Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

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List of Figures 3.1 Tail index estimators ˆν (Hill’s estimator) for the upper tails of index S&P 500 and 9 major assets included plotted in dependence of threshold k on a time interval ranging from July 1985 to April 2008. . . . . . . . 25 3.2 Tail index estimators ˆ b γ n (Gabaix’s estimator) for the upper tails of index S&P 500 and 9 major assets included plotted in dependence of threshold k on a time interval ranging from July 1985 to April 2008. . . . . . . . 26 3.3 Tail index estimators ˆν (Hill’s estimator) for the lower tails of index S&P 500 and 9 major assets included plotted in dependence of threshold k on a time interval ranging from July 1985 to April 2008. . . . . . . . . . . 26 3.4 Tail index estimators ˆ b γ n (Gabaix’s estimator) for the lower tails of index S&P 500 and 9 major assets included plotted in dependence of threshold k on a time interval ranging from July 1985 to April 2008. . . . . . . . 27 3.5 ˆ l(k) = Xk,N/Yk,N with k = 1 . . .150 (k/N = 0% . . .6%) for the lower tail of all assets with the index S&P 500 on the smaller data set ranging 3.6 from January 1991 up to December 2000. . . . . . . . . . . . . . . . . . ˆ � 1 k Xj,N l(k) = k j=1 with k = 1 . . .150 (or k/N = 0% . . .6%) for the lower Yj,N tail of all assets with the index S&P 500 on the smaller data set ranging 32 3.7 from January 1991 up to December 2000. . . . . . . . . . . . . . . . . . ˆl(k)c = 32 1 �k Xj,N k−c+1 j=Y with k = 13 . . .150 (or k/N = 0.5% . . .6%), Yj,N and c = [0.05 · N] ([·] denotes integer numbers) applied to the lower tail of all assets with the index S&P 500 on the smaller data set ranging from 3.8 January 1991 up to December 2000. . . . . . . . . . . . . . . . . . . . . ˆ 1 λ(k) = � �ˆν with k = 1 . . .343 (or k/N = 0% . . .6%), applied to 33 max 1, l ˆβ the lower tail of all assets with the index S&P 500 on the bigger data set ranging from July 1985 to April 2008. . . . . . . . . . . . . . . . . . . . 37 3.9 λ(k) = 1 � �ˆν with k = 1 . . .343 (or k/N = 0% . . .6%), applied to max 1, ˆ l ˆβ the lower tail of all assets with the index S&P 500 on the bigger data set ranging from July 1985 to April 2008. . . . . . . . . . . . . . . . . . . . 37 3.10 λ(k) = 1 � �ˆν with k = 29 . . .343 (or k/N = 0.5% . . .6%) and c = max 1, ˆ lĉ β [0.005 · N] ([·] denotes integer numbers), applied to the lower tail of all assets with the index S&P 500 on the bigger data set ranging from July 1985 to April 2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.11 β(N) estimated for 9 assets X and index S&P 500 Y using the linear single factor model: X = β·Y +ε for rolling time horizon windows of S = 2500 considered data points from N = (1 . . .S), (2 . . .S +1), . . .,(5736− S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . 43 II
3.12 β(N) estimated for 9 assets X and index S&P 500 Y using the linear single factor model: X = β·Y +ε for rolling time horizon windows of S = 1600 considered data points from N = (1 . . .S), (2 . . .S +1), . . .,(5736− S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . 43 3.13 β(N) estimated for 9 assets X and index S&P 500 Y using the linear single factor model: X = β·Y +ε for rolling time horizon windows of S = 800 considered data points from N = (1 . . .S), (2 . . .S + 1), . . .,(5736 − S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . 44 3.14 Constant lU(N) for upper tails plotted in green and constant lL(N) for lower tails plotted in black estimated for 9 assets and index S&P 500 using relation: ˆl(k) = 1 �k j=1 Xj,S/Yj,S with k = 0.04 · S, Xj,S and Yj,S k rank ordered return observations, and rolling time horizon window of S = 2500 considered data points from N = (1 . . .S), (2 . . .S +1), . . .,(5736− S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . 44 3.15 Constant lU(N) for upper tails plotted in green and constant lL(N) for lower tails plotted in black estimated for 9 assets and index S&P 500 using: ˆl(k) = 1 �k j=1 Xj,S/Yj,S with k = 0.04 · S, Xj,S and Yj,S rank k ordered return observations, and rolling time horizon window of S = 1600 considered data points from N = (1 . . .S), (2 . . .S +1), . . .,(5736− S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . 