value Copulas 114
6.5 A Note on Positive Semi-Definiteness 116
6.6 Some Remarks Concerning the Implementation 118
7 How to Deal with Uncertainty Concerning Dependence? 120
7.1 Bounds for the VaR of a Portfolio 122
7.2 What is the Maximal Probability for a Joint Default? 129
7.2.1 Motivation 129
7.2.2 Maximal Coupling 130
8 How to Construct a Portfolio-Default Model? 134
8.1 The Canonical Construction of Default Times 134
8.2 Classical CopulaModels for Dependent Default Times 136
8.2.1 The Portfolio-loss Distribution 138
8.3 A FactorModel for CDO Pricing 143
8.3.1 An Excursion to CDO Pricing 143
8.3.2 Calibrating the Two Portfolio-default Models 151
References 155
Index 166