Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

Eidgenössische Technische Hochschule Zürich 

Swiss Federal Institute of Technology Zurich 

Master Thesis 

Tail Dependence: 

Implementation, Analysis, and Study 

of the most recent concepts 

Sebastian N. Schmuki 

Department of Management, Technology, and Economics - DMTEC 

Chair of Entrepreneurial Risks - ER 

Supervisors: 

Prof. Dr. Didier Sornette 

Prof. Dr. Yannick Malevergne 

October 2008

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.

Contents 

1 Introduction 2 

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 

1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

2 Important Concepts used in Multivariate Extreme Value Theory 4 

2.1 Multivariate Extreme Value Distributions . . . . . . . . . . . . . . . . . 5 

2.2 The Survival Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

2.3 Multivariate Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

2.3.1 Positive Orthant Dependence . . . . . . . . . . . . . . . . . . . 6 

2.3.2 Quadrant Dependence . . . . . . . . . . . . . . . . . . . . . . . 7 

2.3.3 Associated Variables . . . . . . . . . . . . . . . . . . . . . . . . 7 

2.4 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

2.4.1 Copula and Survival Copula . . . . . . . . . . . . . . . . . . . . 9 

2.4.2 Empirical Copula . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

2.5 Tail Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 

3 Concepts for the Estimation of Tail Dependence 11 

3.1 Approaches according to Poon, Rockinger, and Tawn . . . . . . . . . . 12 

3.1.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . 12 

3.1.2 Implementation of Non-Parametric χ and χ . . . . . . . . . . . 15 

3.2 Approaches according to Sornette and Malevergne . . . . . . . . . . . . 21 


3.2.2 Implementation of the Non-Parametric Approach . . . . . . . . 31 

3.2.3 Analysis of TDC estimated by the Non-Parametric Approach . . 34 

3.2.4 Implementation of the Parametric Approach . . . . . . . . . . . 57 

3.2.5 Analysis of TDC estimated by the Parametric Approach . . . . 66 

3.3 β-Smile Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 

3.4 Non-Parametric Approaches according to 

Schmidt & Stadtmüller . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 


3.4.2 Implementation of Non-Parametric Approaches . . . . . . . . . 105 

3.4.3 Analysis of Coefficients . . . . . . . . . . . . . . . . . . . . . . . 107 

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 

4 Application of Concepts to various Data and Purposes 127 

4.1 Application to major Financial Centers in the World . . . . . . . . . . 127 

4.2 Application to Exchange Rates . . . . . . . . . . . . . . . . . . . . . . 142 

4.3 Application to Synthetic Time Series . . . . . . . . . . . . . . . . . . . 147 

5 Concluding Remarks 156 

4

Appendix 160 

M-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 

Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

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 



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 



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 


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







800 considered data points from N = (1 . . .S), (2 . . .S + 1), . . .,(5736 − 


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 = 





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 = 





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 



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 



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

3.20 λU(N) for the upper tails of index S&P 500 and the nine assets plotted 

in blue and λL(N) for the lower tails plotted in red using non-parametric 

approach given by equation ˆ λ +,− � 

= 1/ max 1, l 

�ˆν with coefficients ˆν, 

β 

l and β 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. . . . . . . . . . . . . . 48 




= 1/ max 1, l 


β 







= 1/ max 1, l 


β 




3.23 λU(N) for the upper tails of index S&P 500 and the nine assets for rolling 

time horizon windows of size S = 2500 plotted in black, S = 1600 plotted 

in blue, and S = 800 plotted in red using non-parametric approach given 

by equation ˆ λ +,− � 

= 1/ max 1, l 

�ˆν with coefficients ˆν, l and β from 

β 

N = (2501 − S . . .S), (2502 . . .S + 1), . . .,(5736 − S + 1 . . .5736). . . . 51 

3.24 λL(N) for the lower tails of index S&P 500 and the nine assets for rolling 

time horizon windows of size S = 2500 plotted in black, S = 1600 plotted 

in blue, and S = 800 plotted in red using non-parametric approach given 

by equation ˆ λ +,− � 

= 1/ max 1, l 

�ˆν with coefficients ˆν, l and β from 

β 

N = (2501 − S . . .S), (2502 . . .S + 1), . . .,(5736 − S + 1 . . .5736). . . . 52 

3.25 Return data for index S&P 500 ranging from January 1950 to April 2008 55 

3.26 Autocorrelation for a return series on the left and for a squared return 

series on the right of index S&P 500 ranging from January 1950 to April 

2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 

3.27 Scale factors ĈY (k) = k 

N · (Yk,N) ˆν of the index S&P 500 and Ĉε(k) = 

k 

N · (εk,N) ˆν of the nine assets’ residues plotted for the upper tails of the 

bigger data set ranging from July 1985 to April 2008 with k=1, 2,..., 

458 (k/N = 0% . . .8%). . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

3.28 Scale factors ĈY (k) = k bˆ · (Yk,N) 

N γ n of the index S&P 500 and Ĉε(k) = 

k bˆ · (εk,N) 

N γ n of the nine assets’ residues plotted for the upper tails of the 

bigger data set ranging from July 1985 to April 2008 with k=1, 2,..., 

458 (k/N = 0% . . .8%). . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

3.29 Fraction of scale factors 

� Ĉε(k) 

ĈY (k) 

� 1/α 

= εk,N 

Yk,N 

of the nine assets’ residues 

and the index S&P 500, plotted for the upper tails of the bigger data 

set ranging from July 1985 to April 2008 with k=1, 2,..., 458 (k/N = 

0% . . .8%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.30 Fraction of scale factors 

� �1/α Ĉε 

ĈY 

of the nine assets’ residues and the 

index S&P 500, plotted for the upper tails of the bigger data set ranging 

from July 1985 to April 2008 with k=1, 2,..., 458 (k/N = 0% . . .8%). . 59 

3.31 Empirical complementary cumulative distribution functions F ε,empirical = 

k 

plotted in red and parametric complementary cumulative distribution 

N 

functions F ε,parametric = Ĉε ·ε−ˆν with Ĉε for (k/N = 4%) plotted in black 

for k/N = 4% . . .0% (from left to right) in dependence of returns for the 

upper tails of the nine assets. . . . . . . . . . . . . . . . . . . . . . . . 61 

3.32 Empirical complementary cumulative distribution functions F ε,empirical = 

k plotted in red and parametric complementary cumulative distribu- 

N 

tion functions F ε,parametric = Ĉε,c · ε−ˆν with Ĉε,c = 1 �k k−c j=c Ĉε for 

c = 0.005 · N plotted in black for k/N = 4% . . .0% (from left to right) 

in dependence of returns for the upper tails of the nine assets. . . . . . 62 

3.33 Empirical complementary cumulative distribution functions F εi,empirical = 

k plotted in green and parametric complementary cumulative distribu- 

N 

tion functions F ε,parametric = Ĉε · ε−ˆb γ n with Ĉε for (k/N = 4%) plotted 

in black for the upper tails of k/N = 4% . . .0% in dependence of returns 

for the nine assets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 


k 

plotted in green and parametric complementary cumulative distribu- 

N 

tion functions F ε,parametric = Ĉε,c · ε−ˆb γ n with Ĉε,c = 1 �k k−c j=c Ĉε for 

c = 0.005 · N plotted in black for k/N = 4% . . .0% in dependence of 

returns for the upper tails of the nine assets. . . . . . . . . . . . . . . . 64 

3.35 ˆ λ = with k = 1 . . .458 (or k/N = 0% . . .8%), applied to the 

1 

1+ ˆ β −ˆν · Cε 

Cy 

lower tails of all assets with the index S&P 500 on the bigger data set 


3.36 λ = 

1 

with k = 1 . . .458 (or k/N = 0% . . .8%), applied to the 

1+ ˆ β −ˆν · Ĉε 

Ĉy 

lower tails of all assets with the index S&P 500 on the bigger data set 


3.37 λ = 1 with k = 29 . . .458 (or k/N = 0.5% . . .8%), and c = 

1+ ˆ β −ˆν · Ĉε,c 

Ĉy,c 

[0.005 · N] ([·] denotes integer numbers) applied to the lower tails of all 

assets with the index S&P 500 on the bigger data set ranging from July 

1985 to April 2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 


k plotted in red and parametric complementary cumulative distribution 

N 

functions F ε,parametric = Ĉε · ε−ˆν with Ĉε = 1 �k k j=1 Ĉε plotted in black 

for k/N = 4% . . .0% (from left to right) in dependence of returns for the 

lower tails of the nine assets for a time interval ranging from January 

1991 to December 2000. . . . . . . . . . . . . . . . . . . . . . . . . . . 71



N 

tion functions F εi,parametric = Ĉε, c · ε−ˆν with Ĉε,c = 1 �k k−c j=c Ĉε for 

c = 0.005 · N plotted in black for k/N = 4% . . .0% (from left to right) 

in dependence of returns for the lower tails of the nine assets for a time 

interval ranging from January 1991 to December 2000. . . . . . . . . . 72 


k plotted in red and parametric complementary cumulative distribution 

N 

functions F ε,parametric = Ĉε · ε−ˆν with Ĉε = 1 �k k j=1 Ĉε plotted in black 

for k/N = 4% . . .0% in dependence of returns for the lower tails of the 

nine assets for a time interval ranging from January 1991 to December 

2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 



N 

tion functions F εi,parametric = Ĉε, c · ε−ˆν with Ĉε,c = 1 �k k−c j=c Ĉε for 

c = 0.005 · N plotted in black for k/N = 4% . . .0% in dependence of 

returns for the lower tails of the nine assets for a time interval ranging 

from January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . 74 

3.42 ˆ βSI(k) of the upper tails plotted in blue and of the lower tails plotted in 

black calculated by least square method applied on linear additive single 

factor model: X = β·Y +ε and first SI condition: Y ≥ Y (k)∩X ≥ X(k) 

in dependence of threshold k. Reference ˆ β calculated for all data is given 

in green. Y denotes the index return vector of S&P 500 and Y denotes 

asset return vector of the nine assets for a time interval ranging from 

January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . . . . 80 



factor model: X = β · Y + ε and second SI condition: Y ≥ Y (k) in 

dependence of threshold k. Reference ˆ β calculated for all data is given 






factor model: X = β · Y + ε and third SI condition: X ≥ X(k) in 

dependence of threshold k. Reference ˆ β calculated for all data is given 






factor model: X = β · Y + ε and fourth SI condition: Y ≥ Y (k) ∪ X ≥ 

X(k) in dependence of threshold k. Reference ˆ β calculated for all data 

is given in green. Y denotes the index return vector of S&P 500 and Y 

denotes asset return vector of the nine assets for a time interval ranging 

from January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . 83

3.46 ˆ βSI(k) calculated by the first condition: Y ≥ Y (k) ∩ X ≥ X(k) plotted 

in blue and by the second condition: Y ≥ Y (k) plotted in black for the 

upper tail calculated by least squares method applied on linear additive 

single factor model: X = β · Y + ε in dependence of threshold k for 

k/N = 3% . . .6%. Reference ˆ β calculated for all data is given in green. 

Y denotes the index return vector of S&P 500 and Y denotes asset 

return vector of the nine assets for a time interval ranging from January 

1991 to December 2000. . . . . . . . . . . . . . . . . . . . . . . . . . . 85 



lower tail calculated by least squares method applied on linear additive 




return vector of the nine assets for a time interval ranging from January 

1991 to December 2000. . . . . . . . . . . . . . . . . . . . . . . . . . . 86 



upper tail calculated by least squares method applied on linear additive 




return vector of the nine assets for a time interval ranging from July 

1985 to Mars 2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 



lower tail calculated by least squares method applied on linear additive 




return vector of the nine assets for a time interval ranging from July 

1985 to Mars 2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 

3.50 ˆ λ + (k) for the upper tail of the nine assets and the index S&P 500 is 

plotted in black for ˆ β(k) calculated by least squares method applied to 

linear additive single factor model: X = β·Y +ε and adapted to the first 

condition: Y ≥ Y (k) ∩ X ≥ X(k), ˆν(k = 4%) and ˆ l(k) in dependence 

of threshold k for k/N = 3% . . .10%, in blue for ˆ β(k) and ˆ l(k) adapted 

to the first SI condition for comparison, in green for Reference ˆ β and 

ˆl(k), and for Reference ˆ λ + (k = 4%) is given in red. Y denotes the index 

return vector of S&P 500 and X denotes asset return vector of the nine 

assets for a time interval ranging from January 1991 to December 2000. 90

3.51 ˆ λ + (k) for the lower tail of the nine assets and the index S&P 500 is 


linear additive single factor model: X = β·Y +ε and adapted to the first 

condition: Y ≥ Y (k) ∩ X ≥ X(k), ˆν(k = 4%) and ˆ l(k) in dependence 

of threshold k for k/N = 3% . . .10%, in blue for ˆ β(k) and ˆ l(k) adapted 




assets for a time interval ranging from January 1991 to December 2000. 91 

3.52 ˆ λ + (k) for the upper tail of the nine assets and the index S&P 500 is 


linear additive single factor model: X = β · Y + ε and adapted to the 

second condition: Y ≥ Y (k), ˆν(k = 4%) and ˆ l(k) in dependence of 

threshold k for k/N = 3% . . .10%, in blue for ˆ β(k) and ˆ l(k) adapted 





3.53 ˆ λ + (k) for the lower tail of the nine assets and the index S&P 500 is 


linear additive single factor model: X = β · Y + ε and adapted to the 

second condition: Y ≥ Y (k), ˆν(k = 4%) and ˆ l(k) in dependence of 

threshold k for k/N = 3% . . .10%, in blue for ˆ β(k) and ˆ l(k) adapted 





3.54 ˆ βSI(N) plotted in black for the upper tails and plotted in green for the 

lower tails for 9 assets given by X and index S&P 500 given by Y using 

the linear single factor model: X = β · Y + ε and the first SI condition: 

Y ≥ Y (k) ∩ X ≥ X(k) for rolling time horizon windows of S = 2500 



3.55 ˆ βSI(N) plotted in black for the upper tails and plotted in green for the 

lower tails for 9 assets given by X and index S&P 500 given by Y using 

the linear single factor model: X = β·Y +ε and the second SI condition: 

Y ≥ Y (k) for rolling time horizon windows of S = 2500 considered data 



3.56 λ + (N) for upper tails of index S&P 500 and the nine assets plotted 

in blue and λ − (N) for lower tails plotted in red using non-parametric 

approach given by equation ˆ λ +,− = 1/ max 

� 

1, 

l 

ˆβ +,− 

SI 

� ˆν 

with coefficients 

ˆν, l, and ˆ β +,− 

SI 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. ˆ β +,− 

SI were calculated 

using the linear single factor model: X = β · Y + ε and the first SI 

condition: Y ≥ Y (k) ∩ X ≥ X(k). . . . . . . . . . . . . . . . . . . . . . 102

3.57 λ + (N) for upper tail of index S&P 500 and the nine assets plotted in blue 

and λ− (N) for lower tail plotted in red using non-parametric approach 

given by equation ˆ λ +,− � 

l 

= 1/ max 1, ˆβ +,− 

�ˆν with coefficients ˆν, l, and 

SI 

ˆβ +,− 

SI 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. ˆ β +,− 

SI were calculated using the 

linear single factor model: X = β · Y + ε and the second SI condition: 

Y ≥ Y (k). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 

3.58 ˆ λU,m for upper tails of index S&P 500 and the nine assets plotted in black 

and ÊV T λU,m for upper tails plotted in red, both in dependence of threshold 

k on an interval from k/m = 0% . . .6% for m = 2507 daily return values 

during a time interval ranging from January 1991 to December 2000. . 109 

3.59 ˆ λL,m for lower tails of index S&P 500 and the nine assets plotted in black 

and ÊV T λL,m for lower tails plotted in red, both in dependence of threshold 

k on an interval from k/m = 0% . . .6% for m = 2507 daily return values 

during a time interval ranging from January 1991 to December 2000. . 110 

3.60 Smoothed ˆ λL,m plotted in black and smoothed ˆ λU,m plotted in green for 

the nine assets with the index S&P 500 given for the most extreme 8% 

during a time interval from January 1991 to December 2000. Smoothing 

is performed by locally weighted scatter plot smooth using least squares 

quadratic polynomial fitting filters with 25 considered data points by 

step or [0.125 · 0.08 · m] values with a total of m = 2507 data points and 

[·] denoting integer numbers. . . . . . . . . . . . . . . . . . . . . . . . . 112 

3.61 Smoothed ˆ λL,m plotted in black and smoothed ˆ λU,m plotted in green 

for the nine assets with the index S&P 500 given for the most extreme 

8% during a time interval from July 1985 to April 2008. Smoothing is 

performed by locally weighted scatter plot smooth using least squares 

quadratic polynomial fitting filters with 57 considered data points by 

step or [0.125 · 0.08 · m] values with a total of m = 5736 data points and 

[·] denoting integer numbers. . . . . . . . . . . . . . . . . . . . . . . . . 113 

3.62 Lower tail dependence λL(θ) = 2−1/θ and corresponding asymptotic vari- 

ance σ2 L (θ) = 2−1/θ − 3 

24−1/θ + 1 

28−1/θ for the Pareto copula. . . . . . . . 114 

3.63 λU,m(m) for upper tails of index S&P 500 and the nine assets plotted in 

green and λL,m(m) for lower tails plotted in black using a non-parametric 

approach given by equation ˆ λL,m = 1 �m k j=1 1R (j) 

m1≤k and R(j) 

m2≤k and ˆ λU,m = 

� 1 m 

k j=1 1R (j) 

for rolling time horizon windows of S = 

m1 >m−k and R(j) 

m2 >m−k 








� 1 m 

k j=1 1R (j) 



m2 >m−k 


S + 1 . . .5736) or a total time interval from July 1985 to Mars 2008. . . 120






� 1 m 

k j=1 1R (j) 



m2 >m−k 

800 considered data points from N = (1 . . .S), (2 . . .S + 1), . . .,(5736 − 


3.66 λL,m(N) for lower tails of index S&P 500 and the nine assets for rolling 

time horizon windows of size S = 2500 plotted in blue, S = 1600 

plotted in green, and S = 800 plotted in red using a non-parametric 

approach given by equation ˆ λU,m = 1 �m k j=1 1R (j) 

from 


m2 >m−k 

N = (2501 − S . . .S), (2502 . . .S + 1), . . .,(5736 − S + 1 . . .5736). . . . 122 

3.67 λU,m(N) for lower tails of index S&P 500 and the nine assets for rolling 

time horizon windows of size S = 2500 plotted in blue, S = 1600 plotted 

in green, and S = 800 plotted in red a using non-parametric approach 

given by equation ˆ λL,m = 1 �m k j=1 1R (j) from N = (2501 − 


m2≤k S . . .S), (2502 . . .S + 1), . . ., (5736 − S + 1 . . .5736). . . . . . . . . . . . 123 

4.1 Non-parametric probability density distribution functions of index S&P 

500, assets CVX (Chevron Corp.), and a synthetic sample estimated 

using a Gaussian box kernel for a time interval of N = 2507 data points 

ranging from January 1991 to December 2000. . . . . . . . . . . . . . . 149 

4.2 Non-parametric probability density distribution functions of index S&P 

500, assets CVX (Chevron Corp.), and a synthetic sample plotted on 

a semi-log scale and estimated using a Gaussian box kernel for a time 

interval of N = 2507 data points ranging from January 1991 to December 

2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

List of Tables 

3.1 Estimated values of upper and lower tail dependence for S&P 500 index 

with a set of nine major assets traded on the New York stock Exchange 

calculated by a non-parametric approach: ˆχ +,− = Zk,N ·k 

and correspond- 

N 

� Zk,N 2 k(N−k) 

ing standard deviation: ˆσ (ˆχ) = N3 . The tail represents the 

most extreme 4% of the return values during a time interval from January 

1991 to December 2000. ’tail’ shows the calculation interval of L 

with c = 13 and k = 100, and ∗ denotes ˆχ �= 1 and therefore ˆχ = 0. . . 18 



calculated by a non-parametric approach: ˆχ +,− = Zk,N ·k 

and correspond- 

N 

� Zk,N 2 k(N−k) 

ing standard deviation: ˆσ (ˆχ) = N3 . The tail represents the 

most extreme 4% of the return values during a time interval from July 

1985 to April 2008. ’tail’ shows the calculation interval of L with c = 29 

and k = 229, and ∗ denotes ˆχ �= 1 and therefore ˆχ = 0. . . . . . . . . . 18 

3.3 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 January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . 19 


lower tail dependence coefficients ˆχ by creating 1000 bootstrap samples 




highest 0.5% for χc) of the return values during a time interval 

3.5 

from July 1985 to April 2008. . . . . . . . . . . . . . . . . . . . . . . . 

Comparison of tail index estimators ˆν (Hill’s estimator) and 

20 

ˆb γ n (estimator 

by Gabaix) for positive and negative tails of the S&P 500 index, 9 

included assets, and the residues ε obtained by regressing each asset on 

the S&P 500 index, during the time intervals ranging from January 1991 

to December 2000 with a 4% quantile of k = 100 values and from July 

1985 to April 2008 with a 4% quantile of k = 229 values. Values with ∗ 

represent tail indexes, which can’t be considered equal to the respective 

S&P 500 index at the 95% condfidence level. . . . . . . . . . . . . . . . 28 

XI

3.6 Establishing the uncertainty of estimated upper and lower tail indexes 

ˆν by creating bs = 1000 bootstrap samples of historical return data for 

S&P 500 and included assets and calculation of bootstrap mean value 

νmean with rel. error from the original value, deviation quantiles and 

standard deviation from νmean, and most extreme deviation value. The 

tail represents the most extreme 4 % of the return values during a time 

interval ranging from January 1991 to December 2000 (k = 100 values) 

and from July 1985 to April 2008 (k = 229 values). . . . . . . . . . . . 29 

3.7 Establishing the uncertainty of estimated upper and lower tail indexes ˆ b γ n 

by creating bootstrap samples (bs) of historical return data for S&P 500 

and included assets and calculation of bootstrap mean value νmean with 

rel. error from the original value, deviation quantiles and standard deviation 

from bγ mean , and most extreme deviation value. The tail represents 

the most extreme 4 % of the return values during a time interval ranging 

from January 1991 to December 2000 (k = 100 values, bs = 1000) and 

from July 1985 to April 2008 (k = 229 values, bs = 650). . . . . . . . . 30 



calculated by a non-parametric approach: ˆ λ +,− � 

= 1/ max 1, l 

�ν on β 

the left and by a parametric approach: ˆ λ +,− � 

= 1/ 1 + β−ν · Cε 

� 

on CY 

the right. The tail represents the most extreme 4% of the return values 

during a time interval from January 1991 to December 2000. ’tail’ shows 

the calculation interval of l with c = 13 and k = 100 and ∗ denotes 

negative ˆ β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 




= 1/ max 1, l 

� γ 

bn on β 


= 1/ 1 + β−bγ � 

n Cε · on CY 


during a time interval from January 1991 to December 2000. ’tail’ shows 

the calculation interval of l with c = 13 and k = 100 and ∗ denotes 

negative ˆ β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 



calculated by a non-parametric approach: 

35 

ˆ λ +,− � 

= 1/ max 1, l 

�ν on β 


= 1/ 1 + β−ν · Cε 

� 

on CY 


during a time interval from July 1985 to April 2008. ’tail’ shows the 

calculation interval of l with c = 29 and k = 229. . . . . . . . . . . . . 35




= 1/ max 1, l 

� γ 

bn on β 


= 1/ 1 + ˆ β−bγ � 

n Cε · on CY 


during a time interval from July 1985 to April 2008. ’tail’ shows the 

calculation interval of l with c = 29 and k = 229. . . . . . . . . . . . . 36 


lower tail dependence coefficients ˆ λ by creating 1000 bootstrap samples 




highest 0.5% for λc) of the return values during a time interval 

from January 1991 to December 2000. ˆ βj have been calculated on the 

whole samples. ∗ denotes negative ˆ β and therefore ˆ λ = 0 and ’Inf’ denotes 

·/0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 


lower tail dependence coefficients 

40 

ˆ λ by creating 1000 bootstrap samples 





from July 1985 to April 2008. ˆ βj have been calculated on the whole 

samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

3.14 ARCH statistics applied to daily S&P 500 index return data for an interval 

ranging from January 1950 to April 2008 and for lags of q = 10, 

41 

15, and 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

3.15 Establishing the uncertainty of parametrically estimated upper and lower 

tail dependence coefficients 

56 

ˆ λ 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 λc) of the return values during a time interval from 

January 1991 to December 2000. ˆ βj have been calculated on the whole 

samples. ∗ denotes negative ˆ β and therefore ˆ λ = 0 and ’Inf’ denotes ·/0. 

3.16 Establishing the uncertainty of of parametrically estimated upper and 

lower tail dependence coefficients 

69 

ˆ λ by creating 1000 bootstrap samples 


asset returns and calculation of quantiles, extreme values, and standard 



from July 1985 to April 2008. ˆ βj have been calculated on the whole 

samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.17 Relative deviations between coefficients of upper and lower tail dependence 

estimated by the non-parametric approach of Sornette & Malevergne 

using ˆ β calculated for the whole sample and ˆ βSI calculated for the 

tail according to three different conditions for index S&P 500 and the 

nine assets. The tail represents the most extreme 4 % of data during 

a smaller time interval ranging from January 1991 to December 2000 

(k = 100) and during a bigger time interval ranging from July 1985 to 

Mars 2008 (k = 229). * denotes zero values of tail dependence calculated 

by ˆ βSI, ’NaN’ denotes 0/0, and ’inf’ denotes ·/0. . . . . . . . . . . . . . 76 

3.18 Results for ˆ βSI using the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k) and 

the second SI condition: Y ≥ Y (k) for upper and lower tails calculated 

by least squares method applied on linear additive single factor model: 

X = β · Y + ε and for resulting upper and lower tail dependence coefficients 

ˆ λ + and ˆ λ− with corrected ˆl (c . . .k) and uncorrected ˆl (1 . . .k). 

