Contents -- Preface -- Why Sampling Copulas? -- Why Another Book on Copulas? -- Acknowledgments -- 1. Introduction -- 1.1 Copulas -- 1.1.1 Analytical Properties -- 1.1.1.1 Testing if C is a Copula -- 1.1.2 Sklar's Theorem and Survival Copulas -- 1.1.2.1 Sklar's Theorem -- 1.1.2.2 Survival Copulas -- 1.1.2.3 The Connection between C and C -- 1.1.3 General Sampling Methodology in Low Dimensions -- 1.1.4 Graphical Visualization -- 1.1.5 Concordance Measures -- 1.1.6 Measures of Extremal Dependence -- 1.2 General Classifications of Copulas -- 1.2.1 Radial Symmetry -- 1.2.2 Exchangeability -- 1.2.3 Homogeneous Mixture Models -- 1.2.4 Heterogeneous Mixture Models/Hierarchical Models -- 1.2.5 Extreme-Value Copulas -- 2. Archimedean Copulas -- 2.1 Motivation -- 2.2 Extendible Archimedean Copulas -- 2.2.1 Kimberling's Result and Bernstein's Theorem -- 2.2.2 Properties of Extendible Archimedean Copulas -- 2.2.2.1 Tail Dependence -- 2.2.2.2 Kendall's Tau -- 2.2.2.3 Density of Archimedean Copulas -- 2.2.2.4 Positive Lower Orthant Dependence (PLOD) -- 2.2.3 Constructing Multi-Parametric Families -- 2.2.4 Parametric Families -- 2.2.4.1 Ali{Mikhail{Haq Family -- 2.2.4.2 Frank Family -- 2.2.4.3 Joe Family -- 2.2.4.4 Clayton Family -- 2.2.4.5 Gumbel Family -- 2.2.4.6 Inverse Gaussian Family -- 2.3 Exchangeable Archimedean Copulas -- 2.3.1 Constructing Exchangeable Archimedean Copulas -- 2.3.2 Sampling Exchangeable Archimedean Copulas -- 2.3.3 Properties of Exchangeable Archimedean Copulas -- 2.3.3.1 Exchangeable C are not necessarily PLOD -- 2.3.3.2 Density -- 2.4 Hierarchical (H-Extendible) Archimedean Copulas -- 2.4.1 Compatibility of Generators -- 2.4.2 Probabilistic Construction and Sampling -- 2.4.3 Properties -- 2.4.4 Examples -- 2.5 Other Topics Related to Archimedean Copulas -- 2.5.1 Simulating from the Generator -- 2.5.2 Asymmetrizing Archimedean Copulas.

3. Marshall-Olkin Copulas -- 3.1 The General Marshall Olkin Copula -- 3.1.1 Canonical Construction of the MO Distribution -- 3.1.2 Alternative Construction of the MO Distribution -- 3.1.3 Properties of Marshall Olkin Copulas -- 3.2 The Exchangeable Case -- 3.2.1 Reparameterizing Marshall Olkin Copulas -- 3.2.2 The Inverse Pascal Triangle -- 3.2.3 Efficiently Sampling eMO -- 3.2.4 Hierarchical Extensions -- 3.3 The Extendible Case -- 3.3.1 Precise Formulation and Proof of Theorem 3.1 -- 3.3.2 Proof of Theorem 3.2 -- 3.3.3 Efficient Simulation of Levy-Frailty Copulas -- 3.3.4 Hierarchical (H-Extendible) Levy-Frailty Copulas -- 4. Elliptical Copulas -- 4.1 Spherical Distributions -- 4.2 Elliptical Distributions -- 4.3 Parametric Families of Elliptical Distributions -- 4.4 Elliptical Copulas -- 4.5 Parametric Families of Elliptical Copulas -- 4.6 Sampling Algorithms -- 4.6.1 A Generic Sampling Scheme -- 4.6.2 Sampling Important Parametric Families -- 5. Pair Copula Constructions -- 5.1 Introduction to Pair Copula Constructions -- 5.2 Copula Construction by Regular Vine Trees -- 5.2.1 Regular Vines -- 5.2.2 Regular Vine Matrices -- 5.3 Simulation from Regular Vine Distributions -- 5.3.1 h-Functions for Bivariate Copulas and Their Rotated Versions -- 5.3.2 The Sampling Algorithms -- C-Vines and D-Vines -- R-Vines -- 5.4 Dependence Properties -- Multivariate Gaussian and Student's t-Copulas -- Tail Dependence -- Flexibility vs. Parametric Margins -- 5.5 Application -- 5.5.1 Time Series Model for Each Margin -- 5.5.2 Parameter Estimation -- 5.5.3 Forecasting Value at Risk -- 5.5.4 Backtesting Value at Risk -- 5.5.5 Backtest Results -- 6. Sampling Univariate Random Variables -- 6.1 General Aspects of Generating Random Variables -- 6.2 Generating Uniformly Distributed Random Variables -- 6.2.1 Quality Criteria for RNG -- 6.2.2 Common Causes of Trouble.

6.3 The Inversion Method -- 6.4 Generating Exponentially Distributed Random Numbers -- 6.5 Acceptance-Rejection Method -- 6.6 Generating Normally Distributed Random Numbers -- 6.6.1 Calculating the Cumulative Normal -- 6.6.2 Generating Normally Distributed Random Numbers via Inversion -- 6.6.3 Generating Normal Random Numbers with Polar Methods -- 6.7 Generating Lognormal Random Numbers -- 6.8 Generating Gamma-Distributed Random Numbers -- 6.8.1 Generating Gamma-Distributed RNs with > 1 -- 6.8.2 Generating Gamma-Distributed RNs with < 1 -- 6.8.3 Relations to Other Distributions -- 6.9 Generating Chi-Square-Distributed RNs -- 6.10 Generating t-Distributed Random Numbers -- 6.11 Generating Pareto-Distributed Random Numbers -- 6.12 Generating Inverse Gaussian-Distributed Random Numbers -- 6.13 Generating Stable-Distributed Random Numbers -- 6.14 Generating Discretely Distributed Random Numbers -- 6.14.1 Generating Random Numbers with Geometric and Binomial Distribution -- 6.14.2 Generating Poisson-Distributed Random Numbers -- 7. The Monte Carlo Method -- 7.1 First Aspects of the Monte Carlo Method -- 7.2 Variance Reduction Methods -- 7.2.1 Antithetic Variates -- 7.2.2 Antithetic Variates for Radially Symmetric Copulas -- 7.2.3 Control Variates -- 7.2.4 Approximation via a Simpler Dependence Structure -- 7.2.5 Importance Sampling -- 7.2.6 Importance Sampling via Increasing the Dependence -- 7.2.7 Further Comments on Variance Reduction Methods -- Appendix A Supplemental Material -- A.1 Validating a Sampling Algorithm -- A.2 Introduction to Levy Subordinators -- A.2.1 Compound Poisson Subordinator -- A.2.2 Gamma Subordinator -- A.2.3 Inverse Gaussian Subordinator -- A.2.4 Stable Subordinator -- A.3 Scale Mixtures of Marshall Olkin Copulas -- A.4 Further Reading -- Bibliography -- Index.