Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

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20 m=5736 bs=1000 upper tail lower tail χbs,mean χbs,mean χ k=229 rel err max dev 95% q 90% q std bs k=229 rel err max dev 95% q 90% q std bs BMY 0.92 0.03 0.92 0.04 0.03 0.20 0.74 0.06 0.74 0.71 0.21 0.21 CVX 0.95 0.01 0.89 0.01 0.01 0.05 0.43 0.20 0.52 0.47 0.42 0.22 HPQ 0.95 0.01 0.86 0.01 0.01 0.04 0.71 0.04 0.72 0.71 0.44 0.23 KO 0.87 0.07 0.90 0.88 0.09 0.21 0.51 0.06 0.54 0.52 0.40 0.22 MMM 0.79 0.20 0.82 0.79 0.80 0.40 0.68 0.02 0.68 0.31 0.20 0.23 PG 0.94 0.01 0.91 0.01 0.01 0.08 0.44 0.20 0.52 0.40 0.41 0.34 SGP 0.95 0.01 0.93 0.01 0.01 0.05 0.58 0.20 0.64 0.62 0.59 0.29 TXN 0.94 0.02 0.90 0.02 0.02 0.10 0.59 0.20 0.63 0.62 0.61 0.28 WAG 0.95 0.01 0.94 0.01 0.01 0.04 0.70 0.04 0.70 0.69 0.21 0.27 χc BMY 0.92 0.01 0.93 0.03 0.02 0.07 0.94 0.02 0.91 0.05 0.04 0.13 CVX 0.92 0.00 0.04 0.03 0.02 0.01 0.92 0.05 0.93 0.13 0.07 0.22 HPQ 0.92 0.01 0.05 0.03 0.02 0.01 0.96 0.01 1.01 0.05 0.04 0.10 KO 0.92 0.00 0.92 0.03 0.03 0.05 0.90 0.07 0.93 0.90 0.10 0.24 MMM 0.90 0.04 0.90 0.06 0.05 0.20 0.95 0.02 0.93 0.12 0.04 0.09 PG 0.92 0.01 0.91 0.03 0.02 0.03 0.95 0.02 1.03 0.06 0.05 0.10 SGP 0.92 0.01 0.04 0.02 0.02 0.01 0.95 0.02 1.00 0.09 0.04 0.09 TXN 0.92 0.01 0.87 0.03 0.02 0.06 0.92 0.04 0.93 0.13 0.07 0.22 WAG 0.92 0.01 0.05 0.03 0.02 0.01 0.95 0.02 0.92 0.10 0.05 0.10 Table 3.4: Establishing the uncertainty of non-parametrically estimated upper and lower tail dependence coefficients ˆχ by creating 1000 bootstrap samples of historical return data tables for S&P 500 index and corresponding asset returns and calculation of quantiles, extreme values and standard deviations of the results. The tails represent the most extreme 4% (excluding highest 0.5% for χc) of the return values during a time interval from July 1985 to April 2008.
3.2 Approaches according to Sornette and Malevergne A set of parametric and non-parametric estimators for lower and upper tail dependence is proposed in [19] (2006), [20] (2004), and [21] (2002) assuming a linear factor model between stock returns and market returns. Within this section the implementation and analysis of firstly the non-parametric approach, secondly the parametric approach, and thirdly a so-called β-smile improvement is discussed. 3.2.1 Theoretical Background In this subsection a summary of the papers [20] (2004) and [21] (2002) is provided. First we present a non-parametric approach and secondly a parametric approach to calculate tail dependence between an asset and its market index. Finally, the calculation of tail dependence between two assets related by a factor model is explained. The Non-Parametric Approach by Sornette & Malevergne A linear additive single factor model was introduced to relate asset fluctuations to market fluctuations: X = β · Y + ε, (3.34) where X is a vector of asset returns and Y is the vector of corresponding index returns. β is the regression coefficient of the components of X to its corresponding components Y determined by the least squares method with ε denoting the vector of idiosyncratic noises assumed independent of Y . A general result concerning the tail dependence generated by factor models was derived from the definition of upper tail dependence given by equation (2.31). It is given by: � ∞ with λ + = max{1, l β } f(x)dx, (3.35) l = lim u→1− F −1 X (u) F −1 Y (u), (3.36) t · PY (t · x) f(x) = lim , t→∞ F Y (t) (3.37) where FX and FY are the marginal distribution functions of X and Y , and PY is the probability density function of Y . F Y = 1 − FY is the complementary cumulative distribution function of Y . A similar expression holds for the coefficient of lower tail dependence. A non-vanishing coefficient of tail dependence must have a limit function f(x) that is non-zero and the constant l must remain finite. In [20] (2004) it is explained that (3.35) even holds if factor Y and the idiosyncratic noise ε are dependent, provided that this dependence is not too strong. Another important statement in this paper is the absence of tail dependence for rapidly varying factors, which describes the Gaussian, exponential, and any other distribution decaying faster than any power law, for any arbitrary distribution of the idiosyncratic noise. 21
Page 1 and 2: Eidgenössische Technische Hochschu
Page 3 and 4: Abstract Recent events i.e. severe
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: 19 m=2507 bs=1000 upper tail lower
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
Page 65 and 66: 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
Page 71 and 72: 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
Page 75 and 76: Error Bars in Dependence of Time To
Page 77 and 78: Return 0.1 0.05 0 −0.05 −0.1
Page 79 and 80: 3.2.4 Implementation of the Paramet
Page 81 and 82: (C ε /C Y ) 1/α (C ε,mean /C Y,m
Page 83 and 84: F par ,F emp F par ,F emp 61 F par
Page 85 and 86: F par ,F emp F par ,F emp 63 F par
Page 87 and 88: To summarize for upper tails: tail
Page 89 and 90: λ(k) mean λ(k) mean,c 0.1 0.09 0.
Page 93 and 94:
F par ,F emp F par ,F emp 71 F par
Page 95 and 96:
F par ,F emp F par ,F emp 73 F par
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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
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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
Page 119 and 120:
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
Page 125 and 126:
103 λ λ λ 0.2 0.15 0.1 0.05 BMY
Page 127 and 128:
and the corresponding estimator for
Page 129 and 130:
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
Page 137 and 138:
The right hand side of figure (3.62
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117 m=5736 bs=800 upper tail lower
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119 λ λ λ 0.35 0.3 0.25 0.2 0.15
Page 143 and 144:
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
Page 151 and 152:
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
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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
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
Page 179 and 180:
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
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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