The tail represents the most extreme 4% of data during a time interval 

ranging from January 1991 to December 2000 (k = 100, c = 13). * 

denotes ˆ β < 0 → λ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 

3.19 Results for ˆ βSI using the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k) and 

the second SI condition: Y ≥ Y (k) for upper and lower tails calculated 

by least squares method applied on linear additive single factor model: 

X = β · Y + ε and for resulting upper and lower tail dependence coefficients 

ˆ λ + and ˆ λ− with corrected ˆl (c . . .k) and uncorrected ˆl (1 . . .k). 

The tail represents the most extreme 4% of data during a time interval 

ranging from July 1985 to Mars 2008 (k = 229, c = 29). * denotes 

ˆβ < 0 → λ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 








first SI condition: Y ≥ Y (k) ∩ X ≥ X(k). ’inf’ denotes ·/0. . . . . . . . 95 







from July 1985 to Mars 2008. ˆ βj have been calculated on the first SI 

condition: Y ≥ Y (k) ∩ X ≥ X(k). ’inf’ denotes ·/0. . . . . . . . . . . . 96 








second SI condition: Y ≥ Y (k). ’inf’ denotes ·/0. . . . . . . . . . . . . 97







from July 1985 to Mars 2008. ˆ βj have been calculated on the second SI 

condition: Y ≥ Y (k). ’inf’ denotes ·/0. . . . . . . . . . . . . . . . . . . 98 

3.24 Estimated values of upper and lower tail dependence for index S&P 

500 and a set of nine major assets traded on the New York stock Exchange 

calculated by non-parametric approaches according to Schmidt 

& Stadtmüller. The tail represents the most extreme 4% of the return 

values during a time interval from January 1991 to December 2000 with 

a threshold value of k = 0.04·m = 100.3. Columns denoted by ’w. noise’ 

correspond to columns on their right and are performed by the same approaches 

but with a small random noise of amplitude (10−6 ) added to 

the returns that no two returns are equal anymore. . . . . . . . . . . . 107 

3.25 Estimated values of upper and lower tail dependence for index S&P 

500 and a set of nine major assets traded on the New York stock Exchange 

calculated by non-parametric approaches according to Schmidt & 

Stadtmüller. The tail represents the most extreme 4% of the return values 

during a time interval from July 1985 to April 2008 with a threshold 

value of k = 0.04·m = 229.4. Columns denoted by ’w. noise’ correspond 

to columns on their right and are performed by the same approaches but 

with a small random noise of amplitude (10−6 ) added to the returns that 

no two returns are equal anymore. . . . . . . . . . . . . . . . . . . . . . 108 

3.26 Establishing the uncertainty of non-parametrically estimated upper tail 

dependence coefficients ˆ λU,m and ÊV T λ and lower tail dependence coef- 

ficients ˆ λL,m and ÊV T λL,m U,m 

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% of the return values 

during a time interval from January 1991 to December 2000. . . . . . . 116 

3.27 Establishing the uncertainty of non-parametrically estimated upper tail 

dependence coefficients ˆ λU,m and ÊV T λU,m and lower tail dependence coefficients 

ˆ λL,m and ÊV T λL,m by creating 800 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% of the return values 

during a time interval from July 1985 to April 2008. . . . . . . . . . . . 117 

4.1 Indexes that were used for the implementation of the different concepts 

given by their name, further used abbreviation, and country of origin. . 130 

4.2 Assets included in indexes ’AORD’ (ASX), ’CAC 40’, and ’DAX’ that 

were used for the implementation of the different concepts given by their 

name, further used abbreviation, and country of origin. . . . . . . . . . 131 

4.3 Assets included in indexes ’DOW’ (Dow Jones), ’FTSE 100’, and ’JKSE’ 

that were used for the implementation of the different concepts given by 

their name, further used abbreviation, and country of origin. . . . . . . 132

4.4 Assets included in indexes ’MERVAL’, ’MIBTEL’, ’NASDAQ’, and ’SMI’ 

that were used for the implementation of the different concepts given by 

their name, further used abbreviation, and country of origin. . . . . . . 133 

4.5 Assets included in indexes ’S&P 500’, ’SSEC’, and ’TA 100’ that were 

used for the implementation of the different concepts given by their name, 

further used abbreviation, and country of origin. Additional assets included 

in index S&P 500 that were not part of the reference samples 

presented in chapter (3) are listed below ’new assets’. . . . . . . . . . . 134 

4.6 Estimated upper and lower tail dependence ˆ λ +,− applying the non-parametric 

and parametric approaches according to Sornette & Malevergne to index 

S&P 500 and 16 assets included in dependence of their component 

weights in the index denoted by ’C.W.’ and β. The data samples contain 

N = 2000 daily price observations on a range from 12.04.2000 to 

31.03.2008. Tail index ˆν was calculated using Hill’s estimator, k = 

0.04 · N = 80, c = 0.005 · N = 10, and ∗ denotes negative ˆ β. . . . . . . 135 


approach according to Sornette & Malevergne to index S&P 500 and 16 

assets included using βSI only calculated for the extreme tails by first 

and second β-smile conditions. Component weights of assets within the 

index are denoted by ’C.W.’, βSI for the first and second conditions and 

β calculated for all data are listed to observe their impact on the estimates. 

The data samples contain N = 2000 daily price observations on a 

time interval from 12.04.2000 to 31.03.2008. Tail index ˆν was calculated 

using Hill’s estimator. ’Cond 1’ denotes: Y ≥ Y (k) ∩ X ≥ X(k), ’Cond. 

2’ denotes Y ≥ Y (k), k = 0.04 · N = 80, c = 0.005 · N = 10, and ∗ 

denotes negative ˆ β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 

4.8 Estimated upper and lower tail dependence ˆ λ +,− applying the parametric 

and the non-parametric approach according to Sornette & Malevergne 

to index Dow Jones Industrial Average and 12 assets included using β 

calculated by all data. Component weights of assets within the index 

are denoted by ’C.W.’, and β calculated for all data are listed to observe 

their impact on the estimates. The data samples contain N = 2000 daily 

price observations on a time interval from 16.08.2000 to 31.07.2008. Tail 

index ˆν was calculated using Hill’s estimator, k = 0.04 · N = 80, and 

c = 0.005 · N = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 


approach according to Sornette & Malevergne to index Dow Jones Industrial 

Average and 12 assets included using βSI only calculated for 

the extreme tails by first and second β-smile conditions. Component 

weights of assets within the index denoted by ’C.W.’, βSI for the first 

and second conditions and β calculated for all data are listed to observe 

their impact on the estimates. The data samples contain N = 2000 daily 

price observations on a time interval from 16.08.2000 to 31.07.2008. Tail 

index ˆν was calculated using Hill’s estimator, k = 0.04 · N = 80, and 

c = 0.005 · N = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.10 Estimated upper and lower tail dependence ˆ λ +,− applying the parametric 

and the non-parametric approach according to Sornette & Malevergne to 

index NASDAQ Composite and 9 assets included using β calculated for 

all data and βSI only calculated for the extreme tails by first and second 

β-smile conditions. Component weights of assets within the index denoted 

by ’C.W.’, βSI for the first and second conditions and β calculated 

for all data are listed to observe their impact on the estimates. The data 

samples contain N = 2000 daily price observations on a time interval 

from 15.08.2000 to 31.07.2008. Tail index ˆν was calculated using Hill’s 

estimator, k = 0.04 ·N = 80, c = 0.005 ·N = 10, and ∗ denotes negative ˆ β.139 

4.11 Estimated upper and lower tail dependence according to Poon, Rockinger, 

and Tawn (ˆχ + · , ˆχ − · ) in the upper part and according to Schmidt & 

Stadtmüller in the lower part ( ˆ λ·,m, ÊV T λ·,m ) for index AORD (ASX) and 

14 assets included including error bars estimated by bootstrap sampling 

with replacement for bs = 1000 bootstrap samples shown next to the 

estimates. The data samples contain N = 1398 daily price observations 

on a time interval from begin of February 2003 to end of August 2008. 140 

4.12 Estimated ˆ λ +,− applying the non-parametric approach according to Sornette 

& Malevergne to index AORD (ASX) and 14 assets included using 

βSI only calculated for the extreme tails by first and second β-smile conditions. 

Error bars estimated by bootstrap sampling with replacement 

for bs = 1000 bootstrap samples were large and therefore mostly denoted 

by *. The data samples contain N = 1398 data points from February 

2003 to end of August 2008. Tail index ˆν was calculated using Hill’s 

estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 

4.13 Further used abbreviations of currency exchange rates that were used 

for the estimation of tail dependence. . . . . . . . . . . . . . . . . . . . 142 

4.14 Estimated ˆ λ +,− applying the non-parametric approach according to Sor- 

nette & Malevergne given by: ˆ λ +,− = 1/ max 

� 

1, l 

β 

� ν 

to exchange rates 

of various foreign currencies with the US$. The tail represents the most 

extreme 4% of the return values during a time interval ranging from 

20.10.2002 to 10.04.2008 consisting of N = 2000 daily observations. ’c’ 

means that tails were corrected for 0.5% of most extreme data for the 

estimation of l. Tail index ˆν was calculated using Hill’s estimator. . . . 144 

4.15 Estimated ˆχ +,− applying the non-parametric approach according to Poon, 

Rockinger, and Tawn given by: ˆχ +,− = Zk,N ·k 

N to exchange rates of various 

foreign currencies with the US$. The tail represents the most extreme 

4% of the return values during a time interval ranging from 20.10.2002 

to 10.04.2008 consisting of N = 2000 daily observations. ’c’ means that 

tails were corrected for 0.5% of most extreme data for the estimation of 

L. Tail indexes ˆν were calculated using Hill’s estimator. . . . . . . . . . 145 

4.16 Estimated ˆ λL,m, ˆ λU,m, ÊV T λL,m , and ÊV T λU,m applying the non-parametric 

approaches according to Schmidt & Stadtmüller explained in section 

(3.4) to exchange rates of various foreign currencies with the US$. The 

tail represents the most extreme 4% of the return values during a time 

interval ranging from 20.10.2002 to 10.04.2008 consisting of N = 2000 

daily observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.17 Establishing bias and error bars of ˆ β calculated by all data and effect on 

ˆλ +,− estimated by applying the non-parametric approach according to 

Sornette & Malevergne to 1000 synthetic samples consisting of N = 2507 

data points. The bias is calculated by comparing mean values ˆ β all data 

and ˆ λ +,− 

( ˆ βalldata) to original values denoted by βorig and corresponding 

tail dependence estimator ˆ λ +,− (βorig) and for the estimation of error bars 

standard deviation ’std’ and 95% quantiles of the estimates are provided. 

Relative bias (’rel.bias’) is calculated as percentage of ˆ λ +,− 

(βorig). Tail 

indexes α were set equal to three to avoid a second source of error and 

’inf’ means ·/0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 

4.18 Establishing bias and error bars of ˆ βSI 2 calculated by second β-smile 

condition: � Y ≥ Y (k) � and effect on ˆ λ +,− estimated by applying the 

non-parametric approach according to Sornette & Malevergne to 1000 

synthetic samples consisting of N = 2507 data points. The bias is cal- 

culated by comparing mean values ˆ β alldata and ˆ λ +,− 

( ˆ βalldata) to original 

values denoted by βorig and corresponding tail dependence estimator 

ˆλ +,− (βorig) and for the estimation of error bars standard deviation ’std’ 

and 95% quantiles of the estimates are provided. Tail indexes α were set 

equal to three to avoid a second source of error and ∗ means that relative 

bias (’rel.bias’) calculated as percentage of ˆ λ +,− 

(βorig) ≥ 100%. . . . . . 151 

4.19 Establishing bias and error bars of ˆν and ˆb γ n , estimated for 1000 synthetic 

samples consisting of N = 2507 data points. The bias is calculated by 

comparing mean values ˆν and ˆb γ 

n to original values denoted by α and for 

the estimation of error bars standard deviation ’std’ and 95% quantiles 

of the estimates are provided. Relative bias (’rel.bias’) is calculated as 

percentage of α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 

4.20 Establishing bias and error bars of ˆ λ +,− (ˆν, ˆ βalldata) with ˆ β calculated by 

all data and ˆ λ +,− (ˆν, ˆ βSI2) with ˆ β calculated by second β-smile condition: 

� Y ≥ Y (k) � by applying the non-parametric approach according 

to Sornette & Malevergne to 1000 synthetic samples consisting of 

N = 2507 data points. The bias is calculated by comparing mean 

values ˆ λ +,− 

(ˆν, ˆ βalldata) and ˆ λ +,− 

(ˆν, ˆ βSI 2) to original values denoted by 

ˆλ +,− 

(α, βorig) and for the estimation of error bars standard deviation 

’std’ and 95% quantiles of the estimates are provided. ∗ means that 

relative bias (’rel.bias’) calculated as percentage of ˆ λ +,− 

4.21 Comparison of estimates by non-parametric estimators ˆ λ·,m, ˆ λ 

(βorig) ≥ 100%. 153 

·,m 

cording to Schmidt & Stadtmüller, and ˆχ +,− according to Poon, Rockinger, 

and Tawn with ˆ λ +,− (βalldata) according to Sornette & Malevergne by 

applying the different concepts to 1000 synthetic samples consisting of 

N = 2507 data points created by different βorig. For the estimation of 

error bars standard deviation ’std’ and 95% quantiles of the estimates 

are calculated and for ˆχ +,− the proposed standard deviation ˆσ is given 

for comparison. Tail indexes α were estimated by the Hill estimator. . . 155 

EV T 

ac

5.1 Descriptive statistics of historical return series of S&P 500 and nine 

assets included during a time interval from January 1991 to December 

2000 (smaller reference sample of chapter (3)) . . . . . . . . . . . . . . 171 

5.2 Descriptive statistics of historical return series of S&P 500 and nine assets 

included during a time interval from July 1985 to April 2008 (bigger 

reference sample of chapter (3)) . . . . . . . . . . . . . . . . . . . . . . 171 

5.3 Descriptive statistics of historical return series of S&P 500 and nine assets 

included during a time interval from 12.04.2000 to 31.03.2008 (sample 

consisting of N = 2000 observations used in chapter (4)) . . . . . . 172 

5.4 Descriptive statistics of historical return series of Dow Jones Industrial 

Average and 12 assets included during a time interval from 16.08.2000 to 

31.07.2008 (sample consisting of N = 2000 observations used in chapter 

(4)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 

5.5 Descriptive statistics of historical return series of NASDAQ Composite 

and 9 assets included during a time interval from 15.08.2000 to 31.07.2008 

(sample consisting of N = 2000 observations used in chapter (4)) . . . 173 

5.6 Descriptive statistics of historical return series of AORD (ASX) and 

14 assets included during a time interval from February 2003 to end 

of August 2008 (sample consisting of N = 1398 observations used in 

chapter (4)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 

5.7 Descriptive statistics of historical currency exchange rate series during a 

time interval from 19.10.2002 to 10.04.2008 (sample consisting of N = 

2000 observations used in chapter (4)) . . . . . . . . . . . . . . . . . . 174 

5.8 Descriptive statistics of 6 exemplary synthetic samples (sample consisting 

of N = 2507 observations used in chapter (4)) . . . . . . . . . . . . 174

Chapter 1 

Introduction 

1.1 Motivation 

The importance of extreme events has become a major issue in financial risk management. 

Established risk measures like Pearson’s product moment correlation coefficient 

and Spearman’s rank order correlation coefficient are controlled by small movements 

around the mean and therefor fail to describe dependence between extreme events. Facing 

asset selection and allocation with respect to portfolio management extreme events 

are primarily represented by jump risks and default risks. 

Extreme events appear in the tails of return distributions of assets and because of the 

fact that asset return distributions are heavy tailed, extraordinary downside losses are 

more likely to happen than expected assuming Gaussian distributions. During market 

crashes or recessions, standard portfolio management and optimization strategies may 

break down. Recent events show that impact and frequency of stock market crashes 

have increased during the last two decades. Our target is to diversify away extreme 

risks by minimizing extreme dependence between assets within a portfolio. Therefore we 

focus on the concept of tail dependence, which was first introduced by Sibuya [1] (1960). 

The coefficient of tail dependence between two assets is defined as the probability that 

one of the two assets undergoes a large loss (or gain) assuming that the other asset also 

undergoes a large loss (or gain). 

Several approaches were developed for the estimation of the tail dependence coefficient. 

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 financial risk 

management as well as assets selection and allocation by applying them to a broadbase 

index such as the S&P 500 representing the market and an arbitrary asset that is 

included in the index, to measure extreme dependence between the two. Using historical 

time series and synthetic time series, we will provide an in-depth analysis of the 

different concepts. The results of this thesis should provide the investor with a tool 

that can directly be applied to real financial data and help ancillary to other criteria to 

decide about addition or removal of assets in their portfolios for the purpose of minimizing 

extreme dependence. It is important to add that a special effort has been made 

in writing down and presenting the results in a way that allows a non-expert reader to 

use this methods. Programs in the form of Matlab m-files are enclosed to the appendix 

as well as the attached data CD and throughout the whole work cross-references to the 

respective codes are provided to facilitate traceability and encourage application of the 

concepts. 

2

1.2 Thesis Outline 

The study of extreme co-movements requires the conjunction of the multivariate extreme 

value theory and the notion of copulas. In chapter (2) an introduction to fundamental 

concepts used within this work is provided. 

Chapter (3) is dedicated to the different concepts for the estimation of the coefficient 

of tail dependence. 

Starting in section (3.1) with concepts developed by Poon, Rockinger, and Tawn and 

published in [13] (2004) we will first provide an insight in their most important theoretical 

findings, then give a detailed description how their approaches were implemented, 

and finally analyze the results applying the methods to a bigger and a smaller reference 

data set of the index S&P 500 and nine major assets included in the index and additionally 

provide an estimation of error bars by bootstrap sampling with replacement. 

We will use these two reference data sets throughout the whole chapter. 

In section (3.2) we will come to the main topic of this thesis consisting of concepts 

according to Sornette & Malevergne presented in [21] (2006), [20] (2004), [19] (2002). 

We will also start with a subsection about the theoretical background and then continue 

with implementation and analysis of the different approaches including error bar 

estimations by bootstrap sampling with replacement. In subsection (3.3), we will expand 

the concepts by a so-called β-smile improvement condition. Here, we allude to 

the literature on the ’β-smile’ using a terminology borrowed from the volatility-smile 1 

in financial option theory pointing at the fact that parameter β may exhibit a change 

as a function of the quantiles on which it is estimated. For the non-parametric approach 

and the non-parametric approach including β-smile improvement we performed 

a rolling time window observation to analyze time dynamics and consistency over time 

of the estimators. 

Our last concept according to Schmidt & Stadtmüller was introduced in [24] (2005) 

and is presented in section (3.4). After presenting the theoretical background and 

details concerning the implementation of the concepts, we also provide an analysis of 

coefficients performed on the reference data sets and error bar estimations by bootstrap 

sampling with replacement. Finally, there is also a rolling time window observation performed. 

In section (3.5) we will compare the results yielded by the different methods 

and summarize the findings of the whole chapter. 

Chapter (4) shows the application of the different concepts to various data and 

purposes. Starting with the application of the concepts to major financial centers in the 

world in section (4.1) shows us interesting new insights into the methods and broadens 

our horizon on a global scale. On the other hand, it also reveals some weaknesses of 

our approaches. 

Section (4.2) provides us with applications of the concepts to a very different purpose: 

we use the tail dependence coefficients to estimate extreme dependences of international 

currency exchange rates. 

In section (4.3) the composition of a synthetic time series is presented in order to check 

our methods for bias and lower error bounds. Finally, some concluding remarks are 

provided in chapter (5). 

1 Pattern showing that at-the-money options tend to have lower implied volatilities than other options 

3

Chapter 2 

Important Concepts used in 

Multivariate Extreme Value Theory 

Extreme value theory (EVT) is a branch of statistics, which addresses extreme events. 

These are high impact events that happen with very low probability i.e. earthquakes or 

100 year flood etc. Extreme measures provide estimates for extreme but unlikely events, 

whereas classic general measures tend to focus on the whole empirical distribution and 

therefore often neglect low-probability events. Nevertheless, it is known that extreme 

events account for a large fraction in terms of effect. 

In the financial sector market crashes are extreme realizations of the respective return 

distribution. The traditional measure to assess the risk of an asset is the variance of 

its returns, introduced by Markowitz [2] (1952). But this measure assumes that the 

probability density function (pdf) is of Gaussian nature. Empirical data show that 

there is a ’fatness’ in the tails of pdf functions of returns. That means that extreme 

events occur more often than assumed by Gaussian approximations. 

Anyway, the determination of the precise shape of the tail of the distribution of 

returns has proven to be a challenging task. In order to assess risk of high probability 

levels of above 95 %, parametric and non-parametric estimations are established: 

Parametrization in our context means fitting the parameters of a general model to our 

return data to achieve a good approximation. That becomes necessary when risk at 

very high probability levels is estimated and non-parametric estimates fail because of 

lack of data, which is often the case facing daily return data as we will see later on. 

As mentioned in the title we are concerned with ’multivariate’ EVT. In contrast to 

univariate distributions, multivariate distributions of returns denote joint probabilities 

of realizations of multiple asset return series. This comprises both, risk explained by an 

asset’s univariate marginal distribution and risk due to dependence on other assets. To 

offer to the interested reader a very complete insight into univariate EVT and also on 

the very active and dynamical field of multivariate EVT we refer to [3] (2007) and [4] 

(2006) and references therein. 

Now we want to introduce some fundamental concepts of multivariate EVT to provide 

a wide overview of the field. The explanations will be based on [3] (2007), which 

provides a concise and recent overview of Extreme Value analysis. 

4

2.1 Multivariate Extreme Value Distributions 

In multivariate Extreme Value Analysis it is a standard practice to investigate vectors 

of component wise maxima or minima. Letting {(Xi,1, . . .,Xi,d)} for i = 1, . . ., n be 

independent and identically distributed (iid) random variables with distribution F the 

corresponding row-vector of component wise maxima Mn is given by: 

� 

Mn = (Mn,1, . . .,Mn,d) = max 

1≤i≤n {Xi,1},..., max 

1≤i≤n {Xi,d} 

� 

(2.1) 

and the corresponding row-vector of component wise minima � Mn is given by: 

�Mn = ( � Mn,1, . . ., � � 

Mn,d) = min 

1≤i≤n {Xi,1},..., min 

1≤i≤n {Xi,d} 

� 

(2.2) 

It is important to notice that the maximum or minimum of each of the different random 

variables may occur for different indexes i ∗ 1 , . . .,i∗ d , where ∗ denotes the maximum or 

minimum. Therefore Mn and � Mn do not necessarily correspond to observed values 

in the original series and the analysis of component wise maxima or minima may 

sometimes be of little use. 

A standard way to operate is to look for the existence of sequences of constants an,i 

and bn,i > 1 for 1 ≤ i ≤ d such that for all (x1, . . ., xd) ∈ Rd the function: 

� 

Mn,1 − an,1 

G (x1, . . .,xd) = lim Pr 

≤ x1, . . ., 

n→∞ Mn,d − an,d 

bn,1 

bn,d 

≤ xd 

= F n (an,1 + bn,1 · x1, . . .,an,d + bn,d · xd) (2.3) 

is a proper distribution function with non-degenerate marginals. For minima the following 

function holds: 

� 

�Mn,1 − ãn,1 

G(x1, . . .,xd) = lim Pr 

≥ x1, . . ., 

n→∞ ˜bn,1 � � 

Mn,d − ãn,d 

≥ xd 

˜bn,d � 

n 

= F ãn,1 + ˜bn,1 · x1, . . .,ãn,d + ˜ � 

bn,d · xd , (2.4) 

where G and F denote the survival functions associated with G and F defined below. 

For maxima the distribution F is said to belong to the Maximum Domain of Attraction 

(MDA) of the Multivariate Extreme Value (MEV) distribution G if there exist sequences 

of constants an,i and bn,i > 1 for 1 ≤ i ≤ d such that equation (2.3) is satisfied. 

A similar definition can be given for minima. The following equations hold for all 

(x1, . . .,xd) ∈ R d such that G(x) > 0 and G(x) > 0: 

lim 

n→∞ n[1 − F(an,1 + bn,1 · x1, . . .,an,d + bn,d · xd)] = − log G (x1, . . .,xd) (2.5) 

lim 

n→∞ n[1 − F(ãn,1 + ˜bn,1 · x1, . . .,ãn,d + ˜bn,d · xd)] = − log G(x1, . . .,xd) (2.6) 

From now on we only concentrate on the study of maxima because as we are using 

return series later on with mean around zero we can handle both in the same way by 

multiplying minima with factor (−1). 

5 

�

Setting all the xi’s but one to +∞ in equation (2.3) yields: 

n 

lim Fi (an,i + bn,i · xi) = Gi (xi) , fori = 1, . . ., d, (2.7) 

n→∞ 

where Fi and Gi are the i-th marginals of F and G. In turn Fi ∈ MDA (Gi), where Gi 

is a Gumbel, Fréchet, or Weibull distribution 1 . 

In order to isolate the dependence features from the marginal distribution aspects [6] 

(2000), traditionally the components of both the distribution F and the corresponding 

MEV distribution G are transformed to standard marginals. It is customary to choose 

the standard Fréchet distribution as marginals i.e. the function φ : R → [0, 1] given 

by φ(x) = exp � � 

−1 for x > 0 and zero elsewhere and by its inverse function: y = 

x 

φ−1 (x) = −1/ log(x), where ’log’ denotes the natural logarithm. Defining X as a 

d-variate random row-vector with distribution F and continuous marginals 

Y = φ −1 1 

(F(X)) = − , (2.8) 

log F(X) 

i.e. Yi = φ−1 (Fi(Xi)) = −1/ log F(Xi) for all i ≤ 1 ≤ d. By the Probability Integral 

Transform it follows that Y has standard Fréchet marginals. Letting G be a 

multivariate distribution with continuous marginals Gi’s and defining: 

G ∗ �� (−1) � � � 

(−1) 

−1 

−1 

(y1, . . .,yd) = G 

· yi, . . ., 

· yd , (2.9) 

log G1 

log Gd 

with y1 > 0, . . .,yd > 0, then G ∗ has standard Fréchet marginals, and G is a MEV distribution 

only if G ∗ is also MEV distributed. Thus the marginals of a MEV distribution 

can be standardized, yet preserving the extreme value properties. 

