Simulating Copulas: Stochastic Models, Sampling Algorithms, And Applications (Second Edition)

Simulating Copulas: Stochastic Models, Sampling Algorithms, And Applications (Second Edition)

by Matthias Scherer, Jan-frederik Mai

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Product Details

ISBN-13: 9789813149243
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 08/02/2017
Series: Series In Quantitative Finance Series , #6
Pages: 356
Product dimensions: 6.50(w) x 1.50(h) x 9.50(d)

Table of Contents

Preface vii

1 Introduction 1

1.1 Copulas 4

1.1.1 Analytical Properties 7

1.1.2 Sklar's Theorem and Survival Copulas 14

1.1.3 General Sampling Methodology in Low Dimensions 22

1.1.4 Graphical Visualization 26

1.1.5 Concordance Measures 28

1.1.6 Measures of Extremal Dependence 33

1.2 General Classifications of Copulas 36

1.2.1 Radial Symmetry 36

1.2.2 Exchangeability 39

1.2.3 Homogeneous Mixture Models 41

1.2.4 Heterogeneous Mixture Models/Hierarchical Models 48

1.2.5 Extreme-Value Copulas 52

2 Archimedean Copulas 57

2.1 Motivation 58

2.2 Extendible Archimedean Copulas 61

2.2.1 Kimberling's Result and Bernstein's Theorem 62

2.2.2 Properties of Extendible Archimedean Copulas 65

2.2.3 Constructing Multi-Parametric Families 69

2.2.4 Parametric Families 69

2.3 Exchangeable Archimedean Copulas 76

2.3.1 Constructing Exchangeable Archimedean Copulas 82

2.3.2 Sampling Exchangeable Archimedean Copulas 85

2.3.3 Properties of Exchangeable Archimedean Copulas 87

2.4 Hierarchical (H-Extendible) Archimedean Copulas 89

2.4.1 Compatibility of Generators 90

2.4.2 Probabilistic Construction and Sampling 91

2.4.3 Properties 93

2.4.4 Examples 95

2.5 Other Topics Related to Archimedean Copulas 97

2.5.1 Simulating from the Generator 97

2.5.2 Asymmetrizing Archimedean Copulas 99

3 Marshall-Olkin Copulas 101

3.1 The General Marshall Olkin Copula 102

3.1.1 Canonical Construction of the MO Distribution 104

3.1.2 Alternative Construction of the MO Distribution 110

3.1.3 Properties of Marshall-Olkin Copulas 118

3.2 The Exchangeable Case 122

3.2.1 Reparameterizing Marshall-Olkin Copulas 126

3.2.2 The Inverse Pascal Triangle 129

3.2.3 Efficiently Sampling eMO 131

3.2.4 Hierarchical Extensions 138

3.3 The Extendible Case 140

3.3.1 Precise Formulation and Proof of Theorem 3.1 141

3.3.2 Proof of Theorem 3.2 146

3.3.3 Efficient Simulation of Lévy-Frailty Copulas 150

3.3.4 Hierarchical (H-Extendible) Lévy-Frailty Copulas 153

4 Elliptical Copulas 159

4.1 Spherical Distributions 161

4.2 Elliptical Distributions 166

4.3 Parametric Families of Elliptical Distributions 170

4.4 Elliptical Copulas 174

4.5 Parametric Families of Elliptical Copulas 175

4.6 Sampling Algorithms 179

4.6.1 A Generic Sampling Scheme 179

4.6.2 Sampling Important Parametric Families 181

5 Pair Copula Constructions 185

5.1 Introduction to Pair Copula Constructions 186

5.2 Copula Construction by Regular Vine Trees 191

5.2.1 Regular Vines 191

5.2.2 Regular Vine Matrices 196

5.3 Simulation from Regular Vine Distributions 203

5.3.1 h-Functions for Bivariate Copulas and Their Rotated Versions 204

5.3.2 The Sampling Algorithms 208

5.4 Dependence Properties 218

5.5 Application 223

5.5.1 Time Series Model for Each Margin 224

5.5.2 Parameter Estimation 224

5.5.3 Forecasting Value at Risk 226

5.5.4 Backtesting Value at Risk 227

5.5.5 Backtest Results 228

6 Sampling Univariate Random Variables 231

6.1 General Aspects of Generating Random Variables 231

6.2 Generating Uniformly Distributed Random Variables 232

6.2.1 Quality Criteria for RNG 233

6.2.2 Common Causes of Trouble 234

6.3 The Inversion Method 234

6.4 Generating Exponentially Distributed Random Numbers 235

6.5 Acceptance-Rejection Method 235

6.6 Generating Normally Distributed Random Numbers 238

6.6.1 Calculating the Cumulative Normal 238

6.6.2 Generating Normally Distributed Random Numbers via Inversion 238

6.6.3 Generating Normal Random Numbers with Polar Methods 239

6.7 Generating Lognormal Random Numbers 240

6.8 Generating Gamma-Distributed Random Numbers 240

6.8.1 Generating Gamma-Distributed UNs with β > 1 241

6.8.2 Generating Gamma-Distributed RNs with β < 242

6.8.3 Relations to Other Distributions 243

6.9 Generating Chi-Square-Distributed RNs 243

6.10 Generating t-Distributed Random Numbers 244

6.11 Generating Pareto-Distributed Random Numbers 245

6.12 Generating Inverse Gaussian-Distributed Random Numbers 245

6.13 Generating Stable-Distributed Random Numbers 246

6.14 Generating Discretely Distributed Random Numbers 247

6.14.1 Generating Random Numbers with Geometric and Binomial Distribution 248

6.14.2 Generating Poisson-Distributed Random Numbers 248

7 The Monte Carlo Method 251

7.1 First Aspects of the Monte Carlo Method 251

7.2 Variance Reduction Methods 254

7.2.1 Antithetic Variates 255

7.2.2 Antithetic Variates for Radially Symmetric Copulas 257

7.2.3 Control Variates 258

7.2.4 Approximation via a Simpler Dependence Structure 260

7.2.5 Importance Sampling 262

7.2.6 Importance Sampling via Increasing the Dependence 263

7.2.7 Further Comments on Variance Reduction Methods 265

8 Further Copula Families with Known Extendible Subclass 267

8.1 Exogenous Shock Models 268

8.1.1 Extendible Exogenous Shock Models 271

8.2 Extreme-Value Copulas 285

8.2.1 Multivariate Distributions with Exponential Minima 293

8.2.2 Hierarchical (H-extendible) Extreme-Value Copulas 294

Appendix A Supplemental Material 301

A.1 Validating a Sampling Algorithm 301

A.2 Introduction to Lévy Subordinators 302

A.2.1 Compound Poisson Subordinator 306

A.2.2 Gamma Subordinator 308

A.2.3 Inverse Gaussian Subordinator 309

A.2.4 Stable Subordinator 310

A.3 Scale Mixtures of Marshall-Olkin Copulas 311

A.4 Generalizations of Lévy Subordinators 315

A.4.1 Additive Subordinators 315

A.4.2 IDT Subordinators 316

A.5 Further Reading 319

Bibliography 323

Index 335

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