Modelling Dependence with Copulas - IFOR

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Acknowledgements This thesis was written during my stay at ETH in Zürich the fall 1999. I would like to express my gratitude and appreciation to Alexander McNeil for whom I had the pleasure of working <strong>with</strong> my thesis. I would also like to thank RiskLab for providing me <strong>with</strong> an office and Roger Kaufmann at RiskLab for valuable suggestions regarding the layout. Finally, I would like to thank Paul Embrechts at ETH and Jan Grandell, my supervisor at KTH in Stockholm, for giving me the opportunity to visit ETH. iii
Page 1: Eidgenössische Technische Hochschu
Page 7 and 8: 1 Motivation Consider a portfolio o
Page 9 and 10: 2 Copulas 2.1 Mathematical Introduc
Page 11 and 12: 2.3 The Fréchet-Hoeffding Bounds f
Page 13 and 14: 2.4 Copulas and Random Variables Si
Page 15 and 16: 2.4 Copulas and Random Variables M
Page 17 and 18: 3 Dependence Since the dependence s
Page 19 and 20: 3.3 Concordance and identically dis
Page 21 and 22: 3.3 Concordance Theorem 3.4. Let X
Page 23 and 24: 3.4 Tail Dependence Theorem 3.7. Le
Page 25 and 26: 3.4 Tail Dependence and similarly w
Page 27 and 28: 4 Techniques for Construction of Mu
Page 29 and 30: 4.2 The Marshall-Olkin Family As a
Page 31 and 32: 4.2 The Marshall-Olkin Family Marsh
Page 33: 4.2 The Marshall-Olkin Family Kenda
Page 36 and 37: 5 Archimedean Copulas Let F and G b
Page 38 and 39: 5 Archimedean Copulas Example 5.2.
Page 40 and 41: 5 Archimedean Copulas Proof. Let U
Page 42 and 43: 5 Archimedean Copulas and hence ( h
Page 44 and 45: 5 Archimedean Copulas Theorem 5.8.
Page 46 and 47: 5 Archimedean Copulas (5.2.2) with
Page 48 and 49: 5 Archimedean Copulas Theorem 5.10.
Page 51 and 52: 6 Elliptical Copulas A spherical di
Page 53 and 54:
6.1 The Gaussian Copula This result
Page 55 and 56:
6.2 The t-copula This can be used t
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6.2 The t-copula • • • •
Page 60 and 61:
7 Extreme Value Copulas Example 7.1
Page 62 and 63:
7 Extreme Value Copulas It follows
Page 65 and 66:
8 Mixture of Extremal Distributions
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The copula C J,J c is the copula of
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(8.0.8); that is, we look for a con
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• Generate independent uniform (0
Page 73 and 74:
9 Modelling Extremal Events in Prac
Page 75 and 76:
9.1 Pricing Risky Insurance Contrac
Page 77 and 78:
9.1 Pricing Risky Insurance Contrac
Page 79 and 80:
9.2 The Perfect Storm • • •
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9.2 The Perfect Storm 100 90 80 70
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9.2 The Perfect Storm 30 25 Expecte
Page 87 and 88:
11 Appendix To emphasize the useful
Page 89 and 90:
11.1 Implementations W
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11.2 Examples [1] 0.6886402 > cor(r
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Modelling Dependence with Copulas - IFOR

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