2.2 The Survival Function 

The survival function F associated with multivariate extreme value theory is given by: 

F = Pr {X1 > x1, . . .,Xd > xd} (2.10) 

In the univariate case d = 1 F = 1 − F(x). Unfortunately this does not hold generally 

in multivariate EVT [5] (1988). 

2.3 Multivariate Dependence 

In the multivariate case the notions of dependence becomes numerous and complex. 

Below we introduce some basic concepts. For further information about how to measure 

dependence between random variables we refer to [7] (2006), [8] (2001), and [9] (1997). 

2.3.1 Positive Orthant Dependence 

Assuming that X = (X1, . . .,Xd) is a d-variate random row-vector, X is positively 

lower orthant dependent (PLOD), if for all x = (x1, . . .,xd) ∈ R d , 

Pr {X ≤ x} ≥ 

d� 

Pr {Xi ≤ xi} (2.11) 

i=1 

1 This is shown in Theorem 1.7 of chapter 1.2 of [3] (2007) 

6

and X is positively upper orthant dependent (PUOD) if for all x = (x1, . . .,xd) ∈ R d , 

Pr {X > x} ≥ 

d� 

Pr {Xi > xi} (2.12) 

i=1 

Generally X is positively orthant dependent if for all x = (x1, . . .,xd) ∈ R d both 

equations (2.11) and (2.12) hold. The definition of negative lower orthant dependence 

(NLOD), negative upper orthant dependence (NUOD), and generally negative orthant 

dependence (NOD) can be introduced by reversing the sense of the inequalities. 

If X has joint distribution F with marginals Fi for i = 1, . . ., d then equation (2.11) 

is equivalent to: 

for all x ∈ R d and equation (2.12) is equivalent to: 

for all x ∈ R d . 

2.3.2 Quadrant Dependence 

F (x1, . . .,xd) ≥ F1(x1) · · ·Fd(xd) (2.13) 

F (x1, . . .,xd) ≥ F 1(x1) · · ·Fd(xd) (2.14) 

Quadrant dependence given by positive quadrant dependence (PQD) and negative 

quadrant dependence (NQD) is a bivariate concept of dependence and for d = 2 the 

definitions of PUOD and PLOD are equivalent to PQD. Let X and Y be a pair of 

continuous random variables with joint distribution function FX,Y and marginals FX 

and FY . X and Y are positively quadrant dependent (PQD) if: 

and negatively quadrant dependent (NQD) if: 

∀(x, y) ∈ R 2 FX,Y ≥ FX(x) · FY (y) (2.15) 

∀(x, y) ∈ R 2 FX,Y ≤ FX(x) · FY (y), (2.16) 

whereas the quadrant dependence properties are invariant under strictly increasing 

transformations. Two random variables X and Y are PQD if Cov (f(X), g(Y )) for 

all increasing functions f and g for which the expectations E (f(x)), E (g(y)), and 

E (f(x) · g(y)) exist 2 . 

2.3.3 Associated Variables 

The random variables X1, . . .,Xd are said to be (positively) associated if, for every pair 

of a, b of non-decreasing real-valued functions defined on R d , 

Cov (a (X1, . . .,Xd) , b (X1, . . .,Xd)) ≥ 0 (2.17) 

2 Cov denotes the covariance, which provides a measure of the strength of the correlation between two or 

more random variables and is given by: Cov(X, Y ) = E (X · Y ) − E (X) · E (Y ) assuming that X and Y are 

random variables on R 

7

whenever the relevant expectations exist [10] (1983). If the random variables X1, . . ., Xd 

have a MEV distributions, then they are associated. 

A MEV distribution G satisfies the condition: 

d� 

G (x1, . . .,xd) ≥ Gi (xi) (2.18) 

for all x ∈ R d . The forms of limiting multivariate distributions correspond to the cases 

of (asymptotic) total independence: 

G (x1, . . .,xd) = 

and (asymptotic) total dependence: 

i=1 

d� 

Gi (xi) (2.19) 

i=1 

G (x1, . . .,xd) = min {G1 (x1) , . . .,Gd (xd)} (2.20) 

between the component wise maxima for all x ∈ R d . For further details concerning 

asymptotic total dependence and asymptotic total independence we refer to [6] (2000) 

and [10] (1983). 

Pairwise independent random variables having a joint MEV distribution are mutually 

independent. Thus the study of asymptotic independence can be confined to the 

bivariate case. 

Asymptotic total independence arises only if equation (2.7) holds, and there exists 

x ∈ R d such that 0 < Gi (xi) < 1 for i = 1, . . ., d and 

F n (an,1 + bn,1 · x1, . . .,an,d + bn,d · xd) n→∞ 

−→ G1 (x1) · · ·Gd (xd) (2.21) 

Moreover equation (2.19) only holds for any x ∈ R d if 

G(0, . . .,0) = G1(0), · · ·Gd(0) = exp(−d) (2.22) 

provided that Gi are standard Gumbel distributions or 

provided that Gi are Fréchet distributions or 

G(1, . . .,1) = G1(1), · · ·Gd(1) = exp(−d) (2.23) 

G(−1, . . .,−1) = G1(−1), · · ·Gd(−1) = exp(−d) (2.24) 

(2.25) 

provided that Gi are Weibull distributions 3 (2007). 

Similar conditions hold for the case of asymptotic total dependence. Asymptotic 

dependence arises only if equation (2.7) holds, and there exists x ∈ R d such that 

0 < G1 (x1) = · · · = Gd (xd) < 1 and 

F n (an,1 + bn,1 · x1, . . .,an,d + bn,d · xd) n→∞ 

−→ G1 (x1) (2.26) 

3 The distinction between these three types of distributions belongs to the field of univariate extreme value 

Theory and is explained in i.e. Theorem 1.7 of chapter 1.2 of [3] 

8

We refer to Appendix B of [3] (2007) for a detailed handling of dependence between 

random variables. Now we come to notions of multivariate extreme value theory that 

are fundamental for the comprehension of the concepts that are discussed in chapter 

(3). 

2.4 Copulas 

Facing financial risk management, the diversification of risks between two or more assets 

is the major issue. To achieve diversification, it is fundamental to have notice of an 

accurate description of the dependence between different assets. Copulas introduced by 

Sklar [11] (1959) describe the general dependence structure of several random variables. 

For all further definitions I will focus on the bivariate case only. 

2.4.1 Copula and Survival Copula 

A bivariate distribution function FX,Y of two random variables X and Y with marginal 

distributions Fx(·) and Fy(·) can be written in the form: 

FX,Y (x, y) = Pr(X ≤ x, Y ≤ y) 

= C(Fx(x), Fy(y)), (2.27) 

where C(·, ·) with range in [0, 1] × [0, 1] is the copula of the two random variables X 

and Y , and is unique if the random variables have continuous marginal distributions. 

Moreover, the copula is invariant under strictly increasing transformation of the variables 

and is therefore an intrinsic or scale invariant measure of dependence. Denoting 

the joint survival function Pr(X > x, Y > y) by F X,Y (x, y) we may write: 

F X,Y (x, y) = 1 − FX(x) − FY (y) + FX,Y (x, y) 

= C{Fx(x), Fy(y)}, (2.28) 

where C(·, ·) with range in [0, 1]×[0, 1] is the survival copula of the two random variables 

X and Y . 

2.4.2 Empirical Copula 

If the bivariate distribution function FX,Y of two random variables X and Y and their 

respective marginal distributions Fx(·) and Fy(·) are not known we can calculate the 

empirical copula by counting of the number of pairs that satisfy given constraints. 

Let {(Rk, Sk)} be the ranks of the random sample {(Xk, Yk)}, then the corresponding 

empirical copula is defined as: 

Cn (Fx(x) = u, Fy(y) = v) = 1 

n 

n� 

I 

k=1 

� � 

Rk Sk 

≤ u and ≤ v , (2.29) 

n + 1 n + 1 

where u, v ∈ [0, 1] and I is an indicator function counting the cases, where the ’and’ 

condition is fulfilled. 

For a summary of the most important copula families we refer to Appendix C of [3] 

(2007). 

9

2.5 Tail Dependence 

In the previous section we introduced the concept of copula, which describes the general 

dependence structure between variables. There are many more specific measures 

to estimate dependence between two random variables. But standard measures - as 

mentioned above at the example of univariate distributions - are mostly controlled by 

relatively small moves of the asset prices around their mean. To study large and extreme 

co-movements, we introduce the coefficient of tail dependence between assets 

Xi and Xj, introduced by Sibuya [1] (1960) and advanced by Ledford and Tawn [12] 

(1996). It is defined as the probability for the asset Xi to incur a large loss (or gain) 

assuming that the asset Xj also undergoes a large loss (or gain) at the same probability 

level, in the limit where this probability explores the extreme tails of the distribution 

of returns of the two assets. 

By definition the coefficient of lower tail dependence λ − 

ij 

tail dependence λ + 

ij are defined by: 

and the coefficient of upper 

λ − 

ij = lim 

u→0 + Pr{Xi < F −1 

i (u)|Xj < F −1 

j (u)} (2.30) 

λ + 

ij 

= lim 

u→1 − Pr{Xi > F −1 

i (u)|Xj > F −1 

j (u)} (2.31) 

where F −1 

i (u) and F −1 

j (u) represent the quantiles of assets Xi and Xj at the level u. 

To give an example for the illustration of lower tail dependence: Assuming σ1 and σ2 

represent volatilities of two different stocks, λ − gives the probability that both stocks 

exhibit together very high losses. 

Since the measure of tail dependence can be expressed in terms of the copula of Xi 

and Xj as shown in equations (2.32) for the lower tail dependence and in equation 

(2.33) for the upper tail dependence, it is independent of the marginals and symmetric 

in Xi and Xj. 

λ − 

ij = limu→0 + 

C(u, u) 

u 

λ + 

ij = lim u→1 − 

= lim u→1 − 

C(u, u) 

1 − u 

1 − 2u + C(u, u) 

1 − u 

(2.32) 

(2.33) 

The values for the coefficients of tail dependence are known explicitly for a large number 

of different copulas. This can be found i.e. in [14] (2000) published by J.E. Heffernan. 

10

Chapter 3 

Concepts for the Estimation of Tail 

Dependence 

Many different concepts and approaches have been developed to calculate coefficients of 

tail dependence as described by equation (2.30) and (2.31). This chapter presents and 

describes in detail the most important concepts for the estimation of tail dependence 

as well as their implementation, and a detailed sensitivity analysis of the calculated 

coefficients. 

In section (3.1) an overview of several concepts introduced by Heffernan, Tawn, 

Coles, Pown and Rockinger is provided. Then section (3.2) covers approaches according 

to Sornette and Malevergne and in the third section a non-parametric estimator 

according to Schmidt and Stadtmüller (3.4) is presented. 

All the approaches were analyzed on real data. I downloaded historical price sheets 

of the index S&P 500 and nine major stocks included in the index to estimate and 

analyze the different coefficients of tail dependence. The data can be accessed on 

Yahoo finance 1 and is provided in Microsoft Office Excel-CSV format ranging from 

the first mentioned to the actual price. It is updated daily and therefore allows the 

ongoing adaptation of coefficients. The evaluation of the data sets was performed by 

SPSS 16.0, a commercial statistics software that allows all-in-one-step data analysis. 

All further calculations were performed by Matlab 7.0. 

In the present chapter the methods were applied on two data sets: A smaller data 

set ranging from January 1991 up to December 2000 in accordance with [21] (2002), 

and a bigger data set ranging from July 1985 up to April 2008. The period of the bigger 

data set contained all available data in order to provide as many large fluctuations of 

the returns as possible to achieve meaningful information about extreme dependencies. 

All the assets studied are traded on the New York stock Exchange and had to 

fulfill the following criteria: First they should be among the stocks with the largest 

capitalization and second, each of them should have a weight that is smaller than 1% 

in the S&P 500 index, so that the dependence does not stem from the overlap with the 

index. The nine stocks were composed of: Bristol-Myers Squibb Co. (BMY), Chevron 

Corp. (CVX), Hewlett-Packard Co. (HPQ), Coca-Cola Co. (KO), 3M Co. (MMM), 

Procter & Gamble Co. (PG), Schering-Plough Corp (SGP), Texas Instruments Inc. 

(TXN), and Walgreen Co. (WAG). Starting at this point of the thesis I will refer to the 

stocks by their abbreviations given within brackets. A detailed data analyis is provided 

in the appendix. 

Since Matlab uses vectors, matrices, and tensors for its calculations, the compiler 

1 http://finance.yahoo.com 

11

can directly access the Excel sheets. In a first step the price data provided in the Excel 

sheets was converted into ”historical day-to-day return data” given by equation (3.1) 

to achieve a measure of relative volatility 

return(t) = log(price(t)) − log(price(t − 1)), (3.1) 

where log(·) denotes the natural logarithm, the inverse of the exponential function. 

The different methods are presented structured in subsections as follows: 

The first subsection respectively gives an insight into the underlying theory of the concepts 

or in other words, a summary of the most important findings in the papers. The 

second subsection respectively provides detailed description concerning the implementation, 

analysis and adaptation of variables for better robustness. Within the third 

subsection respectively, tail dependence estimates according to the different methods is 

analyzed i.e. by bootstrap sampling with replacement for an estimation of error bars. 

3.1 Approaches according to Poon, Rockinger, and Tawn 

In the first subsection I will start with some concepts for the estimation of dependence 

measures for multivariate extreme introduced by Poon, Rockinger, and Tawn published 

2004 in [13] (2004) with some further explanations provided by Heffernan 2000 in [14] 

(2000), by Coles, Heffernan and Tawn in [15] (1999), and by Ledford and Tawn in 

[12] (1996). Then I will go into some details concerning non-parametric estimation of 

theses measures and the fitting of parametric models. In the second subsection I will 

provide an overwiev on the implementation of non-parametric estimators for asymptotic 

dependence χ and asymptotic independence χ. 

3.1.1 Theoretical Background 

Within this subsection I will provide an overview on the different concepts and estimators 

that describe asymptotic dependence and asymptotic independence for the 

bivariate case. 

Dependence Measures for Multivariate Extreme 

In order to reduce the information contained in the copula C to a single parameter, two 

measures of extremal dependence χ and χ were introduced in [15] (1999). Both of them 

are needed in order to examine whether two variables are asymptotically dependent or 

asymptotically independent. First the bivariate returns X and Y are transformed to 

unit Fréchet marginals S and T because of fat-tail distributions for risk asset returns: 

S = −1/ log (FX(X)) T = −1/ log (FY (Y )), (3.2) 

where FX and FY are marginal distribution functions for X and Y . Variables S and 

T have the same dependence structure as X and Y and are now on a common scale, 

meaning that events of the form S > s and T > s for large s, correspond to equally 

extreme events for each variable. Asymptotic dependence χ was given by: 

χ = lim 

s→∞ Pr(T > s | S > s) (3.3) 

= lim 

s→∞ 

Pr(T > s, S > s) 

Pr(S > s) 

12

or expressed through copula C: 

1 − 2u + C(u, u) 

= lim 

u→1 1 − u 

= lim 

u→1 2 − 

∼ lim 

→1 2 − 

1 − C(u, u) 

1 − u 

log C(u, u) 

log u 

(3.4) 

with 0 ≤ χ ≤ 1. χ is the same as the upper tail dependence coefficient given in equation 

(2.31). As mentioned in the previous chapter, S and T are asymptotically dependent if 

χ > 0, and they are perfectly dependent if χ = 1. If χ = 0, S and T are asymptotically 

independent. 

A complementary measure χ developed by Ledford and Tawn (1996) was given to 

measure extreme dependence of variables that are asymptotic independent, that is, 

where χ = 0: 

2 log (Pr(S > s)) 

χ = lim 

− 1 

s→∞ log (Pr(S > s, T > s)) 

(3.5) 

2 log(1 − u) 

= lim 

− 1, 

s→∞ 

log Ĉ(u, u) 

(3.6) 

where −1 < χ ≤ 1, and χ is a measure of the rate at which Pr(T > t | S > s) 

approaches zero. For perfect dependence χ = 1 and for independence χ = 0. Hence 

values of χ > 0 indicate that S and T are positively associated, and χ < 0 indicates 

that S and T are negatively associated in the extremes. For the bivariate normal 

dependence structure, χ = ρ, denoting the coefficient of correlation. Other examples 

are listed in Heffernan [14] (2000). 

Before drawing conclusions about asymptotic dependence based on χ, it is important 

to test if χ = 1. For asymptotically dependent variables, χ = 1 with degree of 

dependence given by χ > 0 and for asymptotically independent variables, χ = 0 with 

degree of dependence given by χ. 

Non-Parametric Estimation of χ and χ 

The tail of an univariate heavy-tailed variable Y above a high threshold k is given to: 

Pr(Y > y) = L(y) · y −α 

where L(y) is a slowly varying function of y, that is 

for y > k, (3.7) 

L(k · y) 

lim = 1 forall y > 0. (3.8) 

k→∞ L(k) 

Then tail index α is estimated by Hill’s estimator: 

� 

1 � 

kj=1 

ˆν = log 

k 

Yj,N 

�−1 Yk,N 

13 

(3.9)

and L(y), assumed to be constant for y > k, is estimated by: 

ˆL(y) = k 

N 

· (Yk,N) ˆν 

(3.10) 

with Y1,N ≥ Y2,N ≥ · · · ≥ YN,N denoting ordered statistics of the sample containing N 

independent and identically distributed realizations of the market return vector Y and 

k denoting the number of the threshold value representing the least extreme value still 

counted to the tail of the distribution. 

Ledford and Tawn [12] (1996) characterized the joint tail behavior by constant η 

denoting the coefficient of tail dependence and slowly varying function L(s) and established 

that under weak conditions: 

Pr(S > s, T > s) ∼ L(s) · s −1/η 

with 0 < η ≤ 1. It follows from the representations in [15] (1999) that 

as s → ∞ (3.11) 

χ = 2η − 1 

⎧ 

⎨ c if χ = 1 and L(s) → c > 0 as s → ∞ 

(3.12) 

χ = 0 if χ = 1 and L(s) → 0 as s → ∞ 

⎩ 

0 if χ < 1 

(3.13) 

χ = 1 corresponds to η = 1 and yields χ = lims→∞ L(s). Thus the estimation of η 

and lims→∞ L(s) provides the basis for the estimation of χ and χ. According to the 

modeling assumption for the bivariate case, for Z = min(S, T): 

Pr(Z > z) = Pr (min(S, T) > z) 

= Pr(S > z, T > z) (3.14) 

= L(z) · z −1/η 

for z > Zk,N 

for some high threshold number k. As η is the tail index of the univariate variable Z, it 

can be estimated by equation (3.9) using Hill’s estimator restricted to the interval (0, 1] 

and lims→∞ L(s) can be estimated using equation (3.10). The following estimators are 

based on the assumption of independent observations on Z. χ was estimated to: 

ˆχ = 2 

� 

k� � � 

Zj,N 

log 

k Zk,N 

j=1 

� 

− 1 (3.15) 

var � ˆχ � � 

ˆχ + 1 

= 

�2 (3.16) 

k 

The estimator of χ, only calculated if there is no significant evidence to reject χ = 1, 

was proposed to be: 

ˆχ = Zk,N · k 

N 

var(ˆχ) = Zk,N 2 k(N − k) 

N3 (3.17) 

(3.18) 

ˆχ is a figure that could be compared to other estimators of tail dependence coefficients, 

which are explained in the following sections of chapter (3) because it has the same 

underlying definition as provided in equation (2.33) showing the formula of upper tail 

dependence. 

14

Parametric Joint-Tail Models 

Dependence models were estimated over the whole sample space. Conditional on being 

above the threshold kX and kY a generalized Pareto distribution was used and below 

the empirical distribution function ˜ F(x) was used. So the model for FX(x) (and also 

FY (y)) was given as: 

FX = 

� ˜F(x) if x < kX 

1 − 1 − ˜ F(kX)1 + αX(x − kX)/σX −1/αX if x ≥ kX, 

(3.19) 

where σX and αX are the generalized Pareto distribution scale and shape parameters. 

After estimation of the marginal parameter variables, X and Y are transformed to unit 

Fréchet form S and T by equation (3.2) to model the dependence structure by parametric 

dependence models. Model parameters are fitted by matching their associated 

χ and χ with the non-parametric estimates defined in equations (3.17) and (3.15). 

3.1.2 Implementation of Non-Parametric χ and χ 

Now I will provide a detailed description about the implementation of a non-parametric 

estimator for χ, which is a measure of asymptotic dependence and a non-parametric 

estimator for χ, which is a measure of asymptotic independence 2 . Both estimators 

were described in chapter (3.1.1). 

As already explained, we only calculate ˆχ if there is no significant evidence to reject 

ˆχ = 1. Or in other words, we only estimate χ under the assumption that χ = η = 1. 

Therefore we first calculate ˆχ and its estimated variance given by var � ˆχ � to check 

whether value 1 lies in the respective confidence interval of ˆχ. If that is the case, we 

go on and calculate ˆχ and its variance estimate var(ˆχ) to obtain a measure for tail 

dependence that can be compared to estimates by other methods. If ˆχ �= 1, we set 

ˆχ = 0 assuming asymptotic independence between return vectors X and Y . 

First we transform return vectors X and Y into vectors of respective Fréchet 

marginals given by S and T. Therefore we estimate the survival functions F X(x) 

and F Y (y) of the univariate heavy-tailed variables X an Y above a high threshold 

defined by threshold number k with k/N = 4% of most extreme return data by: 

F X(x) = L(x) · x −α 

(3.20) 

F Y (y) = L(y) · y −α , (3.21) 

where L is a slowly varying function estimated as mean value L in the extreme tail: 

L(x) = 1 

k 

L(y) = 1 

k 

k� 

j=1 

k� 

j=1 

j 

N 

· (Xj,N) α 

(3.22) 

j 

N · (Yj,N) α , (3.23) 

2 The matlab m-file for the implementation of ˆχ is denoted by: lambda porota.m and enclosed to the 

appendix and the data CD 

15

or estimated as mean value Lc, corrected by the most extreme outliers, with c = 

[0.005 ∗ N] and [·] denoting integer numbers: 

Lc(x) = 

Lc(y) = 

1 

k − c + 1 

1 

k − c + 1 

and α estimated by Hill’s estimator given by: 

k� 

j=c 

k� 

j=c 

j 

N 

· (Xj,N) α 

� 

1 � 

kj=1 

ˆνX = log 

k 

Xj,N 

�−1 Xk,N 

� 

1 � 

kj=1 

ˆνY = log 

k 

Yj,N 

�−1 (3.24) 

j 

N · (Yj,N) α , (3.25) 

Yk,N 

(3.26) 

(3.27) 

with X1,N ≥ X2,N ≥ · · · ≥ XN,N and Y1,N ≥ Y2,N ≥ · · · ≥ YN,N denoting ordered 

statistics of the sample containing N realizations of the market return vector Y and 

X. Now transformation to S and T is performed as follows: 

ˆS 

1 

= − � 

log 1 − ˆ � (3.28) 

F X(x) 

ˆT 

1 

= − � 

log 1 − ˆ � (3.29) 

F Y (y) 

Now we come to the estimation of χ and var � ˆχ � by first calculation of Z applying 

relation Z = min(S, T) element-wise to vectors X and Y , and then using: 

ˆχ = 2 

� 

k� � � 

Zj,N 

log 

k Zk,N 

j=1 

� 

− 1 (3.30) 

var � ˆχ � � 

ˆχ + 1 

= 

�2 (3.31) 

k 

The estimator of χ and var(χ), only calculated if {1} ∈ � ˆχ ± var � ˆχ �� by: 

ˆχ = Zk,N · k 

N 

var(ˆχ) = Zk,N 2 k(N − k) 

N3 (3.32) 

(3.33) 

To achieve estimates for the lower tail, we can just sort our data in ascending order 

and multiply the ordered sample by (−1). Tables (3.1) and (3.2) show results of the 

� Zk,N 2 k(N−k) 

estimated χ and ˆσ = N3 for the smaller and the bigger reference sample of 

the index S&P 500 and the nine assets. 

16

First of all we notice that only for one of the nine assets and the index S&P 500 

χ = 1 can be rejected. This can be seen in table (3.1) showing estimates for the smaller 

reference sample, where asset Chevron Corp. (CVX) in the negative uncorrected tail is 

marked with an asterisk. In both tables (3.1) and (3.2) only estimates of χ are provided 

because ˆχ underlies the same definition as the concept of upper tail dependence given in 

equation (2.33) and can therefore later on be compared to estimates by other methods. 

We also find that there is a big difference between tail dependence estimated for 

uncorrected tails on intervals 1, 2 . . .k and tail dependence estimates for corrected tails 

on intervals c, c + 1 . . .k, with c = [0.005 ∗ N] and [·] denoting integer numbers, in the 

lower tail of the smaller data set given in table (3.1). For upper tails of both data 

sets and smaller tails of the bigger data set the estimates for corrected and uncorrected 

tails are more consistent with each other and also more consistent over time because the 

bigger of the two data sets contains the smaller one and the two respective estimates 

are similar for most assets. A reason for these big discrepancies in the lower tail of the 

smaller sample might be distortion caused by extreme outliers. Indeed, looking at the 

return data sets it can be observed that most extreme values or outliers lie primarily 

in the negative tails. What furthermore can be observed is that in case of tail dependence 

coefficient estimates for corrected tails, left-tail dependence or tail dependence 

of negative tails tends to be stronger than right-tail dependence or tail dependence of 

positive tails. This is consistent with previous literature i.e. [13] (2004). Anyway, looking 

at estimates for uncorrected tails, we observe that right-tail dependence tends to be 

stronger than left-tail dependence. Estimated standard deviations ˆσ were calculated to 

be around 6%-7% of the estimates. Thich seems accurate enough. 

For comparison I calculated error bars by bootstrap sample estimates. Results for 

error bars estimated by bootstrap sampling with replacement are shown in table (3.3) 

for the smaller reference data set and in table (3.4) for the bigger reference data set. 

