Simulating copulas: stochastic models, sampling algorithms, and applications
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| Main Authors: | , , , , , |
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| Format: | Book |
| Language: | English |
| Published: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2017]
|
| Edition: | 2nd Edition |
| Series: | Series in quantitative finance
Vol. 6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | xvii, 338 Seiten Illustrationen Diagramme |
| ISBN: | 9789813149991 9789813149243 |
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| 100 | 1 | |a Mai, Jan-Frederik |d 1982- |e Verfasser |0 (DE-588)141790261 |4 aut | |
| 245 | 1 | 0 | |a Simulating copulas |b stochastic models, sampling algorithms, and applications |c Jan-Frederik Mai (XAIA Investment AG, Germany), Matthias Scherer (Technische Universität München, Germany) ; with contributions by Claudia Czado, Elke Korn, Ralf Korn, Jakob Stöber |
| 250 | |a 2nd Edition | ||
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2017] | |
| 300 | |a xvii, 338 Seiten |b Illustrationen Diagramme | ||
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| 490 | 1 | |a Series in quantitative finance |v Vol. 6 | |
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| 700 | 1 | |a Stöber, Jakob |e Verfasser |4 aut | |
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| adam_text | 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
xiii
xiv Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications
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 Levy-Frailty Copulas . . . 150
3.3.4 Hierarchical (H-Extendible) Levy-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
Contents xv
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 ^-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
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 RNs with (3 1 . 241
6.8.2 Generating Gamma-Distributed RNs with /3 1 . 242
6.8.3 Relations to Other Distributions................. 243
6.9 Generating Chi-Square-Distributed RNs................. 243
6.10 Generating ¿-Distributed Random Numbers................244
xvi Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications
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.l Validating a Sampling Algorithm........................301
A.2 Introduction to Levy Subordinators ....................302
A.2.1 Compound Poisson Subordinator ................. 306
Contents xvii
A.2.2 Gamma Subordinates............................... 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
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
xiii
xiv Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications
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 Levy-Frailty Copulas . . . 150
3.3.4 Hierarchical (H-Extendible) Levy-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
Contents xv
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 ^-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
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 RNs with (3 1 . 241
6.8.2 Generating Gamma-Distributed RNs with /3 1 . 242
6.8.3 Relations to Other Distributions................ 243
6.9 Generating Chi-Square-Distributed RNs................ 243
6.10 Generating ¿-Distributed Random Numbers...............244
xvi Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications
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.l Validating a Sampling Algorithm.........................301
A.2 Introduction to Levy Subordinators .....................302
A.2.1 Compound Poisson Subordinator .................. 306
Contents xvii
A.2.2 Gamma Subordinates............................... 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
|
| any_adam_object | 1 |
| author | Mai, Jan-Frederik 1982- Scherer, Matthias Czado, Claudia Korn, Elke Korn, Ralf 1963- Stöber, Jakob |
| author_GND | (DE-588)141790261 (DE-588)133340937 (DE-588)11392853X (DE-588)171321642 |
| author_facet | Mai, Jan-Frederik 1982- Scherer, Matthias Czado, Claudia Korn, Elke Korn, Ralf 1963- Stöber, Jakob |
| author_role | aut aut aut aut aut aut |
| author_sort | Mai, Jan-Frederik 1982- |
| author_variant | j f m jfm m s ms c c cc e k ek r k rk j s js |
| building | Verbundindex |
| bvnumber | BV044641060 |
| classification_rvk | QH 233 QP 890 SK 800 SK 830 SK 980 |
| classification_tum | MAT 603f |
| ctrlnum | (OCoLC)1007865541 (DE-599)BVBBV044641060 |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2nd Edition |
| format | Book |
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| id | DE-604.BV044641060 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:57:59Z |
| institution | BVB |
| isbn | 9789813149991 9789813149243 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030039032 |
| oclc_num | 1007865541 |
| open_access_boolean | |
| owner | DE-384 DE-739 |
| owner_facet | DE-384 DE-739 |
| physical | xvii, 338 Seiten Illustrationen Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | World Scientific |
| record_format | marc |
| series | Series in quantitative finance |
| series2 | Series in quantitative finance |
| spelling | Mai, Jan-Frederik 1982- Verfasser (DE-588)141790261 aut Simulating copulas stochastic models, sampling algorithms, and applications Jan-Frederik Mai (XAIA Investment AG, Germany), Matthias Scherer (Technische Universität München, Germany) ; with contributions by Claudia Czado, Elke Korn, Ralf Korn, Jakob Stöber 2nd Edition New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2017] xvii, 338 Seiten Illustrationen Diagramme txt rdacontent n rdamedia nc rdacarrier Series in quantitative finance Vol. 6 Kopula Mathematik (DE-588)4529954-7 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Kopula Mathematik (DE-588)4529954-7 s Stochastisches Modell (DE-588)4057633-4 s DE-604 Scherer, Matthias Verfasser (DE-588)133340937 aut Czado, Claudia Verfasser (DE-588)11392853X aut Korn, Elke Verfasser aut Korn, Ralf 1963- Verfasser (DE-588)171321642 aut Stöber, Jakob Verfasser aut Erscheint auch als Online-Ausgabe 978-981-3149-25-0 Series in quantitative finance Vol. 6 (DE-604)BV044479602 6 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030039032&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030039032&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mai, Jan-Frederik 1982- Scherer, Matthias Czado, Claudia Korn, Elke Korn, Ralf 1963- Stöber, Jakob Simulating copulas stochastic models, sampling algorithms, and applications Series in quantitative finance Kopula Mathematik (DE-588)4529954-7 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| subject_GND | (DE-588)4529954-7 (DE-588)4057633-4 |
| title | Simulating copulas stochastic models, sampling algorithms, and applications |
| title_auth | Simulating copulas stochastic models, sampling algorithms, and applications |
| title_exact_search | Simulating copulas stochastic models, sampling algorithms, and applications |
| title_full | Simulating copulas stochastic models, sampling algorithms, and applications Jan-Frederik Mai (XAIA Investment AG, Germany), Matthias Scherer (Technische Universität München, Germany) ; with contributions by Claudia Czado, Elke Korn, Ralf Korn, Jakob Stöber |
| title_fullStr | Simulating copulas stochastic models, sampling algorithms, and applications Jan-Frederik Mai (XAIA Investment AG, Germany), Matthias Scherer (Technische Universität München, Germany) ; with contributions by Claudia Czado, Elke Korn, Ralf Korn, Jakob Stöber |
| title_full_unstemmed | Simulating copulas stochastic models, sampling algorithms, and applications Jan-Frederik Mai (XAIA Investment AG, Germany), Matthias Scherer (Technische Universität München, Germany) ; with contributions by Claudia Czado, Elke Korn, Ralf Korn, Jakob Stöber |
| title_short | Simulating copulas |
| title_sort | simulating copulas stochastic models sampling algorithms and applications |
| title_sub | stochastic models, sampling algorithms, and applications |
| topic | Kopula Mathematik (DE-588)4529954-7 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| topic_facet | Kopula Mathematik Stochastisches Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030039032&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030039032&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV044479602 |
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