Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

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Acknowledgements I would like to thank Prof. Dr. Didier Sornette (Chair of Entrepreneurial Risks, ETH Zürich) and Prof. Dr. Yannick Malevergne (EM Lyon Business School, Dept. Economics, Finance & Control) for their motivating way of supervising this thesis. I’am very grateful for their support, suggestions, and guidance, which gave me insights into many new fields and made the completion of this thesis possible. 2
Abstract Recent events i.e. severe market crashes have shown that there is a need for action in financial risk management to quantify dependences between extreme events. So far, there is no generally applied measure for extreme financial risks. The concept of tail dependence represents a powerful means to describe the amount of extreme co-movements between asset prices. Several approaches for the estimation of the tail dependence coefficient were developed. It is the purpose of this work to review and implement the most recent concepts for the estimation of tail dependence and to study their relevance for practical applications providing a complete analysis of the different approaches on historical and synthetic time series. Programs for the implementation of the discussed concepts are enclosed to the appendix as Matlab m-codes allowing the non-expert reader to apply them to his own purposes.
Page 1: Eidgenössische Technische Hochschu
Page 5 and 6: Appendix 160 M-Files . . . . . . .
Page 7 and 8: 3.12 β(N) estimated for 9 assets X
Page 9 and 10: 3.30 Fraction of scale factors �
Page 11 and 12: 3.46 ˆ βSI(k) calculated by the f
Page 13 and 14: 3.57 λ + (N) for upper tail of ind
Page 15 and 16: List of Tables 3.1 Estimated values
Page 17 and 18: 3.11 Estimated values of upper and
Page 19 and 20: 3.23 Establishing the uncertainty o
Page 21 and 22: 4.10 Estimated upper and lower tail
Page 23 and 24: 5.1 Descriptive statistics of histo
Page 25 and 26: 1.2 Thesis Outline The study of ext
Page 27 and 28: 2.1 Multivariate Extreme Value Dist
Page 29 and 30: and X is positively upper orthant d
Page 31 and 32: We refer to Appendix B of [3] (2007
Page 33 and 34: Chapter 3 Concepts for the Estimati
Page 35 and 36: or expressed through copula C: 1
Page 37 and 38: Parametric Joint-Tail Models Depend
Page 39 and 40: First of all we notice that only fo
Page 41 and 42: 19 m=2507 bs=1000 upper tail lower
Page 43 and 44: 3.2 Approaches according to Sornett
Page 45 and 46: is equal to the weakest tail depend
Page 47 and 48: ν(k) 4 3.8 3.6 3.4 3.2 3 2.8 2.6 2
Page 49 and 50: γ b n 3.2 3 2.8 2.6 2.4 2.2 2 1.8
Page 51 and 52: 29 m=2507 bs=1000 upper tail lower
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3.2.2 Implementation of the Non-Par
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l(k) mean,c 4.5 4 3.5 3 2.5 2 1.5 1
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Non-Par Par up lo up lo up lo up lo
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λ(k) λ(k) mean 0.14 0.12 0.1 0.08
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coefficient estimates. For the bigg
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41 m=5736 bs=1000 upper tail lower
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β β β 1 0.9 0.8 0.7 0.6 BMY 0.5
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l l l 2.6 2.4 2.2 2 1.8 BMY 1.6 100
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ν S&P 500 8 7 6 5 4 3 2 1 0 1000 2
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49 λ λ λ 0.2 0.15 0.1 0.05 BMY 0
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51 λ λ λ 0.12 0.1 0.08 0.06 0.04
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Error Bars in Dependence of Time To
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Return 0.1 0.05 0 −0.05 −0.1
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3.2.4 Implementation of the Paramet
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(C ε /C Y ) 1/α (C ε,mean /C Y,m
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F par ,F emp F par ,F emp 61 F par
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Page 87 and 88:
To summarize for upper tails: tail
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λ(k) mean λ(k) mean,c 0.1 0.09 0.
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Page 93 and 94:
Page 95 and 96:
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3.3 β-Smile Improvement As an addi
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77 β β + SI β − SI λ + λ + S
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I also implemented the β-smile imp
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81 β, β SI β, β SI β, β SI 1.
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β, β SI 83 β, β SI β, β SI 1
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85 β SI , β β SI , β β SI , β
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β SI , β 87 β SI , β β SI , β
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As an additional information, ˆν
Page 113 and 114:
91 λ λ λ 0.06 0.05 0.04 0.03 0.0
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93 λ λ λ 0.2 0.15 0.1 0.05 BMY 0
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95 constr 1 m=2507 bs=1000 upper ta
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97 constr 2 m=2507 bs=1000 upper ta
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Observation of λ by Rolling Time W
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101 β β β 1.5 1 0.5 0 −0.5 BMY
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103 λ λ λ 0.2 0.15 0.1 0.05 BMY
Page 127 and 128:
and the corresponding estimator for
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ˆλU,m ˆλL,m ˆλ EV T U,m ˆλ
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EVT λ , λ U,m U,m EVT λ , λ U,m
Page 133 and 134:
Estimation of optimal Threshold k T
Page 135 and 136:
λ BMY λ KO 113 λ SGP 0.4 0.3 0.2
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The right hand side of figure (3.62
Page 139 and 140:
Page 141 and 142:
119 λ λ λ 0.35 0.3 0.25 0.2 0.15
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121 λ λ λ 0.5 0.4 0.3 0.2 0.1 BM
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123 λ λ λ 0.5 0.4 0.3 0.2 0.1 BM
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sometimes were significantly differ
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Chapter 4 Application of Concepts t
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with the index can have much strong
Page 153 and 154:
Index Asset Abbrev. AORD Australian
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Index Asset Abbrev. MERVAL Acindar-
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C. W. β Non-Par Par up lo up lo up
Page 159 and 160:
C. W. β N-Par Par up lo up lo up l
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C. W. β N-Par Par up lo up lo up l
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141 ˆλ + SI1 95% Q 90% Q ˆ λ +
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exchange rate of the US$ and the Sw
Page 167 and 168:
A Eur GBP BrR CHFR CHY indoRu indRu
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4.3 Application to Synthetic Time S
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Density Density 60 50 40 30 20 10 F
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151 βorig ˆβSI 2 bias rel. bias
Page 175 and 176:
153 βorig ˆλ +,− (α,βorig) +
Page 177 and 178:
155 βorig ˆλ +,− EV T (ˆν,β
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synthetic time series we detected a
Page 181 and 182:
[17] Engle, R.F. (1982). Autoregres
Page 183 and 184:
Approach according to Poon, Rocking
Page 185 and 186:
Non-Parametric Approach according t
Page 187 and 188:
%estimate lambda% for j=1:n-1; if(b
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Approaches according to Schmidt & S
Page 191 and 192:
Composition of Synthetic Data Set c
Page 193 and 194:
Descriptive Statistics The kurtosis
Page 195 and 196:
N Mean Std Skew. Kurt. Statistic St
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Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

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