Looking at tails that are not corrected for outliers, deviations calculated by bootstrap 

sampling for both the smaller and the bigger data sample, primarily for negative 

tails, are significantly higher in most cases than calculated by the given relation: 

� Zk,N 2 k(N−k) 

ˆσ = N3 . Comparing standard deviations ’std bs’ with 90% quantiles and 

95%, we observe that error distributions are not Gaussian. Looking at 90% quantiles 

of estimates performed for corrected tails χc, the error bars are in most cases similar to 

ˆσ shown in tables (3.1) and (3.2), but for some assets also higher. I will come back to 

the estimation of error bars by bootstrap sampling with replacement later on and also 

provide an explanation about how it is performed 3 . 

3 The matlab m-file for the bootstrap sampling of ˆχ is denoted by: bootstr porota.m and enclosed to the 

data CD 

17

upper tail ˆσ upper tail ˆσ lower tail ˆσ lower tail ˆσ 

tail 1... k c... k 1... k c... k 

ˆχ 

BMY 0.94 0.09 0.92 0.09 0.50 0.05 0.94 0.09 

CVX 0.94 0.09 0.92 0.09 0.00* 0.00* 0.94 0.09 

HPQ 0.94 0.09 0.91 0.09 0.40 0.04 0.94 0.09 

KO 0.94 0.09 0.92 0.09 0.26 0.03 0.94 0.09 

MMM 0.94 0.09 0.92 0.09 0.42 0.04 0.94 0.09 

PG 0.94 0.09 0.92 0.09 0.46 0.05 0.94 0.09 

SGP 0.94 0.09 0.92 0.09 0.34 0.03 0.94 0.09 

TXN 0.94 0.09 0.92 0.09 0.52 0.05 0.94 0.09 

WAG 0.94 0.09 0.92 0.09 0.36 0.04 0.94 0.09 

Table 3.1: Estimated values of upper and lower tail dependence for S&P 500 index with a set 

of nine major assets traded on the New York stock Exchange calculated by a non-parametric 

approach: ˆχ +,− = Zk,N ·k 

� Zk,N 2 k(N−k) 

N and corresponding standard deviation: ˆσ (ˆχ) = N3 . The 

tail represents the most extreme 4% of the return values during a time interval from January 

1991 to December 2000. ’tail’ shows the calculation interval of L with c = 13 and k = 100, 

and ∗ denotes ˆχ �= 1 and therefore ˆχ = 0. 

upper tail ˆσ upper tail ˆσ lower tail ˆσ lower tail ˆσ 

tail 1... k c...k 1... k c... k 

ˆχ 

BMY 0.96 0.06 0.93 0.06 0.79 0.05 0.97 0.06 

CVX 0.96 0.06 0.93 0.06 0.35 0.02 0.97 0.06 

HPQ 0.96 0.06 0.92 0.06 0.74 0.05 0.97 0.06 

KO 0.93 0.06 0.93 0.06 0.54 0.04 0.97 0.06 

MMM 0.96 0.06 0.93 0.06 0.69 0.04 0.97 0.06 

PG 0.96 0.06 0.93 0.06 0.58 0.04 0.97 0.06 

SGP 0.96 0.06 0.93 0.06 0.74 0.05 0.97 0.06 

TXN 0.96 0.06 0.93 0.06 0.70 0.05 0.97 0.06 

WAG 0.96 0.06 0.93 0.06 0.73 0.05 0.97 0.06 

Table 3.2: Estimated values of upper and lower tail dependence for S&P 500 index with a set 

of nine major assets traded on the New York stock Exchange calculated by a non-parametric 

N and corresponding standard deviation: ˆσ (ˆχ) = N3 . The 

tail represents the most extreme 4% of the return values during a time interval from July 

1985 to April 2008. ’tail’ shows the calculation interval of L with c = 29 and k = 229, and ∗ 

denotes ˆχ �= 1 and therefore ˆχ = 0. 

approach: ˆχ +,− = Zk,N ·k 

18 

� Zk,N 2 k(N−k)

19 

m=2507 bs=1000 

upper tail lower tail 

χbs,mean 

χbs,mean 

χ k=100 rel err max dev 95% q 90% q std bs k=100 rel err max dev 95% q 90% q std bs 

BMY 0.92 0.02 0.91 0.02 0.01 0.08 0.50 0.00 0.52 0.52 0.42 0.32 

CVX 0.93 0.01 0.94 0.01 0.01 0.04 0.35 Inf 0.64 0.61 0.60 0.41 

HPQ 0.93 0.01 0.93 0.01 0.01 0.07 0.35 0.11 0.63 0.53 0.41 0.22 

KO 0.91 0.04 0.90 0.04 0.03 0.20 0.36 0.39 0.64 0.63 0.63 0.30 

MMM 0.91 0.03 0.88 0.03 0.03 0.20 0.56 0.34 0.62 0.52 0.38 0.31 

PG 0.92 0.02 0.93 0.02 0.02 0.10 0.34 0.25 0.60 0.51 0.42 0.28 

SGP 0.92 0.02 0.89 0.02 0.02 0.10 0.34 0.01 0.62 0.44 0.28 0.19 

TXN 0.93 0.01 0.92 0.01 0.01 0.04 0.42 0.20 0.53 0.41 0.41 0.27 

WAG 0.91 0.03 0.94 0.03 0.03 0.10 0.29 0.20 0.60 0.33 0.32 0.21 

χc 

BMY 0.90 0.02 0.92 0.03 0.03 0.03 0.90 0.04 0.92 0.11 0.11 0.20 

CVX 0.90 0.02 0.12 0.03 0.03 0.02 0.83 0.11 0.81 0.82 0.22 0.31 

HPQ 0.90 0.01 0.93 0.03 0.03 0.03 0.91 0.03 0.91 0.14 0.12 0.22 

KO 0.91 0.01 0.14 0.04 0.03 0.02 0.90 0.04 0.88 0.09 0.09 0.21 

MMM 0.91 0.01 0.13 0.04 0.03 0.02 0.90 0.04 0.89 0.13 0.10 0.24 

PG 0.91 0.01 0.12 0.04 0.03 0.02 0.93 0.00 0.91 0.12 0.12 0.13 

SGP 0.90 0.01 0.91 0.04 0.03 0.03 0.87 0.07 0.91 0.90 0.12 0.20 

TXN 0.90 0.02 0.09 0.03 0.03 0.02 0.84 0.10 0.83 0.82 0.21 0.29 

WAG 0.91 0.01 0.10 0.04 0.03 0.02 0.93 0.01 0.87 0.14 0.12 0.10 

Table 3.3: 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 

January 1991 to December 2000.

20 

m=5736 bs=1000 


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 


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. 


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

A general result valid for any regularly varying distribution was provided. Let F Y 

follow a regularly varying distribution with tail index α: 

F Y (y) = L(y) ∗ y −α 

where L(y) is a slowly varying function, i.e. 

then: 

(3.38) 

L(t · y) 

lim = 1 for all y > 0, (3.39) 

t→∞ L(t) 

λ + = 

1 

� 

max 1, l 

β 

� α 

(3.40) 

with l given by equation (3.36). To obtain λ − the limit u → 1 − has to be replaced by 

u → 0 + . 

The Parametric Approach by Sornette & Malevergne 

Following the parametric approach according to Sornette & Malevergne we estimate 

a parametric form of tail distribution, which for extreme market risks is assumed to 

follow a power-law distribution. For ν = νY = νε we have F Y (y) = Ĉy ∗ y −ν and 

F ε(ε) = Ĉε ∗ ε −ν for large y and ε, then the coefficient of tail dependence is a simple 

function of the ratio Cε/CY of the scale factors: 

λ = 

1 

1 + β −α · Cε 

CY 

(3.41) 

If the tail indexes αY and αε of the distribution of the factors and the residue are 

different, then λ = 1 for αY < αε and λ = 0 for αY > αε. 

So far we have only considered a single asset with vector of returns X. Let us now 

consider a portfolio of assets with vectors of returns Xi, where each asset follows the 

linear factor model: 

Xi = βi · Y + εi 

(3.42) 

with independent noises εi, whose scale factors are Cεi . The portfolio X = � wiXi, 

with weights wi, also follows the factor model with a parameter β = � wiβi and noise 

ε, whose scale factor is Cε = � |wi| ν · Cεi . The tail dependence between the portfolio 

and the market index can now be obtained by equation (3.41): 

� � 

|wi| 

λ = 1 + 

α · Cεi 

( � wiβi) α �−1 (3.43) 

· CY 

Tail Dependence between two Assets related by a Factor Model 

The tail dependence between two assets related by a factor model given by: 

X1 = β1 · Y + ε1 

X2 = β2 · Y + ε2 

22 

(3.44) 

(3.45)

is equal to the weakest tail dependence between the assets X1 and X2, and their 

common factor Y : 

λij = min{λi, λj} (3.46) 

Since the dependence between two assets is due to their common factor, this dependence 

cannot be stronger than the weakest dependence between each of the assets and the 

factor. This result shows that it is sufficient to study the tail dependence between the 

assets and their common factor to obtain tail dependence between any pair of assets. 

Estimation of Tail Indexes 

For all our approaches according to Sornette & Malevergne we need to calculate a tail 

index α. This can be conducted independently of the method. As we assume stock 

and market returns to be asymptotically power-law distributed we can calculate the 

tail indexes for both positive and negative tails using Hill’s estimator given by: 

ˆν = 

� 

1 � 

kj=1 

log 

k 

Yj,N 

Yk,N 

� −1 

, (3.47) 

where Y1,N ≥ Y2,N ≥ · · · ≥ YN,N denote the ordered statistics of the sample containing 

N independent and identically distributed realizations of market return vector Y and 

k denotes the number of the threshold value representing the least extreme value still 

counted to the tail of the distribution. Assuming that the tail index of each asset αXi 

is the same as the tail index of the market index αY containing the asset, k denotes the 

number of the thteshold value of the N sorted return observation or the smallest value 

for each, market index and asset, still considered to belong to the tail. Hill’s estimator 

is asymptotically normal distributed with mean α and variance α 2 /k. But for finite k 

it is known that the estimator is biased. 

To find out whether tail indexes calculated by Hill’s estimator are accurate, I performed 

a test under conditions similar to our situation: Assuming heavy tail distributed 

market returns with a tail index of α = 3 and given a general formulation of heavy tail 

probability density distribution functions: 

p(y) ∼ α 

y 1+α, 

(3.48) 

I generated n = 100 random points on an uniformly distributed interval ranging from 

zero to one (p(r) = 1, for r U[0,1]). 

Then I performed a variable transformation applying p(y)dy = p(r)dr to convert the 

uniformly distributed random values into heavy tail distributed values given by relation 

(3.48). Putting in p(r) and p(y) yields: 

α 

y1+αdy = dr (3.49) 

and by integral of both sides: C − y −α = r, where constant C can be calculated using 

the probability condition: 

zeros(C − y −α 1 

− 

) = {C α }, for ∀C > 0 

� � 

Y2 

limY2→∞ p(y)dy = − 1 

yα �∞ = C = 1, for ∀α > 0 

C − 1 α 

23 

C − 1 α

and finally I obtained: 

y(r) = 

� � 1 

α 1 

, (3.50) 

1 − r 

which allowed me to transform our uniformly distributed data into data distributed 

according to relation (3.48) with α = 3. 

Putting now the transformed values into equation (3.47) to calculate Hill’s estimator, 

I could test whether the output was a tail index ˆν close enough to three representing 

the true value α of the tail index for our transformed data. 

Conducting this procedure several times allowed me to check whether the standard 

deviation fulfills the above described relation of: 

std(ˆν) 

α ∼ = 1 

√ k , (3.51) 

which, given α = 3 and n = 100, says that std(ˆν) ∼ = 0.3 and therefore all ˆν should be 

within an interval of 3 ±2∗0.3 if the error follows a Gaussian distribution as requested. 

I performed the above described sampling 10’000 times and received a mean value of 

mean(ˆν) = 3.06, which shows a small bias of 2%, and a 95% deviation-quantile of 0.63, 

which is close to the above calculated 0.6. Therefore I conclude that our smaller sample 

size with about 100 return data points perceived as tail data already agrees well with 

the general assumption of Hill’s estimator to be asymptotically normally distributed. 

However, performing this little affirmation we stressed the assumption of using independent 

and identically distributed (i.i.d.) data samples, which unfortunately is not 

the case for real financial data. Applying Engle’s test for the presence of ARCH effects 

[17] (1982) to our real financial time series the result shows significant evidence in 

support of GARCH effects (heteroscedasticity) 4 . As shown by Kearns and Pagan [22] 

(1997) for heteroskedastic time series the variance of the estimated tail index can be 

seven times larger than the variance given by the asymptotic normality assumption. 

For comparison I also implemented the OLS (ordinary least squares) log-log rank-size 

tail index estimate proposed by Gabaix [23] (2006) defined by: 

log(Rank − 1/2) = a − ˆb γ n · log(Size), (3.52) 

with the tail index given by ˆb γ n . A standard error of the OLS estimate ˆb γ � n of the slope 

2 

was estimated to n ˆb γ n in the paper. Assuming a tail index equal to three as above, 

this yields a standard error of 0.42, which is higher compared to Hill’s estimator. 

We can now perform the same little check as above to see how Gabaix’s ˆb γ n estimator 

performs for i.i.d. data samples. Sampling 10’000 times, I obtained a mean value of 

mean( ˆb γ n) = 3.015, which only constitutes 0.5% bias and a 95% deviation-quantile of 

0.83. This perfectly agrees with the estimated standard error given in the paper [23] 

(2006) but still is a little higher than the standard deviation of Hill’s estimator. 

Now we can apply both, Hill’s estimator ˆν and Gabaix’s ˆb γ n estimator, on real data 

for comparison and subsequently perform bootstrap sampling with replacement (Monte 

Carlo simulation) to find out, whether the fact, that real financial time series exhibit 

internal dependence, in terms of time varying volatility clustering (large price changes 

come in clusters), impact deviation quantiles or in other words: whether Hill’s estimator 

4 Engle’s ARCH test is described in subsection (3.2.3) 

24

ν(k) 

4 

3.8 

3.6 

3.4 

3.2 

3 

2.8 

2.6 

2.4 

S&P 500 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

2.2 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

Figure 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. 

still performs better in a deviation quantile sense compared to Gabaix’s ˆ b γ n estimator 

when it is applied to real financial time series. 

In figures (3.1) and (3.2) I plotted both estimators in dependence of threshold k for 

the upper tails and in figures (3.3) and (3.4) for the lower tails. In the positive tails 

the developing of the values with increasing k looks quite similar for the tail indexes 

calculated by Hill’s estimator and the tail indexes calculated by Gabaix’s estimator, 

although the ˆb γ n (k) estimator is less sensitive to k. In the negative tails ˆb γ n looks quite 

different from ˆν(k). Assets are increasing in k, whereas the index S&P 500 is slightly 

decreasing and at a quite high threshold of 15% they kind of converge. What all the 

is that they 

assets and the index S&P 500 have in common in the negative tails of ˆb γ n 

first tend to rise and then build something like a plateau before they drop. This plateau 

is also apparent in the positive tails, but there it is much less distinctive and at much 

smaller thresholds k. 

Now we have to define a relevant range for the estimation of the tail index in terms 

of threshold k. As k increases, the variance of the estimator decreases while its bias 

increases. It is reported in [20] (2004) that an accurate determination of the optimal k 

by optimization algorithms is rather difficult. Although a relevant range for k between 

1% and 5% of total data for the estimation of the tail index was given, I decided to 

search visually for a k that looks consistent for the 9 assets and the index S&P 500. The 

ˆγ bn estimators shown look quite smooth in the area of 4%. Some assets show something 

like a plateau around there or at least the tail index estimators seem to be stable there. 

Table (3.5) shows the results of tail indexes of all assets for both sample sizes, 

the index S&P 500, and the residues ε obtained by regressing each asset on the S&P 

500 index, assuming a threshold of k = 4%. It is interesting that Hill’s estimator is 

smaller than Gabaix ˆb γ n estimator for the upper tail and is systematically larger for the 

25

γ 

b 

n 

ν(k) 

4.2 

4 

3.8 

3.6 

3.4 

3.2 

3 

S&P 500 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

2.8 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

Figure 3.2: Tail index estimators ˆ b γ n (Gabaix’s estimator) for the upper tails of index S&P 500 



3.5 

3 

2.5 

S&P 500 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

2 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

Figure 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

γ 

b 

n 

3.2 

3 

2.8 

2.6 

2.4 

2.2 

2 

1.8 

1.6 

S&P 500 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

1.4 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

Figure 3.4: Tail index estimators ˆ b γ n (Gabaix’s estimator) for the lower tails of index S&P 500 



lower tail. Furthermore, it is shown in [23] (2006) that although the standard error of 

the Hill’s estimator is less than that of the tail index estimate ˆb γ n, its (small sample) 

bias in the case of deviation from the power law is (typically) greater than that of ˆb γ n . 

Perhaps, we indeed under-estimate the lower tail such that the exponent is probably 

smaller than given by Hill’s estimator. To check whether our assumption of equal tail 

indexes of asset returns, residues, and index returns is accurate with our estimation 

methods and data sets, I performed a simple test of means at the 95% confidence level 

for any given quantile. Values that reject the equality hypothesis are asterisked. As 

we can see, the results were inconclusive. Anyway, the performed tests were based on 

the assumption of asymptotic normality in the distribution of the estimates with the 

standard deviations of � 2/n · ˆb γ n and ˆν/√ k, which might be too restrictive. The true 

distribution is probably ’wilder’ than Gaussian as explained above. This would expand 

the confidence interval. It is nevertheless interesting to notice that in the negative 

tails of ˆ b γ n 

, although the equality hypothesis was rejected for all assets and residues 

because estimates for assets and residues were unexeptionally lower than for the S&P 

500 index, tail index estimates of asset returns are nearly identical to tail indexes of 

their corresponding residues obtained by regression on the index S&P 500. 

As we can obtain in the upper parts of the tables (3.6) and (3.7) where the smaller 

samples with 100 data points considered as tail data are shown, 95% quantiles of the 

upper tails show good accordance to our i.i.d. estimates, whereas 95% quantiles of 

lower tails tend to be somewhat higher. In terms of deviation quantiles Hill’s estimator 

still outperforms Gabaix’s ˆ b γ n estimator. What we can observe in the lower parts of the 

tables is that deviation quantiles become obviously smaller when we augment sample 

sizes. This counts for both estimators. 

27

28 

+tail −tail 

ν b γ n ν b γ n ν b γ n ν b γ n 

k 100 100 100 100 229 229 229 229 100 100 100 100 229 229 229 229 

Asset ε Asset ε Asset ε Asset ε Asset ε Asset ε Asset ε Asset ε 

BMY 3.36 3.14 3.66 3.63 3.31 3.05 3.59 3.38 2.55 2.57 1.80* 1.76* 2.46* 2.46* 2.04* 1.97* 

CVX 3.73 3.67 4.26 3.89 3.49* 3.54* 3.83 3.60 3.61 3.61 2.30* 2.27* 3.14 3.20 2.33* 2.29* 

HPQ 3.42 3.02 3.94 3.42 2.88 2.65* 3.29 3.01 2.82 3.02 1.80* 1.70* 2.91 2.53* 2.35* 2.17* 

KO 3.50 3.08 3.70 3.36 3.48 3.40 3.36 3.24 2.79 3.08 1.80* 1.75* 2.48* 2.59* 1.80* 1.79* 

MMM 3.80 3.34 4.10 3.86 3.56* 3.22 3.69 3.54 2.89 3.34 2.23* 2.21* 2.48* 2.39* 2.04* 1.96* 

PG 3.40 3.14 3.57 3.76 3.19 3.03 3.60 3.39 2.45* 3.14 1.57* 1.55* 2.40* 2.41* 1.69* 1.66* 

SGP 3.37 3.11 3.78 3.72 3.36 2.97 3.88 3.45 2.69 3.11 1.70* 1.65* 2.40* 2.27* 1.84* 1.78* 

TXN 3.51 3.52 3.99 3.99 3.03 3.10 3.34 3.34 2.67 3.52 1.77* 1.77* 2.73 2.68* 2.05* 2.03* 

WAG 3.61 3.99* 3.92 4.22 3.33 3.40 3.81 3.58 2.42* 3.99* 1.46* 1.43* 2.44* 2.65* 1.93* 1.92* 

S&P 500 3.23 − 3.71 − 3.09 − 3.56 − 3.16 − 3.37 − 3.13 − 3.04 − 

Table 3.5: Comparison of tail index estimators ˆν (Hill’s estimator) and ˆ b γ n (estimator by Gabaix) for positive and negative tails of the S&P 500 

index, 9 included assets, and the residues ε obtained by regressing each asset on the S&P 500 index, during the time intervals ranging from January 

1991 to December 2000 with a 4% quantile of k = 100 values and from July 1985 to April 2008 with a 4% quantile of k = 229 values. Values with 

∗ represent tail indexes, which can’t be considered equal to the respective S&P 500 index at the 95% condfidence level.

29 

m=2507 bs=1000 


νmean 

ν k=100 rel err max dev 95% q 90% q std bs k=100 rel err max dev 95% q 90% q std bs 

Index 3.31 0.02 1.01 0.60 0.52 0.32 3.19 0.01 0.82 0.62 0.53 0.34 

BMY 3.42 0.02 1.22 0.64 0.52 0.31 2.62 0.03 1.10 0.72 0.60 0.28 

CVX 3.78 0.01 1.31 0.61 0.53 0.30 3.69 0.02 1.61 1.00 0.82 0.50 

HPQ 3.50 0.02 1.10 0.63 0.51 0.29 2.89 0.02 1.31 0.83 0.71 0.40 

KO 3.55 0.02 1.28 0.62 0.50 0.28 2.86 0.03 1.32 0.84 0.74 0.43 

MMM 3.85 0.01 1.36 0.67 0.58 0.40 2.94 0.02 1.20 0.71 0.62 0.41 

PG 3.45 0.02 1.32 0.59 0.53 0.31 2.51 0.03 1.24 0.70 0.61 0.44 

SGP 3.44 0.02 1.20 0.61 0.50 0.31 2.75 0.02 1.33 0.82 0.61 0.43 

TXN 3.56 0.01 1.21 0.60 0.52 0.32 2.74 0.03 1.24 0.70 0.62 0.42 

WAG 3.68 0.02 1.30 0.63 0.50 0.30 2.48 0.02 1.20 0.71 0.63 0.40 

m=5736 bs=1000 

ν k=229 k=229 

Index 3.10 0.01 0.62 0.33 0.32 0.21 3.16 0.01 0.58 0.42 0.33 0.24 

BMY 3.32 0.00 0.61 0.40 0.31 0.21 2.48 0.01 0.50 0.40 0.34 0.21 

CVX 3.50 0.00 0.71 0.42 0.34 0.21 3.17 0.01 0.82 0.53 0.42 0.22 

HPQ 2.89 0.00 0.50 0.33 0.32 0.22 2.93 0.01 0.71 0.44 0.42 0.20 

KO 3.50 0.01 0.74 0.43 0.41 0.22 2.51 0.01 0.64 0.39 0.31 0.21 

MMM 3.57 0.00 0.73 0.41 0.40 0.21 2.51 0.01 0.63 0.41 0.34 0.22 

PG 3.20 0.00 0.61 0.33 0.32 0.23 2.51 0.01 0.70 0.42 0.32 0.22 

SGP 3.37 0.00 0.61 0.40 0.28 0.23 2.42 0.01 0.63 0.40 0.34 0.20 

TXN 3.04 0.00 0.62 0.33 0.29 0.24 2.75 0.01 0.71 0.41 0.33 0.24 

WAG 3.34 0.00 0.62 0.40 0.31 0.23 2.46 0.01 0.60 0.40 0.32 0.20 

Table 3.6: Establishing the uncertainty of estimated upper and lower tail indexes ˆν by creating bs = 1000 bootstrap samples of historical return 

data for S&P 500 and included assets and calculation of bootstrap mean value νmean with rel. error from the original value, deviation quantiles 

and standard deviation from νmean, and most extreme deviation value. The tail represents the most extreme 4 % of the return values during a 

time interval ranging from January 1991 to December 2000 (k = 100 values) and from July 1985 to April 2008 (k = 229 values). 

