Copula Theory and Its Applications (eBook)

Foreword 5
Preface 6
Contents 7
Contributors 13
Part I Surveys 17
1 Copula Theory: An Introduction 18
Fabrizio Durante and Carlo Sempi 18
1.1 Historical Introduction 18
1.1.1 Outline 21
1.2 Preliminaries on Random Variables and Distribution Functions 21
1.3 Copulas: Definitions and Basic Properties 24
1.4 Sklar's Theorem 27
1.5 Copulas and Random Vectors 29
1.6 Families of Copulas 30
1.6.1 Elliptical Copulas 31
1.6.2 Archimedean Copulas 32
1.6.3 EFGM Copulas 34
1.7 Constructions of Copulas 35
1.7.1 Copulas with Given Lower Dimensional Marginals 35
1.7.2 Copula-to-Copula Transformations 36
1.7.3 Geometric Constructions of Copulas 37
1.8 Copula Theory: What's the Future? 38
References 39
2 Dynamic Modeling of Dependence in Finance via Copulae Between Stochastic Processes 47
Tomasz R. Bielecki, Jacek Jakubowski and Mariusz Nieweg?owski 47
2.1 Introduction 47
2.2 Lévy Copulae 49
2.3 Semimartingale Copulae 53
2.3.1 Copulae for Special Semimartingales 53
2.3.2 Consistent Semimartingale Copulae 62
2.4 Markov Copulae 68
2.4.1 Consistent Markov Processes 69
2.4.2 Markov Copulae: Generator Approach 71
2.4.3 Markov Copulae: Symbolic Approach 77
2.5 Applications in Finance 83
2.5.1 Pricing Rating-Triggered Step-Up Bonds via Simulation 84
2.5.2 Model Calibration and Pricing 86
References 89
3 Copula Estimation 91
Barbara Choros, Rustam Ibragimov and Elena Permiakova 91
3.1 Introduction 91
3.2 Copula Estimation: Random Samples with Dependent Marginals 92
3.2.1 Parametric Models: Maximum Likelihood Methods and Inference from Likelihoods for Margins 92
3.2.2 Semiparametric Estimation 94
3.2.3 Nonparametric Inference and Empirical Copula Processes 95
3.3 Copula-Based Time Series and Their Estimation 96
3.3.1 Copula-Based Characterizations for (Higher-Order) Markov Processes 96
3.3.2 Parametric and Semiparametric Copula Estimation Methods for Markov Processes 97
3.3.3 Nonparametric Copula Inference for Time Series 98
3.3.4 Dependence Properties of Copula-Based Time Series 99
3.4 Further Copula Inference Methods 100
3.5 Empirical Applications 101
References 103
4 Pair-Copula Constructions of Multivariate Copulas 106
Claudia Czado 106
4.1 Introduction 106
4.2 Pair Copula Constructions of D-Vine, Canonical and Regular Vine Distributions 107
4.2.1 Pair-Copula Constructions of D-Vine and Canonical Vine Distributions 107
4.2.2 Regular Vines Distributions and Copulas 109
4.3 Estimation Methods for Regular Vine Copulas 113
4.4 Model Selection Among Vine Specifications 116
4.5 Applications of Vine Distributions 118
4.6 Summary and Open Problems 119
References 120
5 Risk Aggregation 123
Paul Embrechts and Giovanni Puccetti 123
5.1 Motivations and Preliminaries 124
5.1.1 The Mathematical Framework 124
5.2 Bounds for Functions of Risks: The Coupling-Dual Approach 125
5.2.1 Application 1: Bounding Value-at-Risk 127
5.2.2 Application 2: Supermodular Functions 131
5.3 The Calculation of the Distribution of the Sum of Risks 132
5.3.1 Open Problems 136
References 137
6 Extreme-Value Copulas 139
Gordon Gudendorf and Johan Segers 139
6.1 Introduction 139
6.2 Foundations 140
6.3 Parametric Models 143
6.3.1 Logistic Model or Gumbel--Hougaard Copula 144
6.3.2 Negative Logistic Model or Galambos Copula 144
6.3.3 Hüsler--Reiss Model 145
6.3.4 The t-EV Copula 146
6.4 Dependence Coefficients 146
6.5 Estimation 148
6.5.1 Parametric Estimation 149
6.5.2 Nonparametric Estimation 150
6.6 Further Reading 152
References 153
7 Construction and Sampling of Nested Archimedean Copulas 158
Marius Hofert 158
7.1 Introduction 158
7.2 Nested Archimedean Copulas 160
7.3 A Sufficient Nesting Condition 162
7.4 Construction of Nested Archimedean Copulas 164
7.5 Sampling Nested Archimedean Copulas 166
7.6 Conclusion 170
References 170
8 Tail Behaviour of Copulas 172
Piotr Jaworski 172
8.1 Introduction 172
8.2 Tail Expansions of Copulas 174
8.2.1 Characterization and Properties of Leading Parts 178
8.2.2 Relatively Invariant Measures on [0,)n 179
8.3 Examples of Tail Expansions 180
8.3.1 Homogeneous Copulas 180
8.3.2 Diagonal Copulas 180
8.3.3 Absolutely Continuous Copulas 182
8.3.4 Archimedean Copulas 183