45 3.16 Constant lU(N) for upper tails plotted in green and constant lL(N) for lower tails plotted in black estimated for 9 assets and index S&P 500 using: ˆl(k) = 1 �k j=1 Xj,S/Yj,S with k = 0.04 · S, Xj,S and Yj,S rank k ordered return observations, and rolling time horizon window of S = 800 considered data points from N = (1 . . .S), (2 . . .S + 1), . . ., (5736 − S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . . . 45 3.17 νk,U(N) for upper tail of index S&P 500 plotted in red and νk,L(N) for lower tail plotted in black using Hill’s estimator given by equation � kj=1 ˆν = log Yj,S �−1 with k = 0.04 · S and Xj,S and Yj,S rank or- � 1 k Yk,N dered return observations for rolling time horizon windows of S = 2500 considered data points from N = (1 . . .S), (2 . . .S + 1), . . ., (5736 − S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . . . 46 3.18 νk,U(N) for upper tail of index S&P 500 plotted in red and νk,L(N) for lower tail plotted in black using Hill’s estimator given by equation � kj=1 ˆν = log Yj,S �−1 with k = 0.04 · S and Xj,S and Yj,S rank or- � 1 k Yk,N dered return observations for rolling time horizon windows of S = 1600 considered data points from N = (1 . . .S), (2 . . .S + 1), . . ., (5736 − S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . . . 46 3.19 νk,U(N) for upper tail of index S&P 500 plotted in red and νk,L(N) for lower tail plotted in black using Hill’s estimator given by equation ˆν = � kj=1 log Yj,S �−1 with k = 0.04 ·S and Xj,S and Yj,S rank ordered re- � 1 k Yk,N turn observations for rolling time horizon windows of S = 800 considered data points from N = (1 . . .S), (2 . . .S + 1), . . .,(5736 − S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . . . . . . . . . 47
Page 1 and 2: Eidgenössische Technische Hochschu
Page 3 and 4: Abstract Recent events i.e. severe
Page 5: Appendix 160 M-Files . . . . . . .
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
Page 53 and 54: 3.2.2 Implementation of the Non-Par
Page 55 and 56: l(k) mean,c 4.5 4 3.5 3 2.5 2 1.5 1
Page 57 and 58:
Non-Par Par up lo up lo up lo up lo
Page 59 and 60:
λ(k) λ(k) mean 0.14 0.12 0.1 0.08
Page 61 and 62:
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
Page 67 and 68:
l l l 2.6 2.4 2.2 2 1.8 BMY 1.6 100
Page 69 and 70:
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
Page 73 and 74:
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
Page 85 and 86:
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.
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
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
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91 λ λ λ 0.06 0.05 0.04 0.03 0.0
Page 115 and 116:
93 λ λ λ 0.2 0.15 0.1 0.05 BMY 0
Page 117 and 118:
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
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and the corresponding estimator for
Page 129 and 130:
ˆλU,m ˆλL,m ˆλ EV T U,m ˆλ
Page 131 and 132:
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
Page 137 and 138:
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
Page 145 and 146:
123 λ λ λ 0.5 0.4 0.3 0.2 0.1 BM
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sometimes were significantly differ
Page 149 and 150:
Chapter 4 Application of Concepts t
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with the index can have much strong
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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
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C. W. β N-Par Par up lo up lo up l
Page 161 and 162:
C. W. β N-Par Par up lo up lo up l
Page 163 and 164:
141 ˆλ + SI1 95% Q 90% Q ˆ λ +
Page 165 and 166:
exchange rate of the US$ and the Sw
Page 167 and 168:
A Eur GBP BrR CHFR CHY indoRu indRu
Page 169 and 170:
4.3 Application to Synthetic Time S
Page 171 and 172:
Density Density 60 50 40 30 20 10 F
Page 173 and 174:
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
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Non-Parametric Approach according t
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%estimate lambda% for j=1:n-1; if(b
Page 189 and 190:
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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