νmean

30 

m=2507 bs=1000 


b γ mean 

b γ n k=100 rel err max dev 95% q 90% q std bs k=100 rel err max dev 95% q 90% q std bs 

Index 3.71 0.00 1.42 0.72 0.64 0.32 3.43 0.02 2.22 0.92 0.82 0.53 

BMY 3.66 0.00 1.50 0.72 0.52 0.31 1.93 0.07 2.91 1.21 0.81 0.60 

CVX 4.26 0.00 1.52 0.80 0.60 0.40 2.84 0.20 3.02 2.02 1.72 1.22 

HPQ 3.95 0.00 1.24 0.63 0.51 0.32 1.92 0.06 3.50 0.84 0.74 0.51 

KO 3.72 0.01 1.51 0.81 0.68 0.41 1.96 0.09 2.23 1.54 1.00 0.62 

MMM 4.12 0.01 1.52 0.80 0.70 0.43 2.45 0.11 2.00 1.23 1.11 0.74 

PG 3.59 0.00 1.44 0.81 0.61 0.42 1.67 0.06 2.61 0.71 0.64 0.42 

SGP 3.79 0.01 1.58 0.73 0.62 0.42 1.87 0.10 3.03 1.22 0.66 0.61 

TXN 4.00 0.00 1.54 0.79 0.66 0.43 1.86 0.05 2.12 0.93 0.63 0.40 

WAG 3.94 0.01 1.59 0.80 0.70 0.41 1.55 0.06 2.30 0.80 0.60 0.42 

m=5736 bs=650 

b γ n k=229 k=229 

Index 3.56 0.00 0.93 0.50 0.38 0.19 3.07 0.01 1.42 0.81 0.63 0.41 

BMY 3.60 0.00 1.10 0.42 0.42 0.18 2.07 0.02 1.40 0.60 0.52 0.31 

CVX 3.83 0.00 0.92 0.57 0.50 0.30 2.45 0.05 2.11 1.22 0.84 0.58 

HPQ 3.28 0.00 0.62 0.50 0.41 0.22 2.45 0.04 1.44 0.82 0.60 0.39 

KO 3.38 0.01 1.21 0.75 0.71 0.41 1.85 0.03 1.68 0.71 0.52 0.38 

MMM 3.69 0.00 0.84 0.42 0.40 0.21 2.10 0.03 1.39 0.73 0.64 0.39 

PG 3.58 0.01 0.83 0.50 0.39 0.22 1.74 0.03 1.40 0.58 0.40 0.37 

SGP 3.86 0.01 1.00 0.51 0.40 0.30 1.88 0.02 1.13 0.47 0.42 0.30 

TXN 3.34 0.00 0.62 0.42 0.40 0.21 2.13 0.04 1.90 0.70 0.51 0.32 

WAG 3.82 0.00 1.31 0.53 0.42 0.33 1.98 0.02 1.53 0.59 0.51 0.32 

Table 3.7: Establishing the uncertainty of estimated upper and lower tail indexes ˆ b γ n by creating bootstrap samples (bs) of historical return data 

for S&P 500 and included assets and calculation of bootstrap mean value νmean with rel. error from the original value, deviation quantiles and 

standard deviation from b γ mean, and most extreme deviation value. The tail represents the most extreme 4 % of the return values during a time 

interval ranging from January 1991 to December 2000 (k = 100 values, bs = 1000) and from July 1985 to April 2008 (k = 229 values, bs = 650). 

b γ mean

3.2.2 Implementation of the Non-Parametric Approach 

Within this subsection I provide an overwiev of the implementation of the non-parametric 

approach by Sornette & Malevergne. It is denoted non-parametric because of the estimation 

of the constant l by equation (3.36). We don’t give estimates of the scale 

factors of the tail distribution Cε and CY . Instead, we simply consider N sorted realizations 

of return vectors X and Y denoted by X1,N ≥ X2,N ≥ · · · ≥ XN,N and 

Y1,N ≥ Y2,N ≥ · · · ≥N,N and observe that the ratio of the empirical quantiles remains 

remarkably stable for small or large k. 

The coefficients of upper and lower tail dependence were estimated by the following 

equation: 

ˆλ +,− 1 

= � 

max 1, ˆl ˆβ 

where constant l was non-parametrically estimated by: 

ˆl = lim 

u→1− ,0 + 

F −1 

X (u) 

F −1 

Y (u) 

� α 

(3.53) 

Xk,N ∼ = , as k → N, 0 (3.54) 

As we can see already on equation (3.54) and equation (3.47) of Hill’s estimator, 

our non-parametric method is fully compatible with negative tails. Sorting return 

data in increasing order instead of decreasing order, as it is necessary to calculate tail 

dependence in negative tails, yields negative return data in both the index tail and the 

asset’s tail. Therefore the fraction in the definition of Hill’s estimator becomes positive 

and the natural logarithm gives out a real number. Constant ˆ l is accordingly always 

positive and if β < 0 for any asset and the index relation (3.53) cannot be applied 

since it assumes β > 0 and therefore λ = 0. To check whether constant ˆ li are sensitive 

to threshold k, I first plotted: 

ˆ l(k) = Xk,N/Yk,N for k = 1, 2, . . ., 150 for the smaller data set of all lower tales 

of the assets with the index. The 150 most extreme return values of the upper tails 

correspond to k/N = 6%. 

Figure (3.5) shows that ˆ l(k) becomes stable after about k/N = 0.5% and remains 

within narrow limits up to 6%. 

Using the average value of ˆ l(k), ˆ l(k) = 1 

k 

Yk,N 

�k Xj,N 

j=1 Yj,N 

on the one hands constitutes in 

a shift of emphasis to the extreme tail and on the other hand increases the robustness 

of constant ˆ l and ˆ λ to changes of threshold k, which also makes the necessity of finding 

k ∗ , the optimal threshold value, obsolete. Instead it becomes sufficient to determine 

an optimal interval, which in our case is: k = 3% . . .5%. Figure (3.6) shows on the 

example of negative tails, because there the effect is particularly strong, that in the 

area of the most extreme return values of an asset when k/N → 0, ˆ l(k) determined as 

average value over k becomes unstable. This could lead to distortion of our estimats. 

We can avoid this by calculation of ˆlc(k) = 1 �k Xj,N 

k−c+1 j=Y , where we chose an interval 

Yj,N 

ranging from c = [0.05·N] to k with [·] denoting integer numbers. This can be observed 

in figure (3.7). However ˆ l(k) seems to be stable on the interval we’re interested in. 

To summarize for upper tails: Tail indexes α were calculated by Hill’s estimator ˆν 

31

l(k) 

l(k) mean 

18 

16 

14 

12 

10 

8 

6 

4 

2 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 

k/N 

Figure 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 from January 1991 up to 

December 2000. 

12 

10 

8 

6 

4 

2 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 

k/N 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

Figure 3.6: ˆl(k) = 1 �k Xj,N 

k j=1 with k = 1...150 (or k/N = 0% ...6%) for the lower tail of 

Yj,N 

all assets with the index S&P 500 on the smaller data set ranging from January 1991 up to 

December 2000. 

32

l(k) mean,c 

4.5 

4 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 

k/N 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

Figure 3.7: ˆl(k)c = 1 �k Xj,N 

k−c+1 j=Y with k = 13... 150 (or k/N = 0.5% ... 6%), and c = 

Yj,N 

[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 January 1991 up to December 2000. 

and Gabaix’s ordinary least squares log-log rank-size tail index estimate ˆ b γ n 

ˆν = 

� 

1 � 

kj=1 

log 

k 

Yj,N 

Yk,N 

� −1 

given by: 

(3.55) 

log(Rank(Yj,N) − 1/2) = a − ˆ b γ n ∗ log(Yj,N), for j = 1 . . .k, (3.56) 

where Y1,N ≥ Y2,N ≥ · · · ≥ YN,N denote the ordered statistics of the sample containing 

N realizations of the market return vector Y and threshold number k assumed to 

represent the 4% quantile of Y . 

Constant ˆl was non-parametrically estimated by: 

k� 

ˆ 

1 Xj,N 

l(k) = (3.57) 

k 

ˆ lc(k) = 

j=1 

Yj,N 

1 

k − c + 1 

k� 

j=c 

Xj,N 

Yj,N 

(3.58) 

and ˆ β was calculated by ordinary least squares regression to the linear factor model: 

X = β · Y + ε with X denoting asset returns in descending chronologic order, with Y 

denoting corresponding index returns, and with ε denoting idiosyncratic noise. 

Putting all these parameters in equation (3.59) for upper tail dependence: 

ˆλ + 1 

= � 

max 

ˆλ + 1 

= � 

max 

33 

1, ˆ �ˆν l· 

ˆβ 

1, ˆ �ˆγ bn l· 

ˆβ 

(3.59) 

(3.60)

Non-Par Par 

up lo up lo up lo up lo 

ˆν 3.23 3.16 3.23 3.16 3.23 3.16 3.23 3.16 

tail 1... k 1...k Y ...k Y ... k 1... k 1... k Y ...k Y ...k 

ˆλ 

BMY 0.06 0.05 0.06 0.07 0.07 0.01 0.07 0.08 

CVX 0.02 0.02 0.02 0.02 0.02 0.00 0.02 0.02 

HPQ 0.10 0.07 0.10 0.09 0.13 0.02 0.14 0.12 

KO 0.06 0.05 0.06 0.07 0.07 0.01 0.08 0.08 

MMM 0.04 0.03 0.04 0.04 0.04 0.01 0.04 0.04 

PG 0.03 0.02 0.03 0.04 0.03 0.00 0.03 0.04 

SGP 0.04 0.04 0.04 0.05 0.05 0.01 0.05 0.06 

TXN 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 

WAG 0.09 0.06 0.09 0.11 0.11 0.01 0.10 0.12 

Table 3.8: Estimated values of upper and lower tail dependence for S&P 500 index with 

a set of nine major assets traded on the New York stock Exchange calculated by a nonparametric 

approach: ˆ λ +,− � 

= 1/max 1, l 

�ν β on the left and by a parametric approach: 

ˆλ +,− � 

= 1/ 1 + β−ν · Cε 

� 

on the right. The tail represents the most extreme 4% of the 

CY 

return values during a time interval from January 1991 to December 2000. ’tail’ shows the 

calculation interval of l with c = 13 and k = 100 and ∗ denotes negative ˆ β. 

and applying the approach to our reference data sets yields the estimates reported in 

tables (3.8) and (3.10) on the left side using Hill’s estimator ν and reported in tables 

(3.9) and (3.11) on the left side using Gabaix’s estimator b γ n 5 . 


and, for the calculation of Gabaix’s tail index estimator, multiply the ordered sample 

by (-1). ˆ β remains the same for negative tails, as it is calculated for the whole data 

sample within this approach. 

3.2.3 Analysis of TDC estimated by the Non-Parametric Approach 

Within this subsection I provide a sensitivity analysis of TDC estimated by the nonparametric 

approach of Sornette & Malevergne performed on the two reference data 

sets: the robustness of the results to the choice of threshold number k is first analyzed 

visually by plotting ˆ λ in dependence of k. In a second step estimation of error bars by 

bootstrap sampling with replacement of ˆ λ is provided. The aim is to find confidence 

intervals for the estimates and to get an insight into the underlying error distributions. 

Then as a third step variations of ˆ λ in time are studied by rolling time windows. Different 

sample sizes are shifted along the time axis in order to measure time dependence 

of ˆ λ. 

Figure (3.8) shows ˆ λ(k) for k 

k 

= 0% . . .6%, figure (3.9) λ(k) for = 0% . . .6%, 

N N 

and (3.10) shows λc(k) for k 

= 0.5% . . .6% for the lower tail of the smaller data 

N 

sample. Results for the positive tail look more stable. But also for negative tails the 

charts look consistent and stable. Looking at the interval of k = 3% . . .5% that is of 

N 

primary interest to us, the curve of λ is not stiff to moderate changes in k. Using the 

5 The matlab m-file for the implementation of λ +,− estimated by the non-parametric approach according to 

Sornette & Malevergne is denoted by: lambda np sor ma.m and enclosed to the appendix and the data CD 

34

Non-Par Par 


ˆ b γ n 3.71 3.37 3.71 3.37 3.71 3.37 3.71 3.37 


ˆλ 

BMY 0.04 0.04 0.04 0.05 0.05 0.00 0.05 0.07 

CVX 0.01 0.01 0.01 0.02 0.01 0.00 0.01 0.02 

HPQ 0.07 0.06 0.07 0.08 0.10 0.01 0.11 0.11 

KO 0.04 0.04 0.04 0.06 0.05 0.00 0.05 0.07 

MMM 0.02 0.02 0.02 0.03 0.03 0.00 0.03 0.03 

PG 0.02 0.02 0.02 0.03 0.02 0.00 0.02 0.03 

SGP 0.03 0.03 0.03 0.04 0.03 0.00 0.03 0.05 

TXN 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 

WAG 0.06 0.05 0.06 0.10 0.08 0.00 0.08 0.11 




= 1/max 1, l 

� γ 

bn β on the left and by a parametric approach: 

ˆλ +,− � 

= 1/ 1 + β−bγn · Cε 

� 


CY 

return values during a time interval from January 1991 to December 2000. ’tail’ shows the 

calculation interval of l with c = 13 and k = 100 and ∗ denotes negative ˆ β. 

Non-Par Par 


ˆν 3.09 3.13 3.09 3.13 3.09 3.13 3.09 3.13 


ˆλ 

BMY 0.05 0.04 0.05 0.05 0.06 0.02 0.06 0.06 

CVX 0.04 0.04 0.04 0.04 0.05 0.02 0.05 0.05 

HPQ 0.09 0.08 0.09 0.09 0.12 0.07 0.13 0.13 

KO 0.10 0.08 0.10 0.11 0.13 0.02 0.13 0.13 

MMM 0.06 0.05 0.06 0.06 0.07 0.02 0.07 0.08 

PG 0.02 0.02 0.02 0.03 0.03 0.00 0.03 0.03 

SGP 0.03 0.02 0.03 0.03 0.03 0.01 0.03 0.03 

TXN 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

WAG 0.07 0.06 0.07 0.08 0.09 0.02 0.09 0.09 




= 1/max 1, l 

�ν β on the left and by a parametric approach: 

ˆλ +,− � 

= 1/ 1 + β−ν · Cε 

� 


CY 

return values during a time interval from July 1985 to April 2008. ’tail’ shows the calculation 

interval of l with c = 29 and k = 229. 

35

Non-Par Par 


ˆ b γ n 3.56 3.04 3.56 3.04 3.56 3.04 3.56 3.04 


ˆλ 

BMY 0.03 0.05 0.03 0.05 0.04 0.03 0.04 0.07 

CVX 0.03 0.04 0.03 0.04 0.03 0.02 0.03 0.05 

HPQ 0.06 0.09 0.06 0.10 0.09 0.08 0.10 0.14 

KO 0.07 0.09 0.07 0.12 0.10 0.02 0.10 0.14 

MMM 0.04 0.05 0.04 0.06 0.05 0.02 0.05 0.08 

PG 0.01 0.02 0.01 0.03 0.02 0.01 0.02 0.04 

SGP 0.02 0.02 0.02 0.03 0.02 0.01 0.02 0.04 

TXN 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

WAG 0.05 0.07 0.05 0.08 0.06 0.03 0.06 0.10 




= 1/max 1, l 

� γ 

bn β on the left and by a parametric approach: 

ˆλ +,− � 

= 1/ 1 + ˆ β−bγn · Cε 

� 


CY 

return values during a time interval from July 1985 to April 2008. ’tail’ shows the calculation 

interval of l with c = 29 and k = 229. 

uncorrected mean value λ(k), estimates tend to be slightly lower than by choosing the 

corrected estimate λc(k). Nevertheless, I think that neglecting most extreme values as 

we did by corrected values does not have to yield better estimates because they might 

have high impanct on distributions as it is most notable for negative tails. This can be 

observed at the example of figure (3.10). Therefore I will give both estimators λ(k) and 

λc(k) in the result charts. Significant differences between these two estimators generally 

signalise strong impact of outliers, the values that are to a high degree uncommon. 

Estimation of Error Bars 

For the estimation of error bars or in other words: the level of uncertainty specified 

by a particular estimation method, I performed bootstrap sampling with replacement. 

The bootstrap method is a procedure that involves choosing random samples with 

replacement from a data set and analyzing each sample in the same way. Sampling 

with replacement means that every sample is returned to the data set after sampling or 

after being chosen. So a particular data point from the original data set could appear 

multiple times in a particular bootstrap sample. The number of elements in each 

bootstrap sample equals the number of elements in the original data set. The range 

of sample estimates obtained enables you to establish the uncertainty of the quantity 

you’re estimating. 

Considering the linear factor model: X = β·Y +ε, I created bootstrap samples (with 

replacement) of the index data table (i.e. S&P 500) and then picked out the respective 

data points of the assets (according in time to S&P 500 index data). Like that I only 

performed bootstrap sampling of one vector (S&P 500 index data) and searched for 

the corresponding asset data by algorithm. ’bootstrp’ is a predefined function handle 

in Matlab, which allows one to perform sampling by replacement. ˆ β, ˆν, and ˆ l as the 

36

λ(k) 

λ(k) mean 

0.14 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 

k/N 

Figure 3.8: ˆ λ(k) = 

1 

� 

max 1, l 

ˆβ 

� ˆν with k = 1...343 (or k/N = 0% ... 6%), 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. 

0.08 

0.07 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 

k/N 

Figure 3.9: λ(k) = 

1 

� 

max 1, ˆl ˆβ 

� ˆν with k = 1... 343 (or k/N = 0% ... 6%), 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. 

37

λ(k) mean,c 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 

k/N 

Figure 3.10: λ(k) = 

1 

� 

max 1, ˆlĉ β 

� ˆν with k = 29... 343 (or k/N = 0.5% ... 6%) and c = [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. 

mean value of lj = Xj,N 

Yj,N 

for j = 1 . . .k and j = c . . .k with c = 0.005 · N then were 

calculated independently for each sample. Table (3.12) shows results of bootstrap 

sampling with replacement applied to the smaller data set and (3.13) shows results of 

bootstrap sampling with replacement applied to the bigger data sample 6 . The mean 

values of the tail dependence coefficients calculated for the 1000 bootstrap samples were 

close to the ˆ λ of the original sample. 

I calculated relative deviations between average of estimated bootstrap sample tail 

dependence coefficients and original values (λ of original data sample) of maximally 

7%, which can be observed in columns denoted by ’rel err’. The quantiles describe 

the width of the error bar or in other words the extent of uncertainty: a 95% quantile 

for example gives the absolute deviation of 95% of the bootstrap tail dependence 

estimates from their mean value. Taking a look at estimated error bars we determine 

that standard deviations of the difference between tail dependence coefficient estimates 

calculated for the bs = 1000 bootstrap samples most of times are approximately half 

of the respective 95% quantiles. This is an indicator for the error distributions to be 

asymptitically Gaussian. Furthermore with respect to relative sizes of the quantiles 

compared to absolute values of estimators we can detect a certain pattern. Looking at 

the smaller sample bootstrap estimates, provided in table (3.12), we observe that most 

of the 95% quantiles calculated in the positive tails tend towards 70% to 80% of the 

corresponding total values of tail dependence coefficient estimates and for negative tails 

95% quantiles are about the size of the total value of corresponding tail dependence 

6 The matlab m-file for the bootstrap sampling with replacement of λ +,− estimated by the non-parametric 

approach according to Sornette & Malevergne is denoted by: bootstr nonpar alld.m and enclosed to the data 

CD 

38

coefficient estimates. For the bigger data sample quantiles are significantly smaller. 

95% quantiles here are mostly about 50% to 60% of corresponding total value of tail 

dependence coefficient estimates in the positive tails and slightly higher in the negative 

tails. This shows us that either the only way to improve our λ estimates by the given 

approach is to aggregate more data, or that error bars simply depend on the chosen 

time window and evolve according an ARCH process (autoregressive conditional heteroscedasticity) 

considering the variance of the current error term to be a function of 

the variances of the previous time period’s error terms. 

39

40 

m=2507 bs=1000 


λbs,mean 

λbs,mean 

λ k=100 rel err max dev 95% q 90% q std bs k=100 rel err max dev 95% q 90% q std bs 

BMY 0.06 0.01 0.10 0.04 0.03 0.02 0.04 0.07 0.09 0.04 0.03 0.02 

CVX 0.02 0.00 0.04 0.02 0.01 0.01 0.02 0.01 0.05 0.02 0.02 0.01 

HPQ 0.10 0.00 0.20 0.07 0.06 0.04 0.07 0.05 0.10 0.06 0.05 0.03 

KO 0.06 0.01 0.10 0.04 0.04 0.02 0.05 0.02 0.20 0.06 0.05 0.03 

MMM 0.04 0.02 0.10 0.03 0.03 0.02 0.03 0.03 0.07 0.03 0.02 0.02 

PG 0.03 0.03 0.10 0.03 0.02 0.01 0.02 0.03 0.08 0.03 0.02 0.02 

SGP 0.04 0.01 0.10 0.03 0.02 0.02 0.03 0.05 0.08 0.03 0.03 0.02 

TXN 0.00* Inf 0.00 0.00 0.00 0.00 0.00* Inf 0.00 0.00 0.00 0.00 

WAG 0.09 0.02 0.20 0.08 0.06 0.04 0.06 0.03 0.20 0.06 0.05 0.03 

λc 

BMY 0.06 0.01 0.08 0.04 0.03 0.02 0.06 0.04 0.10 0.04 0.04 0.02 

CVX 0.02 0.00 0.04 0.02 0.01 0.01 0.02 0.03 0.04 0.02 0.02 0.01 

HPQ 0.10 0.00 0.20 0.07 0.06 0.04 0.09 0.03 0.20 0.07 0.05 0.04 

KO 0.06 0.01 0.08 0.04 0.04 0.02 0.07 0.03 0.10 0.06 0.05 0.03 

MMM 0.04 0.02 0.09 0.03 0.03 0.02 0.04 0.02 0.08 0.03 0.03 0.02 

PG 0.03 0.04 0.07 0.03 0.02 0.01 0.04 0.00 0.10 0.04 0.03 0.02 

SGP 0.04 0.01 0.08 0.03 0.02 0.02 0.05 0.03 0.10 0.04 0.03 0.02 

TXN 0.00* Inf 0.00 0.00 0.00 0.00 0.00* Inf 0.00 0.00 0.00 0.00 

WAG 0.09 0.02 0.20 0.08 0.06 0.04 0.11 0.03 0.20 0.08 0.07 0.04 

Table 3.12: Establishing the uncertainty of non-parametrically estimated upper and lower tail dependence coefficients ˆ λ by creating 1000 bootstrap 


deviations of the results. The tails represent the most extreme 4% (excluding highest 0.5% for λc) of the return values during a time interval from 

January 1991 to December 2000. ˆ βj have been calculated on the whole samples. ∗ denotes negative ˆ β and therefore ˆ λ = 0 and ’Inf’ denotes ·/0.

41 

m=5736 bs=1000 


λbs,mean 

λbs,mean 


BMY 0.05 0.01 0.06 0.03 0.02 0.01 0.04 0.02 0.08 0.03 0.02 0.01 

CVX 0.04 0.01 0.05 0.02 0.02 0.01 0.04 0.05 0.08 0.03 0.03 0.02 

HPQ 0.09 0.00 0.07 0.04 0.03 0.02 0.09 0.02 0.12 0.04 0.04 0.02 

KO 0.10 0.02 0.10 0.05 0.04 0.03 0.09 0.05 0.15 0.07 0.06 0.04 

MMM 0.07 0.03 0.07 0.04 0.03 0.02 0.05 0.06 0.08 0.03 0.03 0.02 

PG 0.03 0.20 0.08 0.03 0.02 0.02 0.02 0.10 0.06 0.02 0.02 0.01 

SGP 0.03 0.10 0.05 0.03 0.02 0.02 0.02 0.10 0.06 0.02 0.02 0.01 

TXN 0.00 0.20 0.00 0.00 0.00 0.00 0.00 0.40 0.01 0.00 0.00 0.00 

WAG 0.07 0.00 0.07 0.04 0.03 0.02 0.06 0.03 0.08 0.03 0.03 0.02 

λc 

BMY 0.05 0.01 0.06 0.03 0.02 0.01 0.05 0.03 0.07 0.03 0.02 0.01 

CVX 0.04 0.01 0.05 0.02 0.02 0.01 0.04 0.05 0.07 0.03 0.02 0.02 

HPQ 0.09 0.00 0.07 0.04 0.03 0.02 0.10 0.03 0.11 0.04 0.03 0.02 

KO 0.10 0.01 0.12 0.05 0.04 0.03 0.11 0.02 0.13 0.07 0.06 0.04 

MMM 0.07 0.03 0.07 0.04 0.03 0.02 0.06 0.06 0.07 0.03 0.03 0.02 

PG 0.03 0.20 0.08 0.03 0.02 0.01 0.03 0.20 0.07 0.03 0.03 0.02 

SGP 0.03 0.10 0.05 0.03 0.02 0.01 0.03 0.10 0.08 0.03 0.02 0.01 

TXN 0.00 0.20 0.00 0.00 0.00 0.00 0.00 0.30 0.01 0.00 0.00 0.00 

WAG 0.07 0.00 0.07 0.04 0.03 0.02 0.08 0.03 0.08 0.04 0.03 0.02 



deviations of the results. The tails represent the most extreme 4% (excluding highest 0.5% for λc) of the return values during a time interval from 

July 1985 to April 2008. ˆ βj have been calculated on the whole samples.

Observation of λ by Rolling Time Window 

As mentioned, it is the aim of the determination of tail dependence in terms of portfolio 

analysis to find assets with low tail dependence coefficients in order to diversify 

away extreme risks. Looking at tables (3.8), (3.10), (3.9), and (3.11) asset Texas 

Instruments Inc. is maybe the most interesting case. Depending on the sample size, ˆ β 

becomes either zero, which yields zero tail dependence by default criteria or diminutive, 

which yields vanishing tail dependence. The sample size reflects the time window of the 

estimate. It would be interesting to determine, how tail dependence estimates evolve 

with time. The hypothesis is that λ is not fixed but perhaps is evolving according to an 

ARCH process. Therefore I used a rolling time window to observe the time dependence 

of the estimate λ. The size of the time window should on the one hand be sufficiently 

large to allow for good precision according to the bootstrap procedure and should on 

the other hand be sufficiently small to reflect the possible change of values as a function 

of time. This had to be tested on different window sizes while I had to keep an 

eye on the individual parameters tail index α, β, and l to control whether the window 

size or generally the size of the sample allows for appropriate determination. I first 

plotted ˆ β, ˆ l, ˆν in dependence of N for three window sizes: S = 2500, S = 1600, and 

S = 800 shown in figures (3.11),. . .,(3.19). X-axis N has the dimension of observation 

days (daily observations) starting from N = 1 representing July 1 st 1985 and ending 

by N = 5736 representing Mars 31 st 2008 

First of all it is important to take notice of the different plot intervals in N given 

the different window sizes. When I chose S = 2500, I have to wait for the first 2500 

daily observations to take place, which literally corresponds to a time interval of 2500 

days from the first observation. Choosing S = 800 and a step size of one between 

successive measurements, indicating that the time window moves one data point to 

the right hand side per step (or chronologically one day), means that I already have 

2500 − 800 = 1700 measurements at observation day N = 2500, which is the starting 

point of measurements by choosing the biggest window size. That’s why at first sight 

the plots look kind of different. 

Looking at figures (3.11), (3.12), and (3.13) for the three window sizes S, we observe 

that the behavior of ˆ β looks consistent over time. The plots for different window sizes 

only differ in detailedness of the mapping of changes. The smaller we go, the bigger the 

peaks become. The same counts for constant ˆ l, which is shown in figures (3.14), (3.15), 

and (3.16) for the different window sizes. Sample peaks in ˆ l sometimes are smoothed 

out extensively when we use bigger window sizes S. This makes ˆ l look very stable over 

time when we plot it by choosing window size S = 2500. As furthermore observable for 

most of the assets, ˆ β, after having reached a peak, seems to have a declining tendency 

over time. The only asset that is completely incompatible with the declining trend 

of ˆ β is Texas Instruments Inc.(TXN). It reflects on the others by showing a countertrend. 