8.3.5 Multivariate Extreme Value Copulas 188
8.4 Applications 189
8.4.1 Tail Conditional Copulas 189
8.4.2 Extreme Value Copulas of a Given Copula 191
8.4.3 Regularly Varying Random Vectors with a Given Copula 192
8.4.4 Value at Risk 193
References 196
9 Copulae in Reliability Theory (Order Statistics, Coherent Systems) 198
Tomasz Rychlik 198
9.1 Coherent Systems 198
9.2 Signatures 200
9.2.1 Components with i.i.d. Lifetimes 200
9.2.2 Mixed Systems 201
9.2.3 Components with Exchangeable Lifetimes 203
9.3 Bounds for Exchangeable Lifetime Components 205
9.3.1 Distribution Bounds 205
9.3.2 Expectation Bounds 207
9.4 Characterizations of k-Out-of-n System Lifetime Distributions 209
9.4.1 General Copula Joint Distribution 210
9.4.2 Absolute Continuous Copula Joint Distribution 211
9.4.3 Variance Bounds 214
9.5 Final Remarks 216
References 217
10 Copula-Based Measures of Multivariate Association 220
Friedrich Schmid, Rafael Schmidt, Thomas Blumentritt, Sandra Gaißer and Martin Ruppert 220
10.1 Introduction and Definitions 220
10.2 Aspects of Multivariate Association 223
10.3 Multivariate Generalizations of Spearman's Rho, Kendall's Tau, Blomqvist's Beta, and Gini's Gamma 226
10.3.1 Spearman's Rho 226
10.3.2 Kendall's Tau 228
10.3.3 Blomqvist's Beta 230
10.3.4 Gini's Gamma 231
10.4 Information-Based Measures of Multivariate Association 232
10.5 Measures of Multivariate Association Based on Lp-Distances 235
10.5.1 2 as a L2-Distance-Based Measure 236
10.5.2 as a L1-Distance-Based Measure 238
10.5.3 as a L-Distance-Based Measure 238
10.6 Multivariate Tail Dependence 239
References 243
11 Semi-Copulas and Interpretations of Coincidences Between Stochastic Dependence and Ageing 248
Fabio Spizzichino 248
11.1 Introduction 248
11.2 Univariate Ageing and Dependence Properties of Archimedean Semi-Copulas 251
11.3 Dependence and Univariate Ageing in Schur-Constant Models 254
11.4 Level Curves, B functions, Duality, and Interpretation of Coincidence Between Ageing and Dependence 258
11.5 Summary and Concluding Remarks 262
References 263
Part II Contributed Papers 266
12 A Copula-Based Model for Spatial and Temporal Dependence of Equity Markets 267
Umberto Cherubini, Fabio Gobbi, Sabrina Mulinacci and Silvia Romagnoli 267
12.1 Introduction 267
12.2 A market Model in Discrete Time 268
12.3 The Martingale Property 269
12.4 Applications 271
12.4.1 Multivariate Digital Options 271
12.4.2 Basket and Spread Options 273
References 274
13 Nonparametric and Semiparametric Bivariate Modeling of Petrophysical Porosity-Permeability Dependence from Well Log Data 276
Arturo Erdely and Martin Diaz-Viera 276
13.1 Introduction 276
13.2 Methodology 277
13.3 Data Analysis 280
13.4 Final Remarks 285
References 287
14 Testing Under the Extended Koziol-Green Model 288
Auguste Gaddah and Roel Braekers 288
14.1 Introduction 288
14.2 Asymptotic Results 291
14.3 Test Statistics 294
14.4 Data Example: Survival with Malignant Melanoma 295
References 297
15 Parameter Estimation and Application of the Multivariate Skew t-Copula 298
Tõnu Kollo Gaida Pettere 298
15.1 Introduction 298
15.2 Preliminary Notions and Notation 299
15.3 Construction of a Skew t-Copula 301
15.4 Parameter Estimation 302
15.5 Simulation 304
15.6 Application 305
References 307
16 On Analytical Similarities of Archimedean and Exchangeable Marshall-Olkin Copulas 308
Jan-Frederik Mai and Matthias Scherer 308
16.1 Introduction 308
16.2 Complete Monotonicity and d-Monotonicity 310
16.2.1 Definitions and Examples 310
16.2.2 Probabilistic Interpretations 311
16.2.3 d-Monotonicity 312
16.3 Probabilistic Models and Sampling 314
16.3.1 The Completely Monotone Case 314
16.3.2 The Proper d-Monotone Case 315
References 317
17 Relationships Between Archimedean Copulas and Morgenstern Utility Functions 319
Jaap Spreeuw 319
17.1 Introduction 319
17.2 Archimedean Copulas 320
17.3 Utility Functions 320
17.4 Relationships Between Properties of Utility Functions and Properties of Generators 323
17.5 Examples 326
17.5.1 Classical Cases 326
17.5.2 The HARA Family 327
17.5.3 The Expo Power Utility 327
17.5.4 Other Examples of Decreasing Absolute Risk Aversion (DARA) as in Pratt [9] 327
17.6 Conclusion 329
References 329
Index 331