Asset TXN indicates a overall counter-movement with the index S&P 500 in the 

early phases indicated by negative ˆ β and recently a co-movement with the index, as ˆ β 

becomes positive or even close to one. The trend shown by the other assets would be 

the opposite of this: decreasing overall co movement of assets and index with increasing 

time. 

Looking at figures (3.17), (3.18), and (3.19) of ˆν for the three window sizes, estimates 

oscillate strongly but withing stable intervals depending on the window size. We 

know that Hill’s estimator performs better with bigger data sets. Therefore it is suggestive 

to choose a window size, such that upper and lower bounds of these oscillations 

remain small enough. This is the case for samples with S >1000. 

42

β 

β 

β 

1 

0.9 

0.8 

0.7 

0.6 

BMY 

0.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1.4 

1.2 

1 

0.8 

0.6 

KO 

0.4 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1 

0.9 

0.8 

0.7 

0.6 

SGP 

0.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

β 

β 

β 

0.8 

0.7 

0.6 

0.5 

0.4 

CVX 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.9 

0.8 

0.7 

0.6 

MMM 

0.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.8 

0.6 

0.4 

0.2 

0 

TXN 

−0.2 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

β 

β 

β 

1.4 

1.3 

1.2 

1.1 

HPQ 

1 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1 

0.8 

0.6 

0.4 

PG 

0.2 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1.4 

1.2 

1 

0.8 

0.6 

WAG 

0.4 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 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. 

β 

β 

β 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

BMY 

0.4 

1000 2000 3000 4000 5000 6000 

N 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

1000 2000 3000 4000 5000 6000 

N 

1.2 

1 

0.8 

0.6 

0.4 

KO 

SGP 

0.2 

1000 2000 3000 4000 5000 6000 

N 

β 

β 

β 

0.8 

0.7 

0.6 

0.5 

0.4 

CVX 

1000 2000 3000 4000 5000 6000 

N 

0.9 

0.8 

0.7 

0.6 

0.5 

MMM 

0.4 

1000 2000 3000 4000 5000 6000 

N 

1.5 

1 

0.5 

0 

TXN 

−0.5 

1000 2000 3000 4000 5000 6000 

N 

β 

β 

β 

1.6 

1.4 

1.2 

1 

HPQ 

0.8 

1000 2000 3000 4000 5000 6000 

N 

1 

0.8 

0.6 

0.4 

PG 

0.2 

1000 2000 3000 4000 5000 6000 

N 

1.4 

1.2 

1 

0.8 

0.6 

WAG 

0.4 

1000 2000 3000 4000 5000 6000 

N 





43

β 

β 

β 

1.4 

1.2 

1 

0.8 

0.6 

BMY 

0.4 

0 1000 2000 3000 

N 

4000 5000 6000 

1.2 

1 

0.8 

0.6 

0.4 

KO 

0.2 

0 1000 2000 3000 

N 

4000 5000 6000 

1.5 

1 

0.5 

SGP 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

β 

β 

β 

1 

0.8 

0.6 

0.4 

0.2 

CVX 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

MMM 

0.4 

0 1000 2000 3000 

N 

4000 5000 6000 

1.5 

1 

0.5 

0 

TXN 

−0.5 

0 1000 2000 3000 

N 

4000 5000 6000 

β 

β 

β 

2 

1.5 

1 

HPQ 

0.5 

0 1000 2000 3000 

N 

4000 5000 6000 

1 

0.8 

0.6 

0.4 

0.2 

PG 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

1.5 

1 

0.5 

WAG 

0 

0 1000 2000 3000 

N 

4000 5000 6000 





l 

l 

l 

2.4 

2.2 

2 

1.8 

BMY 

1.6 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1.9 

1.8 

1.7 

1.6 

1.5 

1.4 

1.3 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2.8 

2.6 

2.4 

2.2 

2 

KO 

SGP 

1.8 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

l 

l 

l 

1.9 

1.8 

1.7 

1.6 

1.5 

1.4 

1.3 

CVX 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1.8 

1.7 

1.6 

1.5 

1.4 

MMM 

1.3 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

4 

3.5 

3 

TXN 

2.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

l 

l 

l 

2.8 

2.6 

2.4 

2.2 

HPQ 

2 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2.2 

2 

1.8 

1.6 

1.4 

PG 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2.6 

2.4 

2.2 

2 

1.8 

WAG 

1.6 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 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: 

ˆ 1 �k l(k) = k j=1 Xj,S/Yj,S with k = 0.04 · S, Xj,S and Yj,S 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

l 

l 

l 

2.6 

2.4 

2.2 

2 

1.8 

BMY 

1.6 

1000 2000 3000 4000 5000 6000 

N 

3 

2.5 

2 

1.5 

KO 

1 

1000 2000 3000 4000 5000 6000 

N 

3 

2.8 

2.6 

2.4 

2.2 

2 

SGP 

1.8 

1000 2000 3000 4000 5000 6000 

N 

l 

l 

l 

2.5 

2 

1.5 

CVX 

1 

1000 2000 3000 4000 5000 6000 

N 

2.5 

2 

1.5 

MMM 

1 

1000 2000 3000 4000 5000 6000 

N 

4.5 

4 

3.5 

3 

2.5 

TXN 

2 

1000 2000 3000 4000 5000 6000 

N 

l 

l 

l 

3.5 

3 

2.5 

2 

HPQ 

1.5 

1000 2000 3000 4000 5000 6000 

N 

2.5 

2 

1.5 

1 

PG 

0.5 

1000 2000 3000 4000 5000 6000 

N 

3 

2.5 

2 

1.5 

WAG 

1 

1000 2000 3000 4000 5000 6000 

N 

Figure 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 k j=1 Xj,S/Yj,S with 

k = 0.04 ·S, Xj,S and Yj,S rank 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. 

l 

l 

l 

3 

2.5 

2 

1.5 

BMY 

1 

0 1000 2000 3000 

N 

4000 5000 6000 

3 

2.5 

2 

1.5 

1 

0.5 

0 1000 2000 3000 

N 

4000 5000 6000 

4 

3.5 

3 

2.5 

2 

KO 

SGP 

1.5 

0 1000 2000 3000 

N 

4000 5000 6000 

l 

l 

3.5 

3 

2.5 

2 

1.5 

1 

CVX 

0.5 

0 1000 2000 3000 

N 

4000 5000 6000 

3.5 

3 

2.5 

2 

1.5 

1 

MMM 

0.5 

0 1000 2000 3000 

N 

4000 5000 6000 

l 

6 

5 

4 

3 

2 

TXN 

1 

0 1000 2000 3000 

N 

4000 5000 6000 

l 

l 

l 

4.5 

4 

3.5 

3 

2.5 

2 

HPQ 

1.5 

0 1000 2000 3000 

N 

4000 5000 6000 

3.5 

3 

2.5 

2 

1.5 

1 

PG 

0.5 

0 1000 2000 3000 

N 

4000 5000 6000 

3.5 

3 

2.5 

2 

1.5 

WAG 

1 

0 1000 2000 3000 

N 

4000 5000 6000 

Figure 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 k j=1 Xj,S/Yj,S with 

k = 0.04 ·S, Xj,S and Yj,S rank 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) 


45

ν S&P 500 

4 

3.5 

3 

2.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 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 ˆν = kj=1 log Yj,S 

�−1 with k = 

0.04 · S and Xj,S and Yj,S rank ordered return observations for rolling time horizon windows 



ν S&P 500 

5 

4.5 

4 

3.5 

3 

2.5 

2 

1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 

N 


� 


�−1 with k = 




46 

� 

1 

k 

� 

1 

k 

Yk,N 

Yk,N 

ν U 

ν L 

ν U 

ν L

ν S&P 500 

8 

7 

6 

5 

4 

3 

2 

1 

0 1000 2000 3000 

N 

4000 5000 6000 


� 


�−1 with k = 


of S = 800 considered data points from N = (1... S),(2... S + 1),... ,(5736 − S + 1...5736) 


Figures (3.20), (3.21), and (3.22) show plots of ˆ λU and ˆ λL for the different assets 

and for the three different window sizes. What most of these plots have in common is 

that upper tail dependence estimates ˆ λU show an increasingly constant behavior over 

time and lower tail dependence estimates ˆ λL overall show a decreasing tendency, which 

is more distinctive for window sizes with S >1000. Figures (3.23) and (3.24) show tail 

dependence estimated for rolling windows of the three different sizes altogether in the 

same figure and for the same range. It can be observed that tail dependence estimates 

for the different window sizes sometimes seem inconsistent with each other. It is not 

only because the plots with wider window sizes look smoothed, but also that some 

peaks, which are distinctive in red (S = 800) don’t even appear in blue (S = 1600) or 

black (S = 2500). This counts first of all for more recent estimates. The three assets 

Chevron Corp.(CVX), 3M Co.(MMM), and Texas Instruments Inc.(TXN) show a peak 

in the upper tail dependence at the end of the measurement period N >5000. The only 

asset that is very much contradictory with overall trends is again TXN, what makes it 

particularly interesting. 

Although it is difficult to draw conclusions because of the inconsistency of the estimates 

obtained by the different window sizes, something is clearly apparent in all of 

the plots: Whereas lower tail dependence ˆ λL in the beginning used to dominate over 

upper tail dependence ˆ λU, by some point typically around N = 4000 ˆ λU becomes bigger 

than ˆ λL. This is very interesting because it would mean that the tendency of extreme 

positive co-movement between the assets and index S&P 500 has become stronger than 

the tendency of extreme negative co-movements or in other words: extraordinary high 

gains of assets and index S&P 500 are more likely to happen simultaneously than it is 

the case that high losses of assets and index S&P 500 occur at the same time. 

47 

� 

1 

k 

Yk,N 

ν U 

ν L

48 

λ 

λ 

λ 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

BMY 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.4 

0.3 

0.2 

0.1 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.08 

0.06 

0.04 

0.02 

KO 

SGP 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.08 

0.06 

0.04 

0.02 

CVX 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

MMM 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

x 10−3 

8 

6 

4 

2 

TXN 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

HPQ 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

PG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.2 

0.15 

0.1 

0.05 

WAG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 3.20: λU(N) for the upper tails of index S&P 500 and the nine assets plotted in blue and λL(N) for the lower tails plotted in red using 

non-parametric approach given by equation ˆ λ +,− � 

= 1/max 1, l 

�ˆν β with coefficients ˆν, l and β 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.

49 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

BMY 

0 

1000 2000 3000 4000 5000 6000 

N 

0.4 

0.3 

0.2 

0.1 

0 

1000 2000 3000 4000 5000 6000 

N 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

KO 

SGP 

0 

1000 2000 3000 4000 5000 6000 

N 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

CVX 

0 

1000 2000 3000 4000 5000 6000 

N 

0.2 

0.15 

0.1 

0.05 

MMM 

0 

1000 2000 3000 4000 5000 6000 

N 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

TXN 

0 

1000 2000 3000 4000 5000 6000 

N 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

HPQ 

0 

1000 2000 3000 4000 5000 6000 

N 

0.2 

0.15 

0.1 

0.05 

PG 

0 

1000 2000 3000 4000 5000 6000 

N 

0.25 

0.2 

0.15 

0.1 

0.05 

WAG 

0 

1000 2000 3000 4000 5000 6000 

N 



= 1/max 1, l 



50 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

BMY 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

0.8 

0.6 

0.4 

0.2 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

0.2 

0.15 

0.1 

0.05 

KO 

SGP 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

λ 

λ 

λ 

0.25 

0.2 

0.15 

0.1 

0.05 

CVX 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

0.25 

0.2 

0.15 

0.1 

0.05 

MMM 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

0.1 

0.08 

0.06 

0.04 

0.02 

TXN 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

λ 

λ 

λ 

0.4 

0.3 

0.2 

0.1 

HPQ 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

0.25 

0.2 

0.15 

0.1 

0.05 

PG 

0 

0 1000 2000 3000 

N 

4000 5000 6000 

0.4 

0.3 

0.2 

0.1 

WAG 

0 

0 1000 2000 3000 

N 

4000 5000 6000 



= 1/max 1, l 



51 

λ 

λ 

λ 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

BMY 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.25 

0.2 

0.15 

0.1 

0.05 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

KO 

SGP 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.25 

0.2 

0.15 

0.1 

0.05 

CVX 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.2 

0.15 

0.1 

0.05 

MMM 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.08 

0.06 

0.04 

0.02 

TXN 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

HPQ 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

PG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.4 

0.3 

0.2 

0.1 

WAG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 3.23: λU(N) for the upper tails of index S&P 500 and the nine assets for rolling time horizon windows of size S = 2500 plotted in black, 

S = 1600 plotted in blue, and S = 800 plotted in red using non-parametric approach given by equation ˆ λ +,− = 1/max 

ˆν, l and β from N = (2501 − S ... S),(2502... S + 1),... ,(5736 − S + 1... 5736). 

� 

1, l 

β 

� ˆν 

with coefficients

52 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

BMY 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.4 

0.3 

0.2 

0.1 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.2 

0.15 

0.1 

0.05 

KO 

SGP 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.25 

0.2 

0.15 

0.1 

0.05 

CVX 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

MMM 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.1 

0.08 

0.06 

0.04 

0.02 

TXN 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

HPQ 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.25 

0.2 

0.15 

0.1 

0.05 

PG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.4 

0.3 

0.2 

0.1 

WAG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 3.24: λL(N) for the lower tails of index S&P 500 and the nine assets for rolling time horizon windows of size S = 2500 plotted in black, 

S = 1600 plotted in blue, and S = 800 plotted in red using non-parametric approach given by equation ˆ λ +,− = 1/max 

ˆν, l and β from N = (2501 − S ... S),(2502... S + 1),... ,(5736 − S + 1... 5736). 

� 

1, l 

β 

� ˆν 

with coefficients

Error Bars in Dependence of Time 

To establish uncertainty of estimated upper and lower tail dependence I performed 

bootstrap sampling over the whole area: N = (1 . . .S), (2 . . .(S + 1)), . . .,(5736 − S + 

1 . . .5736) for the nine assets with index S&P 500 and all of the 3 window sizes 7 . 

Time order of the results is given top down. That means that the latest estimate is 

on top representing Mars 31 st , 2008. For the biggest window size of S = 2500 daily 

observations I calculated bs = 800 bootstrap sample per window and for each of the 

two smaller ones S = 1600 and S = 800 I calculated bs = 1000 bootstrap samples. 

As step size between the successive windows I choose five days. That means that the 

window moves five daily observations to the right per step considering the time axis 

(or five days into the future). This provides the window with five new observations per 

step, meanwhile it ignores the five oldest ones. As it can be observed in the Excel files, 

this makes [(5736 −2500)/5] = 647 windows for the biggest window size S = 2500 data 

points, 827 windows for samples of window size S = 1600, and 987 windows for the 

smallest window size S = 800. 

In terms of quantile estimates on the bootstrap samples in order to estimate error 

bars, bootstrap estimates of the biggest window size S = 2500 should theoretically yield 

similar results as obtained by bootstrap sampling of our smaller reference data set that 

ranges from January 1991 to December 2000 and consists of 2506 daily observations. 

In fact quantiles and standard deviations of the bootstrap estimates are in very good 

accordance with these further results given in table (3.12), whereas for upper tails on 

average 95% quantiles between 70% and 80% of the corresponding total values of tail 

dependence coefficient estimates were calculated and for lower tails 95% quantiles were 

about the size of the total values (100% of corresponding tail dependence coefficient 

estimates). Choosing a window size of S = 2500 yields average 95% quantiles of 70% 

up to 85% of the tail dependence estimates bs L np k up m (notation explained below) 

for positive tails and around 80% to 150% of corresponding bs L np k lo m for negative 

tails. Choosing corrected ˆ l, quantiles calculated for negative tails become lower. 

Here again the assumption of error distributions to be approximately Gaussian seems to 

hold well, as calculated standard deviations within and between the different bootstrap 

samples are very close of being equal to half of 95% quantiles. This counts for all 

window sizes. Looking at the smallest window size of S = 800 daily observations, 

95% quantiles on average tend to be 150% to 200% of corresponding tail dependence 

coefficient estimates for positive tails and up to 250% for negative tails. This means 

that standard deviations are around the size of the estimates, which is after my opinion 

too high. In between we have window size S = 1600 with average 95% quantiles of 

100% to 140% of corresponding tail dependence estimates for positive tails and 120% 

to 180% of corresponding tail dependence estimates for negative tails. This reveals 

that sample sizes shouldn’t be smaller than S = 1600 to achieve at least more or less 

accurate estimates. 

In the following list explanatory notes on the results of bootstrap sampling for the rolling 

window tail dependence estimations attached in Excel files provided at the example of 

non-parametric tail dependence estimates for upper tails and uncorrected ˆ l: 

bs L np k up m: mean value of the bootstrap tail dependence estimators ˆ λU by nonparametric 

approach calculated for each individual window consisting of bs samples 

and all of the nine assets given from left to right in the following order: 

BMY, CVX, HPQ, KO, MMM, PG, SGP, TXN, WAG. ’k’ means that ˆ l were 

7 Results are on the data CD submitted with the thesis in the folder: window results 

53

estimated for the whole threshold k given by X1,N ≥ X2,N ≥ · · · ≥ Xk,N and 

Y1,N ≥ Y2,N ≥ · · · ≥ Yk,N with k = 0.04 · N and . . .y . . . means that ˆ l were 

estimated for the corrected threshold from Xc,N ≥ Xc+1,N ≥ · · · ≥ Xk,N and 

Yc,N ≥ Yc+1,N ≥ · · · ≥ Yk,N with k = 0.04 · N and c = [0.05 · k] where [·] denotes 

integer numbers. 

rel 99k up: value of 99% quantile divided by bs L np k up m shows the magnitude 

of the quantile relative to the absolute average estimate. To obtain the absolute 

value of the quantile one simply multiplies the quantile by bs L np k up m. 







For 95% quantiles I additionally calculated mean values over time and bootstrap deviations 

over time (shown below data sets on Excel sheets). 

Overall the deviation estimates of the rolling time windows look consistent along the 

time axis. But first of all with respect to negative tails there are certain fluctuations 

in the quantile estimates in time. Therefore the time series were checked for serial 

correlation. This is shown at the example of a return series of index S&P 500 for an 

interval from January 1950 to April 2008 with a total of N = 14654 data points plotted 

over time in figure (3.25). It seems that large and small changes are clustered together. 

To confirm this, autocorrelation proposed by Box & Jenkins [16] (1994) for different 

lags is plotted in figure (3.26) for returns and squared returns 8 . There seems to be no 

autocorrelation in the return series itself but the squarred returns exhibit significant 

autocorrelation at least up to lag 5. Since the squared returns measure the second 

order moment of the original time series, this result indicates that the variance of S&P 

500 conditional on its past history may change over time, or that the time series may 

exhibit time varying conditional heteroskedasticity or volatility clustering. The serial 

correlation in squared returns, or conditional heteroskedasticity, can be expressed using 

an autoregressive process for squared residuals i.e. by the autoregressive conditional 

heteroscedasticity ARCH model of Engle [17] (1982) usually referred to as the ARCH(p) 

model given by: 

Yt = c + εt 

σ 2 t = α0 + 

p� 

αj · ε 

(3.61) 

2 t−j where α0 > 0, αj ≥ 0 (3.62) 

j=1 

Yt denote a stationary time series such as financial returns and is expressed by its mean 

c and iid Gaussian white noise εt with mean zero. p denotes the length of ARCH lags 

(number of time steps considered). Since εt has zero mean, Vart−1(ε) = Et−1 (ε 2 ) = σ 2 t , 

the above equation can be rewritten as: 

rk = 

ε 2 t = α0 + 

p� 

j=1 

αj · ε 2 t−j 

+ ut 

(3.63) 

8 Autocorrelation denotes the correlation of a variable with itself over successive time intervals defined by: 

� N −k 

i=1 (Yi−Y )·(Yi+k−Y ) 

� 

Ni=1(Y 

i−Y ) 2 with successive measurements Y1, . . . , YN and lag k 

54

Return 

0.1 

0.05 

0 

−0.05 

−0.1 

−0.15 

−0.2 

S&P 500 Daily Returns 

−0.25 

Jan 1950 Jan 1969 Jan 1988 Apr 2008 

Figure 3.25: Return data for index S&P 500 ranging from January 1950 to April 2008 

where ut = ε2 t − Et−q (ε2 t ) is a zero mean white noise process. 

To check the hypothesis that λ evolves according an ARCH process I applied Engle’s 

test [17] (1982) for the presence of ARCH effects to my historical day-to-day return 

time series. ’archtest’ is a predefined function handle in Matlab, which allows one to 

test for autocorrelation of univariate time series. Since an ARCH model can be written 

as an AR (autoregressive) model in terms of squared residuals as in equation (3.63), 

a Lagrange Multiplier (LM) test for ARCH effects can be constructed based on the 

auxiliary regression (3.63). Under the null hypothesis that there are no ARCH effects: 

α1 = α2 = . . . = αp = 0, the test statistic follows a χ2 distribution with p degrees of 

freedom: 

LM = T · R 2 ∼ χ 2 (p), (3.64) 

where T is the sample size and R 2 is computed from the regression (3.63) using estimated 

residuals 9 . The alternative hypothesis is that, in the presence of ARCH 

components, at least one of the estimated α coefficients must be significantly different 

from zero. Generally spoken if (T · R 2 ) is greater than the χ 2 table value, we reject the 

null hypothesis and conclude there are ARCH effects. If (T · R 2 ) is smaller than the 

χ 2 value, we accept the null hypothesis assuming there are no ARCH effects. These 

explainations are drawn on Ziwot & Wang [18] (2007). They based their explainations 

on S-Plus, which is a statistics software similar to Matlab. 

Test statistics at the example of index S&P 500 for an interval from January 1950 

to April 2008 with a total of N = 14654 is given in table (3.14). The results show 

significant evidence in support of GARCH effects for all the three lags. 

p: size of lags 

9 We refer to Engle [17] (1982) for details 

55

Sample Autocorrelation 

0.8 

0.6 

0.4 

0.2 

0 

ACF with Bounds for Raw Return Series 

−0.2 

0 5 10 

Lag 

15 20 

Sample Autocorrelation 

0.8 

0.6 

0.4 

0.2 

0 

ACF of the Squared Returns 

−0.2 

0 5 10 

Lag 

15 20 

Figure 3.26: Autocorrelation for a return series on the left and for a squared return series on 

the right of index S&P 500 ranging from January 1950 to April 2008 

H: Boolean decision variable: 0 indicates acceptance of null hypothesis that no ARCH 

effects exist; i.e. there is no heteroskedasticity at the corresponding length of lags. 

1 indicates rejection of the null hypothesis. 

P-Value: P-values (significance levels) at which the ARCH test rejects the null hypothesis 

of no existing ARCH effects 

ARCH Test Stat: ARCH test statistics for each number of lags equal to (T · R 2 ) with 

T denoting the sample size and R 2 computed from the regression (3.63) using 

estimated residuals 

Critical Value: critical values of the χ 2 distribution for comparison with the corresponding 

ARCH test statistics 

p H P-Value ARCH Test Stat. Critical Value 

10 1 0 805 18 

15 1 0 815 25 

20 1 0 824 31 

Table 3.14: ARCH statistics applied to daily S&P 500 index return data for an interval 

ranging from January 1950 to April 2008 and for lags of q = 10, 15, and 20 

As we can see on the example of index S&P 500 for the chosen time interval, the p-value 

is zero for all of the three lags, which is smaller than the conventional 5% rejection level, 

so we reject the null hypothesis that there are no ARCH effects. 

56

3.2.4 Implementation of the Parametric Approach 

Now I come to the Implementation of the parametric approach by Sornette & Malevergne. 

This approach assumes that in the extreme tails the survival distributions 

functions of asset returns X and the corresponding residues ε, assumed as idiosyncratic 

noise independent of index returns Y and given by the linear factor model: 

X = β · Y + ε, can be approximated by power-laws given by: Pr{X > x} ∼ C · x −α , 

which constitutes the parametrization. 

Considering N sorted realizations of return vector Y and residual vector ε with 

Y1,N ≥ Y2,N ≥ · · · ≥ YN,N and ε1,N ≥ ε2,N ≥ · · · ≥ εN,N, the coefficient of tail 

dependence is given as function of the ratio of the scale factors: 

as: 

ˆλ = 

CY 

ˆ = k 

N · (Yk,N) α , as k → N, 0 (3.65) 

Ĉε = k 

N · (εk,N) α , as k → N, 0 (3.66) 

1 

1 + ˆ β −α · Ĉε 

Ĉy 

= 

1 + 

1 

� ˆεk,N 

ˆβ·Yk,N 

� α, for k → N, 0 (3.67) 

where ε and β are estimated in a least squares sense. If β < 0 for any asset relation, 

relation (3.67) cannot be applied, since it assumes β > 0 and therefore ˆ λ = 0. 

We can check whether the scale factors Ĉ can be consistently estimated given a 

certain threshold k by plotting ĈY and Ĉε for the nine assets in dependence of k using 

Hill’s estimator ˆν and Gabaix’s estimator ˆ b γ n. Figure (3.27) shows ĈY (k) and Ĉε(k) for 

k=1, 2,..., 458 (this constitutes the interval k/N = 0% . . .8% of the bigger data set) 

calculated by Hill’s estimator, and figure (3.28) shows ĈY (k) and Ĉε(k) for k=1, 2,..., 

458 calculated by Gabaix’s estimator, both for the upper tails of the bigger data set. 

Both plots show robust behaviour in the range of 3 to 5%. 

To provide a coefficient that is independent of the choice of the tail coefficient, I also 

plotted: 

� Ĉε(k) 

ĈY (k) 

� 1/α 

= εk,N 

. Figure (3.29) shows 

Yk,N 

� Ĉε(k) 

ĈY (k) 

� 1/α 

for k=1, 2,..., 458 also 

for the upper tails of the bigger data set. All these plots show that asset TXN (Texas 

Instruments Inc.) seems to show somewhat a different behavior than the other assets 

and the index. 

Also for the parametric approach, in order to put emphasis on the extreme tails and 

in order to increase robustness of ĈY and Ĉε and hence of tail dependence coefficients 

ˆλ to changes of threshold k, I used the average values of ĈY , ĈY = 1 

dependence of k and the average value of Ĉε, Ĉε = 1 

k 

k 

� k 

j=1 Ĉy,j in 

� k 

j=1 Ĉε,j in dependence of k. 

Figure (3.30) shows on the example of the positive tails of the bigger sample that 

� �1/α Ĉε 

ĈY 

for the nine assets is stable in k for k 

N 

= 3% . . .5%. 

Figure (3.28) shows that for small k/N values of ĈY and Ĉε could be comparatively 

� �1/α Ĉε(k) 

large, which also has a certain, even though weaker effect on . To avoid 

ĈY (k) 

distortion of our results it might be better to exclude the most extreme values (k → 

0, N) in order to estimate the mean of the scale factors of the power law distributions. 

Therefore we calculate ĈY,c = 1 �k k−c+1 j=c ĈY,j and Ĉε,c = 1 �k k−c+1 j=c Ĉε,j with c = 

[0.05 · k] where [·] denotes integer numbers. 

57

C Y , C ε 

C Y , C ε 

x 10−6 

6 

5 

4 

3 

2 

1 

S&P 500 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 

k/N 

Figure 3.27: Scale factors ĈY (k) = k 

N · (Yk,N) ˆν of the index S&P 500 and Ĉε(k) = k 

N · (εk,N) 

ˆν 

of the nine assets’ residues plotted for the upper tails of the bigger data set ranging from July 

1985 to April 2008 with k=1, 2,. . ., 458 (k/N = 0% ... 8%). 

1.8 

1.6 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

x 10−6 

2 

S&P 500 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 

k/N 

Figure 3.28: Scale factors ĈY (k) = k 

N 

·(Yk,N) ˆ 

b γ n of the index S&P 500 and Ĉε(k) = k 

N 

·(εk,N) ˆ 

b γ n 

of the nine assets’ residues plotted for the upper tails of the bigger data set ranging from July 

1985 to April 2008 with k=1, 2,. . ., 458 (k/N = 0% ... 8%). 

58

(C ε /C Y ) 1/α 

(C ε,mean /C Y,mean ) 1/α 

3.5 

3 

2.5 

2 

1.5 

1 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 

Figure 3.29: Fraction of scale factors 

� Ĉε(k) 

ĈY (k) 

k/N 

�1/α = εk,N 

Yk,N 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

of the nine assets’ residues and the 

index S&P 500, plotted for the upper tails of the bigger data set ranging from July 1985 to 

April 2008 with k=1, 2,. .., 458 (k/N = 0% ... 8%). 

3.5 

3 

2.5 

2 

1.5 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

1 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 

k/N 

�1/α Figure 3.30: Fraction of scale factors 

� Ĉε 

ĈY 

of the nine assets’ residues and the index S&P 

500, plotted for the upper tails of the bigger data set ranging from July 1985 to April 2008 

with k=1, 2,. .., 458 (k/N = 0% ...8%). 

59

In general it is an advantage to follow a parametric approach, if the assumed parametric 

form of the distributions is not too far from the true one. We can check this 

visually by comparing the empirical complementary cumulative distribution functions 

F i,empirical = k 

for k/N = 0% . . .4% with the parametric complementary cumulative 

N 

distribution functions F i,parametric = C· · x−α of the tails. Figure (3.31) shows the two 

functions in dependence of return values for the upper tails of the bigger sample of εi 

for all assets using Hill’s estimator and Ĉε for (k/N = 4%) and figure (3.32) using 

Ĉε,c the mean value of Ĉε(k) for k/N = 0.5% . . .4%. Figure (3.33) and figure (3.34) 

show the same functions using Gabaix’s estimator. Results are consistent and show 

that by building the mean values of constants C the parametrization becomes more 

precise independent of the tail index estimator we chose. Anyway, it is difficult to say 

in general, which tail index estimator works better. Therefore it makes sense to still 

calculate both for comparison. 

60

F par ,F emp 

F par ,F emp 

61 

F par ,F emp 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

BMY 

0 

0.03 0.04 0.05 0.06 

returns 

KO 

0 

0.025 0.03 0.035 

returns 

0.04 0.045 

0 

SGP 

0.04 0.05 

returns 

0.06 0.07 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

CVX 

0 

0.025 0.03 0.035 

returns 

0.04 0.045 

MMM 

0 

0.025 0.03 0.035 0.04 0.045 

returns 

0 

TXN 

0.06 0.07 0.08 0.09 

returns 

0.1 0.11 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.03 

0.02 

0.01 

0 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

HPQ 

0.04 0.05 0.06 

returns 

0.07 0.08 

PG 

0 

0.025 0.03 0.035 0.04 0.045 

returns 

WAG 

0 

0.03 0.035 0.04 0.045 0.05 0.055 

returns 

Figure 3.31: Empirical complementary cumulative distribution functions F ε,empirical = k 

N plotted in red and parametric complementary cumulative 

distribution functions F ε,parametric = Ĉε · ε−ˆν with Ĉε for (k/N = 4%) plotted in black for k/N = 4% ...0% (from left to right) in dependence of 

returns for the upper tails of the nine assets.

F par ,F emp 

F par ,F emp 

62 

F par ,F emp 

0.04 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

0.04 

0.03 

0.02 

0.01 

BMY 

0 

0.03 0.04 0.05 0.06 

returns 

KO 

0 

0.025 0.03 0.035 

returns 

0.04 0.045 

0 

SGP 

0.04 0.05 

returns 

0.06 0.07 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.03 

0.02 

0.01 

0.04 

0.03 

0.02 

0.01 

0.04 

0.03 

0.02 

0.01 

CVX 

0 

0.025 0.03 0.035 

returns 

0.04 0.045 

MMM 

0 

0.025 0.03 0.035 0.04 0.045 

returns 

0 

TXN 

0.06 0.07 0.08 0.09 

returns 

0.1 0.11 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.04 

0.03 

0.02 

0.01 

0 

0.04 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

HPQ 

0.04 0.05 0.06 

returns 

0.07 0.08 

PG 

0 

0.025 0.03 0.035 0.04 0.045 

returns 

WAG 

0 

0.03 0.035 0.04 0.045 0.05 0.055 

returns 

Figure 3.32: Empirical complementary cumulative distribution functions F ε,empirical = k 


distribution functions F ε,parametric = Ĉε,c · ε−ˆν with Ĉε,c = 1 �k k−c j=c Ĉε for c = 0.005 · N plotted in black for k/N = 4% ...0% (from left to right) 

in dependence of returns for the upper tails of the nine assets.

F par ,F emp 

F par ,F emp 

63 

F par ,F emp 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

BMY 

0 

0.03 0.04 0.05 0.06 

returns 

KO 

0 

0.025 0.03 0.035 

returns 

0.04 0.045 

0 

SGP 

0.04 0.05 

returns 

0.06 0.07 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

CVX 

0 

0.025 0.03 0.035 

returns 

0.04 0.045 

MMM 

0 

0.025 0.03 0.035 0.04 0.045 

returns 

0 

TXN 

0.06 0.07 0.08 0.09 

returns 

0.1 0.11 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.03 

0.02 

0.01 

0 

0.03 

0.02 

0.01 

0.03 

0.02 

0.01 

HPQ 

0.04 0.05 0.06 

returns 

0.07 0.08 

PG 

0 

0.025 0.03 0.035 0.04 0.045 

returns 

WAG 

0 

0.03 0.035 0.04 0.045 0.05 0.055 

returns 

Figure 3.33: Empirical complementary cumulative distribution functions F εi,empirical = k 

N plotted in green and parametric complementary cumulative 

distribution functions F ε,parametric = Ĉε ·ε−ˆb γ n with Ĉε for (k/N = 4%) plotted in black for the upper tails of k/N = 4% ... 0% in dependence 

of returns for the nine assets.

F par ,F emp 

F par ,F emp 

64 

F par ,F emp 

0.04 

0.03 

0.02 

0.01 

0.04 

0.03 

0.02 

0.01 

0.04 

0.02 

BMY 

0 

0.03 0.04 0.05 0.06 

returns 

KO 

0 

0.025 0.03 0.035 

returns 

0.04 0.045 

0 

SGP 

0.04 0.05 

returns 

0.06 0.07 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.04 

0.03 

0.02 

0.01 

0.04 

0.03 

0.02 

0.01 

0.04 

0.03 

0.02 

0.01 

CVX 

0 

0.025 0.03 0.035 

returns 

0.04 0.045 

MMM 

0 

0.025 0.03 0.035 0.04 0.045 

returns 

0 

TXN 

0.06 0.07 0.08 0.09 

returns 

0.1 0.11 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.04 

0.02 

0 

0.04 

0.03 

0.02 

0.01 

0.04 

0.03 

0.02 

0.01 

HPQ 

0.04 0.05 0.06 

returns 

0.07 0.08 

PG 

0 

0.025 0.03 0.035 0.04 0.045 

returns 

WAG 

0 

0.03 0.035 0.04 0.045 0.05 0.055 

returns 


N plotted in green and parametric complementary cumulative 

distribution functions F ε,parametric = Ĉε,c ·ε−ˆb γ n with Ĉε,c = 1 �k k−c j=c Ĉε for c = 0.005 ·N plotted in black for k/N = 4% ... 0% in dependence 

of returns for the upper tails of the nine assets.

To summarize for upper tails: tail indexes α were calculated by Hill’s estimator ˆν 

and Gabaix’s OLS log-log rank-size tail index estimate ˆb γ n given by: 

ˆν = 

� 

1 � 

kj=1 

log 

k 

Yj,N 

�−1 Yk,N 

(3.68) 

log(Rank(Yj,N) − 1/2) = a − ˆ b γ n · log(Yj,N), for j = 1 . . .k (3.69) 

where Y1,N ≥ Y2,N ≥ · · · ≥ YN,N denote ordered statistics of the sample consisting of 

N realizations of the market return vector Y and threshold k is assumed to represent 

the 4% quantile of Y . 

β and idiosyncratic noise ε were calculated by ordinary least squares regression to 

the linear factor model: X = β · Y + ε with X denoting asset returns in descending 

chronologic order, and with Y denoting corresponding index returns. 

Constants ĈY and Ĉε were parametrically estimated by: 

and 

ĈY = 1 

k 

ĈY,c = 

Ĉε = 1 

k 

Ĉε,c = 

k� 

j=1 

j 

N 

1 

k − c + 1 

k� 

j=1 

j 

N 

1 

k − c + 1 

· (Yj,N) α 

k� 

j=c 

j 

N 

· (εj,N) α 

k� 

j=c 

j 

N 

· (Yj,N) α 

· (εj,N) α 

(3.70) 

(3.71) 

(3.72) 

(3.73) 

considering N sorted realizations of index return vector Y and residual vector ε denoted 

by Y1,N ≥ Y2,N ≥ · · · ≥ YN,N and ε1,N ≥ ε2,N ≥ · · · ≥ εN,N. 

Putting all these parameters in equation (3.74) for upper tail dependence: 

ˆλ + = 

ˆλ + = 

1 

1 + ˆ β−ˆν · Ĉε 

ĈY 

1 

1 + ˆ β −ˆ b γ n · Ĉε 

ĈY 

(3.74) 

(3.75) 

and applying the approach to our reference data sets yields the estimates reported in 

tables (3.8) and (3.10) on the left side using Hill’s estimator ν and reported in tables 

(3.9) and (3.11) on the left side using Gabaix’s estimator b γ n 10 . 


and multiply all of the ordered samples by (-1). β and ε remains the same for negative 

tails, as it is calculated for the whole data sample within this approach. 

10 The matlab m-file for the implementation of λ +,− estimated by the parametric approach according to 

Sornette & Malevergne is denoted by: lambda par sor ma.m and enclosed to the appendix and the data CD 

65

λ(k) 

0.16 

0.14 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 

k/N 

Figure 3.35: ˆ λ = 

1 

1+ ˆ β −ˆν · Cε 

Cy 

with k = 1... 458 (or k/N = 0% ...8%), applied to the lower tails 

of all assets with the index S&P 500 on the bigger data set ranging from July 1985 to April 

2008. 

3.2.5 Analysis of TDC estimated by the Parametric Approach 

Also for the parametric approach by Sornette & Malevergne I provide a sensitivity 

analysis of TDC estimates performed on the two reference data sets: the robustness 

of the results to the choice of threshold k is first analyzed visually by plotting ˆ λ in 

dependence of k. In a second step estimation of error bars of ˆ λ is provided by bootstrap 

sampling with replacement. The aim is to find confidence intervals for the estimates 

and to get an insight into the underlying error distributions. 

Figure (3.35) shows ˆ λ(k) for k 

k 

= 0% . . .8%, figure (3.36) λ(k) for = 0% . . .8%, 

N N 

and figure (3.37) shows λc(k) for k = 0.5% . . .8% for the lower tail of the smaller data 

N 

sample. Results of estimators for positive tails look less fluctuating because in positive 

tails outliers are less extreme. For the parametric approach also counts that on the 

interval: k = 3% . . .5%, that is of primary interest to us, the curve of λ is not stiff to 

N 

moderate changes in k. Using the uncorrected mean value λ(k), estimates tend to be 

slightly lower than by choosing the corrected estimate λc(k), which means neglecting 

most extreme outliers. 

The results of the bootstrap sampling are shown in table (3.15) for the smaller sample 

and in table (3.16) for the bigger sample 11 . bs denotes the number of samples, which 

has been set to thousand for both sample sizes and N shows the number of daily observations 

of the samples. The procedure is exactly the same as for the non-parametric 

approach. Especially conspicuous is the column of relative errors in the uncorrected 

negative tail calculated for the smaller sample and located in the upper right parts of 

table (3.15). ’rel err’ are calculated as follows: |λoriginal − 1/bs · �bs j=1 λj|/λoriginal. An 

explanation for this could be that extreme outliers, as already discussed at the example 

of the non-parametric approach, have an even higher stake in terms of distorting tail 

11 The matlab m-file for the bootstrap sampling with replacement of λ +,− estimated by the parametric 

approach according to Sornette & Malevergne is denoted by: bootstr par alld.m and enclosed to the data CD 

66

λ(k) mean 

λ(k) mean,c 

0.1 

0.09 

0.08 

0.07 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 

k/N 

Figure 3.36: λ = 

1 

1+ ˆ β −ˆν · Ĉε 

Ĉy 

with k = 1... 458 (or k/N = 0% ... 8%), applied to the lower tails 

of all assets with the index S&P 500 on the bigger data set ranging from July 1985 to April 

2008. 

0.14 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

BMY 

CVX 

HPQ 

KO 

MMM 

PG 

SGP 

TXN 

WAG 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 

k/N 

Figure 3.37: λ = 

1 

1+ ˆ β −ˆν · Ĉε,c 

Ĉy,c 

with k = 29... 458 (or k/N = 0.5% ... 8%), and c = [0.005 · N] 

([·] denotes integer numbers) applied to the lower tails of all assets with the index S&P 500 

on the bigger data set ranging from July 1985 to April 2008. 

67

dependence estimates in the parametric approach. If we correct for these most extreme 

outliers results become accurate. To have a clear advantage by using a parametric 

approximation, a certain amount of data is needed to determine the coefficients with 

satisfying accuracy. This amount might be higher than for a non-parametric approach. 

Looking at the relative errors of the bigger sample we see that relative errors are 

much lower compared to the smaller sample. We can check how the parametrization 

works for the lower tail of the smaller sample by comparing the empirical complementary 

cumulative distribution functions F i,empirical = k 

for k/N = 0% . . .4% with the 

N 

parametric complementary cumulative distribution functions F i,parametric = C· ∗ x−α of the tails. Figure (3.38) shows the two functions in dependence of absolute return 

values for the lower tails of the smaller sample of εi for all assets using Hill’s estimator 

and Ĉε the mean value of Ĉε(k) for k/N = 0% . . .4% and figure (3.39) using Hill’s 

estimator and Ĉε,c the mean value of Ĉε(k) for k/N = 0.5% . . .4%. 

Comparing the two plots shows that the extraordinary high outliers in the negative 

tails of our data samples in connection with a sample size that seems to be less 

than the critical or necessary size indeed seems to distort the approximation of the 

complementary cumulative distribution function in the region of k/N ≈ 4%. Looking 

at figure (3.39) we can detect that in the domain of k/N → 0 in most cases 

F εi,parametric = Ĉε,c · ε −ˆν seems to overestimate F εi,empirical = k/N. To see this more 

clearly we can plot the complementary cumulative distribution functions on a logarithmic 

scale to the base ten. Figures (3.40) and (3.41) show the semi-log plots 

corresponding to figures (3.38) and (3.39). As we can observe on the semi-log plots 

the uncorrected parametric complementary cumulative distribution functions perform 

better in the area k/N → 0 and the corrected parametric complementary cumulative 

distribution functions perform better in the area k/N → 4%. This is intuitively clear 

because if we do not correct for the most extreme outliers we shift the emphasis of 

the parametrization towards these most extreme outliers. This shows the limitations 

of our parametrization approach but anyway only poses a problem in the negative tails 

because of the extraordinary outliers we find there. Because of the fact that both approaches 

have a domain where they dominate, it still makes sense to calculate both of 

them first of all for negative tails. 

Anyway, for the non-parametric approach sample sizes should be chosen around the 

size of the bigger reference data sample that consists of 5736 data points to achieve 

satisfying accuracy. In spite of relative errors the magnitude of quantile and standard 

deviation estimates relative to the total tail dependence estimates is comparable to 

estimates by the non-parametric approach. 

Because we need very high sample sizes here to obtain trustworthy results and because 

it has been shown that for bigger sample sizes there is a loss of significant peaks, I 

think that the non-parametric approach is better to analyze tail dependence estimates 

in dependence of time. Therefore I don’t go into observations of λ by rolling time windows 

within this approach. Error bars are of similar extent as for the non-parametric 

approach but the procedure seems to request more data to provide accurate results. 

68

69 

m=2507 bs=1000 


λbs,mean 

λbs,mean 


BMY 0.07 0.01 0.12 0.04 0.04 0.02 0.02 0.90 0.10 0.06 0.04 0.03 

CVX 0.02 0.01 0.04 0.02 0.02 0.01 0.01 1.60 0.10 0.03 0.02 0.01 

HPQ 0.13 0.02 0.20 0.08 0.07 0.04 0.03 0.50 0.20 0.08 0.05 0.04 

KO 0.07 0.01 0.10 0.06 0.05 0.03 0.02 2.00 0.20 0.09 0.07 0.04 

MMM 0.04 0.01 0.09 0.03 0.03 0.02 0.02 0.90 0.09 0.04 0.03 0.02 

PG 0.03 0.02 0.05 0.03 0.02 0.02 0.01 1.40 0.09 0.02 0.01 0.01 

SGP 0.05 0.01 0.07 0.04 0.03 0.02 0.01 0.90 0.10 0.04 0.02 0.02 

TXN 0.00 Inf 0.00 0.00 0.00 0.00 0.00* Inf 0.00 0.00 0.00 0.00 

WAG 0.11 0.00 0.15 0.08 0.06 0.04 0.01 0.90 0.20 0.05 0.02 0.02 

λc 

BMY 0.07 0.01 0.20 0.04 0.04 0.02 0.08 0.03 0.13 0.05 0.05 0.03 

CVX 0.02 0.01 0.06 0.02 0.02 0.01 0.02 0.01 0.08 0.03 0.02 0.01 

HPQ 0.13 0.02 0.20 0.08 0.07 0.04 0.12 0.04 0.19 0.07 0.06 0.04 

KO 0.08 0.02 0.10 0.05 0.05 0.03 0.08 0.02 0.13 0.06 0.05 0.03 

MMM 0.04 0.01 0.08 0.03 0.03 0.02 0.04 0.00 0.10 0.04 0.03 0.02 

PG 0.03 0.02 0.09 0.03 0.02 0.02 0.04 0.02 0.10 0.04 0.03 0.02 

SGP 0.05 0.01 0.09 0.04 0.03 0.02 0.06 0.02 0.09 0.05 0.04 0.02 

TXN 0.00 Inf 0.01 0.00 0.00 0.00 0.00* inf 0.02 0.00 0.00 0.00 

WAG 0.10 0.00 0.10 0.07 0.06 0.04 0.12 0.02 0.20 0.08 0.07 0.04 

Table 3.15: Establishing the uncertainty of parametrically estimated upper and lower tail dependence coefficients ˆ λ 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 λc) of the return values during a time 

interval from January 1991 to December 2000. ˆ βj have been calculated on the whole samples. ∗ denotes negative ˆ β and therefore ˆ λ = 0 and ’Inf’ 

denotes ·/0.

70 

m=5736 bs=1000 


λbs,mean 

λbs,mean 


BMY 0.06 0.01 0.06 0.03 0.03 0.02 0.03 0.40 0.08 0.04 0.03 0.02 

CVX 0.05 0.00 0.05 0.03 0.02 0.01 0.02 0.50 0.09 0.04 0.03 0.02 

HPQ 0.12 0.00 0.08 0.05 0.04 0.03 0.08 0.10 0.10 0.07 0.06 0.04 

KO 0.13 0.02 0.10 0.06 0.05 0.03 0.03 1.10 0.30 0.09 0.05 0.04 

MMM 0.08 0.02 0.10 0.04 0.04 0.02 0.03 0.80 0.20 0.06 0.04 0.03 

PG 0.03 0.10 0.06 0.03 0.03 0.02 0.01 1.00 0.07 0.02 0.01 0.01 

SGP 0.03 0.10 0.06 0.03 0.03 0.02 0.01 0.60 0.08 0.02 0.01 0.01 

TXN 0.00 0.30 0.01 0.00 0.00 0.00 0.00 0.80 0.01 0.00 0.00 0.00 

WAG 0.09 0.09 0.07 0.04 0.04 0.02 0.03 0.50 0.08 0.03 0.03 0.02 

λc 

BMY 0.06 0.01 0.05 0.03 0.03 0.02 0.07 0.04 0.06 0.03 0.03 0.02 

CVX 0.05 0.00 0.05 0.03 0.02 0.01 0.05 0.05 0.07 0.04 0.03 0.02 

HPQ 0.13 0.00 0.08 0.05 0.04 0.03 0.13 0.02 0.09 0.05 0.04 0.03 

KO 0.13 0.01 0.10 0.06 0.05 0.03 0.14 0.02 0.10 0.07 0.06 0.04 

MMM 0.08 0.02 0.09 0.04 0.04 0.02 0.08 0.05 0.10 0.04 0.04 0.02 

PG 0.03 0.10 0.06 0.03 0.03 0.02 0.04 0.10 0.09 0.04 0.03 0.02 

SGP 0.03 0.10 0.06 0.03 0.03 0.02 0.04 0.10 0.06 0.03 0.03 0.02 

TXN 0.00 0.20 0.01 0.00 0.00 0.00 0.00 0.40 0.01 0.00 0.00 0.00 

WAG 0.09 0.01 0.07 0.04 0.03 0.02 0.10 0.02 0.08 0.04 0.04 0.02 

Table 3.16: Establishing the uncertainty of of parametrically estimated upper and lower tail dependence coefficients ˆ λ by creating 1000 bootstrap 



interval from July 1985 to April 2008. ˆ βj have been calculated on the whole samples.

F par ,F emp 

F par ,F emp 

71 

F par ,F emp 

0.4 

0.3 

0.2 

0.1 

BMY 

0 

0 0.2 0.4 

returns 

0.6 0.8 

0.8 

0.6 

0.4 

0.2 

0 

0 0.2 0.4 

returns 

0.6 0.8 

0.8 

0.6 

0.4 

0.2 

KO 

SGP 

0 

0 0.2 0.4 

returns 

0.6 0.8 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.2 

0.15 

0.1 

0.05 

CVX 

0 

0 0.2 0.4 

returns 

0.6 0.8 

0.4 

0.3 

0.2 

0.1 

MMM 

0 

0 0.2 0.4 

returns 

0.6 0.8 

0.2 

0.15 

0.1 

0.05 

TXN 

0 

0 0.2 0.4 

returns 

0.6 0.8 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.4 

0.3 

0.2 

0.1 

HPQ 

0 

0 0.2 0.4 

returns 

0.6 0.8 

0.8 

0.6 

0.4 

0.2 

PG 

0 

0 0.2 0.4 

returns 

0.6 0.8 

1 

0.5 

WAG 

0 

0 0.2 0.4 

returns 

0.6 0.8 



distribution functions F ε,parametric = Ĉε · ε−ˆν with Ĉε = 1 �k k j=1 Ĉε plotted in black for k/N = 4% ... 0% (from left to right) in dependence of 

returns for the lower tails of the nine assets for a time interval ranging from January 1991 to December 2000.

F par ,F emp 

F par ,F emp 

72 

F par ,F emp 

0.06 

0.04 

0.02 

0.06 

0.04 

0.02 

0.06 

0.04 

0.02 

BMY 

0 

0 0.2 0.4 

returns 

0.6 0.8 

KO 

0 

0 0.2 0.4 

returns 

0.6 0.8 

SGP 

0 

0 0.2 0.4 

returns 

0.6 0.8 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.04 

0.03 

0.02 

0.01 

0.06 

0.04 

0.02 

0.06 

0.04 

0.02 

CVX 

0 

0 0.2 0.4 

returns 

0.6 0.8 

MMM 

0 

0 0.2 0.4 

returns 

0.6 0.8 

TXN 

0 

0 0.2 0.4 

returns 

0.6 0.8 

F par ,F emp 

F par ,F emp 

F par ,F emp 

0.06 

0.04 

0.02 

0.06 

0.04 

0.02 

0.06 

0.04 

0.02 

HPQ 

0 

0 0.2 0.4 

returns 

0.6 0.8 

PG 

0 

0 0.2 0.4 

returns 

0.6 0.8 

WAG 

0 

0 0.2 0.4 

returns 

0.6 0.8 



distribution functions F εi,parametric = Ĉε, c ·ε−ˆν with Ĉε,c = 1 �k k−c j=c Ĉε for c = 0.005 ·N plotted in black for k/N = 4% ... 0% (from left to right) 

in dependence of returns for the lower tails of the nine assets for a time interval ranging from January 1991 to December 2000.

F par ,F emp 

F par ,F emp 

73 

F par ,F emp 

10 0 

BMY 

10 

0 0.2 0.4 0.6 0.8 

−5 

returns 

10 0 

KO 

10 

0 0.2 0.4 0.6 0.8 

−5 

returns 

10 0 

SGP 

10 

0 0.2 0.4 0.6 0.8 

−5 

returns 

F par ,F emp 

F par ,F emp 

F par ,F emp 

10 0 

10 −2 

10 −4 

CVX 

10 

0 0.2 0.4 0.6 0.8 

−6 

returns 

10 0 

10 −5 

MMM 

10 

0 0.2 0.4 0.6 0.8 

−10 

returns 

10 0 

TXN 

10 

0 0.2 0.4 0.6 0.8 

−5 

returns 

F par ,F emp 

F par ,F emp 

F par ,F emp 

10 0 

HPQ 

10 

0 0.2 0.4 0.6 0.8 

−5 

returns 

10 0 

PG 

10 

0 0.2 0.4 0.6 0.8 

−5 

returns 

10 0 

WAG 

10 

0 0.2 0.4 0.6 0.8 

−5 

returns 



distribution functions F ε,parametric = Ĉε · ε−ˆν with Ĉε = 1 �k k j=1 Ĉε plotted in black for k/N = 4% ... 0% in dependence of returns for the lower 

tails of the nine assets for a time interval ranging from January 1991 to December 2000.

F par ,F emp 

F par ,F emp 

74 

F par ,F emp 

10 0 

10 −2 

10 −4 

BMY 

10 

0 0.2 0.4 0.6 0.8 

−6 

returns 

10 0 

10 −5 

KO 

10 

0 0.2 0.4 0.6 0.8 

−10 

returns 

10 0 

10 −5 

SGP 

10 

0 0.2 0.4 0.6 0.8 

−10 

returns 

F par ,F emp 

F par ,F emp 

F par ,F emp 

10 0 

10 −5 

CVX 

10 

0 0.2 0.4 0.6 0.8 

−10 

returns 

10 0 

10 −5 

MMM 

10 

0 0.2 0.4 0.6 0.8 

−10 

returns 

10 0 

TXN 

10 

0 0.2 0.4 0.6 0.8 

−5 

returns 

F par ,F emp 

F par ,F emp 

F par ,F emp 

10 0 

10 −2 

10 −4 

HPQ 

10 

0 0.2 0.4 0.6 0.8 

−6 

returns 

10 0 

10 −5 

PG 

10 

0 0.2 0.4 0.6 0.8 

−10 

returns 

10 0 

10 −5 

WAG 

10 

0 0.2 0.4 0.6 0.8 

−10 

returns 



distribution functions F εi,parametric = Ĉε, c · ε−ˆν with Ĉε,c = 1 �k k−c j=c Ĉε for c = 0.005 · N plotted in black for k/N = 4% ...0% in dependence of 

returns for the lower tails of the nine assets for a time interval ranging from January 1991 to December 2000.

3.3 β-Smile Improvement 

As an addition to my Sornette & Malevergne approaches, where the regression coefficient 

β of the components of vector Y on its corresponding components of vector X 

was determined by the least squares method applying a linear additive single factor 

model to the whole return data samples of a given time interval given by: 

X = β · Y + ε (3.76) 

with ε denoting the vector of error terms, a simple improvement was proposed. Considering 

that the linear factor model given by equation (3.76) only holds for components 

X and Y large enough on a certain condition, typically when both X and corresponding 

Y belong to the extreme tails of their probability density distributions, which by 

assumption denotes the most extreme 4 % of the return data samples. ˆ βSI is then 

calculated only for extreme tail data in dependence of threshold value k. There are 

basically four possible conditions for the calculation of ˆ βSI of the upper tails 12 : 

1 st condition : Y ≥ Y (k) ∩ X ≥ X(k) 

2 nd condition : Y ≥ Y (k) 

3 rd condition : X ≥ X(k) 

4 th condition : Y ≥ Y (k) ∪ X ≥ X(k) 

The first condition is to calculate β by only considering return values, where X and Y 

belong to the most extreme 4% of the sample, the second and the third conditions are 

to calculate β by either considering Y large enough or X large enough. As a fourth 

condition I wanted to use the Boolean ’or’ (∪) as an extension to the first condition given 

by Boolean ’and’ (∩). Table (3.17) shows relative deviations between tail dependence 

coefficient estimates calculated using β of the whole data sample and tail dependence 

coefficient estimates that were calculated using ˆ βSI applying the first, second, or third 

condition. The fourth condition yields β < 0 and therefor zero tail dependence for all 

the nine assets. 

In terms of implementation, we just have to build vectors of data that fulfill the 

chosen condition and then calculate ˆ βSI (smile improvement) with this data. The 

other coefficients are calculated in the same way as described in section (3.4.2), where 

the implementation of the non-parametric approach was discussed. 

What we can observe in table (3.17) is that the first condition: Y ≥ Y (k)∩X ≥ X(k) 

seems to fit best or at least yields values that are closer to my classic β approach than 

the two other conditions. Anyway for some assets the second condition: Y ≥ Y (k) 

yields estimates, which are more accurate than estimated by the first condition in 

terms of being consistent with the classic β approach. Therefor I decided to further 

analyze the first and the second condition. Table (3.18) shows results of ˆ βSI calculated 

by first and second SI condition in comparison to results of ˆ β calculated by all data and 

results of ˆ λ +,− calculated by the first and second SI conditions in comparison to results 

of ˆ λ +,− with ˆ β calculated by all data for the smaller data set and table (3.19) shows 

the same for the bigger data set 13 . Tail indexes were calculated by Hill’s estimator ˆν. 

12 I will describe the procedure only on the example of upper tails. For ˆ βSI of the lower tails the same 

relation hold using absolute values of return data. 

13 The matlab m-file for the implementation of λ +,− estimated by the non-parametric approach according 

to Sornette & Malevergne with ˆ β calculated by SI conditions is denoted by: Lambda np SI.m and enclosed to 

the data CD 

75

condition {X ≥ X(k)} ∩ X ≥ X(k) Y ≥ Y (k) {X ≥ X(k)} ∩ X ≥ X(k) Y ≥ Y (k) 

{Y ≥ Y (k)} {Y ≥ Y (k)} 


m=2507 

ν 3.23 3.23 3.23 3.16 3.16 3.16 

λ k=100 k=100 k=100 k=100 k=100 k=100 

BMY 0.27 0.96 0.19 0.92 1.00* 2.78 

CVX 1.00* 1.00* 0.93 0.73 1.00* 2.06 

HPQ 1.00 1.00* 0.33 1.00* 1.00* 0.10 

KO 0.86 0.99 0.68 0.21 1.00* 1.36 

MMM 1.00* 1.00* 0.91 0.20 0.97 7.76 

PG 9.11 0.95 0.96 0.35 1.00* 2.38 

SGP 0.26 0.93 0.32 0.97 1.00* 1.91 

TXN Inf Inf NaN* NaN* NaN* NaN* 

WAG 0.32 0.96 0.79 1.00* 0.85 0.33 

m=5736 

ν 3.09 3.09 3.09 3.13 3.13 3.13 

λ k=229 k=229 k=229 k=229 k=229 k=229 

BMY 1.26 0.99 1.00* 0.88 1.00 0.54 

CVX 1.17 1.00* 1.00* 0.24 1.00* 0.82 

HPQ 0.68 1.00 0.81 0.90 1.00 0.60 

KO 8.78 0.79 0.09 1.93 1.00* 3.35 

MMM 0.06 0.98 1.00 1.00 1.00* 0.77 

PG 1.74 1.00* 1.00* 19.58 1.00* 1.00* 

SGP 0.93 1.00* 1.00* 0.27 1.00* 1.00* 

TXN 93.63 0.70 1.00* 0.96 1.00* 1.00* 

WAG 0.39 1.00* 0.81 0.94 0.91 0.69 

Table 3.17: Relative deviations between coefficients of upper and lower tail dependence estimated 

by the non-parametric approach of Sornette & Malevergne using ˆ β calculated for the 

whole sample and ˆ βSI calculated for the tail according to three different conditions for index 

S&P 500 and the nine assets. The tail represents the most extreme 4 % of data during a 

smaller time interval ranging from January 1991 to December 2000 (k = 100) and during a 

bigger time interval ranging from July 1985 to Mars 2008 (k = 229). * denotes zero values of 

tail dependence calculated by ˆ βSI, ’NaN’ denotes 0/0, and ’inf’ denotes ·/0. 

76

77 

β β + 

SI β − 

SI λ + λ + 

SI λ + 

SI λ− λ − 

SI λ − 

ν 

condition all data 

3.23 3.23 3.23 3.16 3.16 

SI 

3.16 

� X ≥ X(k) � ∩ � X ≥ X(k) � ∩ 1... k Y ≥ Y (k) � X ≥ X(k) � ∩ 1... k Y ≥ Y (k) � X ≥ X(k) � m=2507 

� � 

Y ≥ Y (k) 

� � 

Y ≥ Y (k) 

� � 

Y ≥ Y (k) 

� 

∩ 

� 

Y ≥ Y (k) 

BMY 0.82 0.74 0.37 0.06 0.07 0.04 0.05 0.18 0.00 

CVX 0.45 0.02 0.30 0.02 0.00 0.00 0.02 0.06 0.01 

HPQ 1.10 0.22 -0.16 0.10 0.13 0.00 0.07 0.06 0* 

KO 0.69 0.38 0.64 0.06 0.02 0.01 0.05 0.12 0.04 

MMM 0.55 -0.18 0.51 0.04 0.00 0.00* 0.03 0.26 0.02 

PG 0.58 1.20 0.50 0.03 0.00 0.30 0.02 0.07 0.01 

SGP 0.85 0.78 0.29 0.04 0.03 0.03 0.04 0.10 0.00 

TXN -0.09 5.10 -0.65 0.00* 0.00* 1.00 0.00* 0.00 0.00* 

WAG 

condition 

0.97 

all data 

1.10 

Y ≥ Y (k) 

-0.35 

Y ≥ Y (k) 

0.09 

c... k 

0.16 

Y ≥ Y (k) 

0.12 0.06 0.04 0* 

� X ≥ X(k) � ∩ c... k Y ≥ Y (k) � X ≥ X(k) � m=2507 

� � 

Y ≥ Y (k) 

� 

∩ 

� 

Y ≥ Y (k) 

BMY 0.82 0.87 1.20 0.06 0.07 0.04 0.07 0.25 0.01 

CVX 0.45 0.20 0.64 0.02 0.00 0.00 0.02 0.07 0.01 

HPQ 1.10 1.30 1.10 0.10 0.13 0.00 0.09 0.09 0.00* 

KO 0.69 0.48 0.91 0.06 0.02 0.01 0.07 0.17 0.06 

MMM 0.55 0.26 1.10 0.04 0.00 0.00* 0.04 0.31 0.03 

PG 0.58 0.22 0.85 0.03 0.00 0.30 0.04 0.13 0.03 

SGP 0.85 0.76 1.20 0.04 0.03 0.03 0.05 0.15 0.00 

TXN -0.09 -1.40 -0.10 0.00* 0.00* 1.00 0.00* 0.00 0.00* 

WAG 0.97 1.20 0.86 0.09 0.16 0.12 0.11 0.07 0.00* 

Table 3.18: Results for ˆ βSI using the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k) and the second SI condition: Y ≥ Y (k) for upper and lower tails 

calculated by least squares method applied on linear additive single factor model: X = β ·Y +ε and for resulting upper and lower tail dependence 

coefficients ˆ λ + and ˆ λ− with corrected ˆl (c... k) and uncorrected ˆl (1... k). The tail represents the most extreme 4% of data during a time interval 

ranging from January 1991 to December 2000 (k = 100, c = 13). * denotes ˆ β < 0 → λ = 0.

78 

β β + 

SI β − 

SI λ + λ + 

SI λ + 

SI λ− λ − 

SI λ − 

ν 

condition all data 

3.09 3.09 3.09 3.13 3.13 

SI 

3.13 

� X ≥ X(k) � ∩ � X ≥ X(k) � ∩ 1... k Y ≥ Y (k) � X ≥ X(k) � ∩ 1... k Y ≥ Y (k) � X ≥ X(k) � m=5736 

� � 

Y ≥ Y (k) 

� � 

Y ≥ Y (k) 

� � 

Y ≥ Y (k) 

� 

∩ 

� 

Y ≥ Y (k) 

BMY 0.69 0.90 0.35 0.05 0.00* 0.12 0.04 0.02 0.00 

CVX 0.51 0.66 0.47 0.04 0.00* 0.09 0.04 0.07 0.03 

HPQ 1.10 0.74 0.51 0.09 0.02 0.03 0.08 0.03 0.01 

KO 0.70 1.60 0.98 0.10 0.09 1.00 0.08 0.36 0.24 

MMM 0.58 0.56 0.09 0.06 0.00 0.06 0.05 0.01 0.00 

PG 0.44 0.61 1.20 0.02 0.00* 0.07 0.02 0.00* 0.39 

SGP 0.64 0.27 0.58 0.03 0.00 0.00 0.02 0.00* 0.01 

TXN 0.30 1.30 0.10 0.00 0.00* 0.08 0.00 0.00 0.00 

WAG 

condition 

0.77 

all data 

0.85 

Y ≥ Y (k) 

0.31 

Y ≥ Y (k) 

0.07 

c... k 

0.01 

Y ≥ Y (k) 

0.10 0.06 0.02 0.00 

� X ≥ X(k) � ∩ c... k Y ≥ Y (k) � X ≥ X(k) � m=5736 

� � 

Y ≥ Y (k) 

� 

∩ 

� 

Y ≥ Y (k) 

BMY 0.69 -0.10 0.54 0.05 0.00* 0.12 0.05 0.02 0.01 

CVX 0.51 -0.61 0.62 0.04 0.00* 0.09 0.04 0.07 0.03 

HPQ 1.10 0.62 0.79 0.09 0.02 0.03 0.09 0.04 0.01 

KO 0.70 0.68 1.10 0.10 0.09 1.00 0.11 0.48 0.32 

MMM 0.58 0.10 0.36 0.06 0.00 0.06 0.06 0.01 0.00 

PG 0.44 -0.17 -0.06 0.02 0.00* 0.06 0.03 0.00* 0.57 

SGP 0.64 0.01 -0.11 0.03 0.00 0.00 0.03 0.00* 0.02 

TXN 0.30 -0.41 0.09 0.00 0.00* 0.08 0.00 0.00 0.00 

WAG 0.77 0.45 0.53 0.07 0.01 0.10 0.08 0.02 0.00 

Table 3.19: Results for ˆ βSI using the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k) and the second SI condition: Y ≥ Y (k) for upper and lower tails 

calculated by least squares method applied on linear additive single factor model: X = β ·Y +ε and for resulting upper and lower tail dependence 

coefficients ˆ λ + and ˆ λ− with corrected ˆl (c... k) and uncorrected ˆl (1... k). The tail represents the most extreme 4% of data during a time interval 

ranging from July 1985 to Mars 2008 (k = 229, c = 29). * denotes ˆ β < 0 → λ = 0.

I also implemented the β-smile improvement to the estimation of tail dependence 

by the parametric approach according to Sornette & Malevergne discussed in section 

(3.2.4). But to estimate ˆ λ by the parametric approach we first need to calculate: 

ε = X − β · Y with ε, X, Y as vectors of N elements in chronologic order and scale 

factor Cε estimated by: Ĉε = 1 

k 

� k 

j=1 

j 

N · εν j,N with ε1,N ≥ ε2,N ≥ . . . ≥ εk,N and k 

denoting the threshold. The problem now is that when ˆ βSI by one of the SI conditions 

is calculated, the yielded ˆ βSI is only valid for tail data that fulfills the respective SI 

condition. Therefore values for ε can only be calculated for return values where ˆ βSI is 

valid, which is data that fulfills the respective SI condition. Looking at the equation 

for the estimation of scale factor Cε we need rank ordered ε-values of the whole data 

sample. But return data, which fulfills the SI conditions does not necessarily have 

high ε-values and therefore the scale factor Ĉε adapted to the SI conditions is not the 

correct one because with ˆ βSI ε cannot be calculated for the whole sample and the 

ε-elements that we can calculate do not have to be the most extreme ones. Because 

of that application of β-smile improvement to the parametric approach did not yield 

meaningful results. That’s why I don’t present it. 

Analysis of Coefficients 

It is interesting to see the convergence of ˆ βSI(k) to β (calculated for the whole data set) 

if we plot it according to the different conditions in dependence of increasing threshold k. 

Figures (3.42), (3.43), (3.44), and (3.45) show ˆ βSI(k) plotted for the smaller reference 

sample on first, on second, on third, and on fourth condition. 

79

β, β SI 

β, β SI 

80 

β, β SI 

1 

0.5 

0 

−0.5 

−1 

BMY 

−1.5 

0 0.2 0.4 0.6 0.8 1 

k/N 

1.5 

1 

0.5 

0 

−0.5 

−1 

0 0.2 0.4 0.6 0.8 1 

k/N 

2.5 

2 

1.5 

1 

0.5 

0 

KO 

SGP 

−0.5 

0 0.2 0.4 0.6 0.8 1 

k/N 

β, β SI 

β, β SI 

β, β SI 

0.6 

0.4 

0.2 

0 

−0.2 

CVX 

−0.4 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0 

−1 

−2 

MMM 

−3 

0 0.2 0.4 0.6 0.8 1 

k/N 

60 

40 

20 

0 

TXN 

−20 

0 0.2 0.4 0.6 0.8 1 

k/N 

β, β SI 

β, β SI 

β, β SI 

2 

0 

−2 

−4 

−6 

HPQ 

−8 

0 0.2 0.4 0.6 0.8 1 

k/N 

2 

1.5 

1 

0.5 

0 

−0.5 

PG 

−1 

0 0.2 0.4 0.6 0.8 1 

k/N 

2 

1 

0 

−1 

WAG 

−2 

0 0.2 0.4 0.6 0.8 1 

k/N 

Figure 3.42: ˆ βSI(k) of the upper tails plotted in blue and of the lower tails plotted in black calculated by least square method applied on linear 

additive single factor model: X = β ·Y + ε and first SI condition: Y ≥ Y (k) ∩ X ≥ X(k) in dependence of threshold k. Reference ˆ β calculated for 

all data is given in green. Y denotes the index return vector of S&P 500 and Y denotes asset return vector of the nine assets for a time interval 

ranging from January 1991 to December 2000.

81 

β, β SI 

β, β SI 

β, β SI 

1.4 

1.2 

1 

0.8 

0.6 

BMY 

0.4 

0 0.2 0.4 0.6 0.8 1 

k/N 

1.5 

1 

0.5 

0 

0 0.2 0.4 0.6 0.8 1 

k/N 

1.5 

1 

0.5 

0 

KO 

SGP 

−0.5 

0 0.2 0.4 0.6 0.8 1 

k/N 

β, β SI 

β, β SI 

β, β SI 

1 

0.8 

0.6 

0.4 

0.2 

CVX 

0 

0 0.2 0.4 0.6 0.8 1 

k/N 

1.5 

1 

0.5 

MMM 

0 

0 0.2 0.4 0.6 0.8 1 

k/N 

2 

1 

0 

−1 

TXN 

−2 

0 0.2 0.4 0.6 0.8 1 

k/N 

β, β SI 

β, β SI 

β, β SI 

1.5 

1 

0.5 

0 

HPQ 

−0.5 

0 0.2 0.4 0.6 0.8 1 

k/N 

2 

1.5 

1 

0.5 

PG 

0 

0 0.2 0.4 0.6 0.8 1 

k/N 

2.5 

2 

1.5 

1 

0.5 

WAG 

0 

0 0.2 0.4 0.6 0.8 1 

k/N 


additive single factor model: X = β · Y + ε and second SI condition: Y ≥ Y (k) in dependence of threshold k. Reference ˆ β calculated for all data 

is given in green. Y denotes the index return vector of S&P 500 and Y denotes asset return vector of the nine assets for a time interval ranging 

from January 1991 to December 2000.

β, β SI 

82 

β, β SI 

β, β SI 

1 

0.5 

0 

−0.5 

−1 

BMY 

−1.5 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0 

−1 

−2 

−3 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0.5 

0 

−0.5 

KO 

SGP 

−1 

0 0.2 0.4 0.6 0.8 1 

k/N 

β, β SI 

β, β SI 

β, β SI 

0.5 

0 

−0.5 

CVX 

−1 

0 0.2 0.4 0.6 0.8 1 

k/N 

0.6 

0.4 

0.2 

0 

−0.2 

MMM 

−0.4 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0.5 

0 

−0.5 

−1 

TXN 

−1.5 

0 0.2 0.4 0.6 0.8 1 

k/N 

β, β SI 

β, β SI 

β, β SI 

1.5 

1 

0.5 

0 

−0.5 

−1 

HPQ 

−1.5 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0.5 

0 

−0.5 

PG 

−1 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0.5 

0 

−0.5 

WAG 

−1 

0 0.2 0.4 0.6 0.8 1 

k/N 


additive single factor model: X = β · Y + ε and third SI condition: X ≥ X(k) in dependence of threshold k. Reference ˆ β calculated for all data 

is given in green. Y denotes the index return vector of S&P 500 and Y denotes asset return vector of the nine assets for a time interval ranging 

from January 1991 to December 2000.

β, β SI 

83 

β, β SI 

β, β SI 

1 

0.5 

0 

−0.5 

−1 

−1.5 

BMY 

−2 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0 

−1 

−2 

−3 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0.5 

0 

−0.5 

−1 

−1.5 

KO 

SGP 

−2 

0 0.2 0.4 0.6 0.8 1 

k/N 

β, β SI 

β, β SI 

β, β SI 

0.5 

0 

−0.5 

−1 

CVX 

−1.5 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0.5 

0 

−0.5 

MMM 

−1 

0 0.2 0.4 0.6 0.8 1 

k/N 

0 

−1 

−2 

−3 

TXN 

−4 

0 0.2 0.4 0.6 0.8 1 

k/N 

β, β SI 

β, β SI 

β, β SI 

2 

1 

0 

−1 

−2 

HPQ 

−3 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0.5 

0 

−0.5 

−1 

−1.5 

PG 

−2 

0 0.2 0.4 0.6 0.8 1 

k/N 

1 

0 

−1 

−2 

WAG 

−3 

0 0.2 0.4 0.6 0.8 1 

k/N 


additive single factor model: X = β · Y + ε and fourth SI condition: Y ≥ Y (k) ∪ X ≥ X(k) in dependence of threshold k. Reference ˆ β calculated 

for all data is given in green. Y denotes the index return vector of S&P 500 and Y denotes asset return vector of the nine assets for a time interval 

ranging from January 1991 to December 2000.

ˆβSI(k) calculated by the first and the second condition seem to converge best. For 

some assets we have strong fluctuations in the extreme tails. Anyway, relative deviations 

calculated for k/N = 4% given in table (3.17) agree well with the convergence plots. 

To see it in detail I also plotted ˆ βSI for the first and for the third condition just in 

the threshold area of extreme tails, which we determined to be around 3% to 5% of 

total return data. Figure (3.46) shows ˆ βSI plotted in dependence of threshold k for 

k/N = 3% . . .6% calculated by first and third condition with ˆ β calculated for all data 

for comparison for upper tails and figure (3.47) shows the same for lower tails both for 

the smaller data sample. Figures (3.48) and (3.49) show ˆ βSI calculated by the first 

and by the second condition for the upper and for the lower tails of the bigger reference 

sample also for k/N = 3% . . .6%. The green line in all of these plots denotes the 

reference ˆ β calculated for the whole data sample and is therefore constant on the whole 

interval. As we can see, there is no general pattern between ˆ β calculated for all data 

and ˆ βSI calculated by first and second SI conditions for the different assets. For some 

cases the three are similar and for others not. But in many cases there is something 

like a convergence of the ˆ βSI and ˆ β around k/N = 4%. What we observe furthermore is 

that in most cases the differences between ˆ βSI calculated by first and second conditions 

and ˆ β calculated for all data is within certain limits. It is interesting to see to what 

extent ˆ λ +,− is affected by these discrepancies. Therefore I plotted ˆ λ +,− (k) calculated 

by the first and by the second conditions for k/N = 3% . . .10%. Figure (3.50) shows 

plots of ˆ λ(k) calculated by ˆ βSI according to the first condition: Y ≥ Y (k) ∩ X ≥ X(k) 

for upper tails and figure (3.51) for lower tails. Figures (3.52) and (3.53) show plots 

of ˆ λ(k) calculated by ˆ βSI according to the second condition: Y ≥ Y (k) for upper and 

lower tails. Also here differences between ˆ λ calculated by ˆ βSI plotted in black and 

ˆλ calculated by ˆ β plotted in green in most cases are not ’huge’. In the positive tails 

Texas Instruments Inc. (TXN), Hewlett Packard Co. (HPQ), and Procter & Gamble 

Co. (PG) are exceptions, where we have big differences of ˆ λ calculated using ˆ βSI by 

first condition and ˆ λ using ˆ β of all data. In the negative tails we don’t find these big 

discrepancies around k/N = 4%. Looking at ˆ λ calculated by the second condition we 

find some large discrepancies for asset 3M Co. (MMM) and Bristol-Myers Squibb Co. 

(BMY) in the negative tails and in the positive tails we don’t find ’huge’ discrepancies 

around k/N = 4%. 

84

85 

β SI , β 

β SI , β 

β SI , β 

1.2 

1 

0.8 

0.6 

BMY 

0.4 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

0.8 

0.6 

0.4 

0.2 

0 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1 

0.8 

0.6 

0.4 

KO 

SGP 

0.2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

β SI , β 

β SI , β 

β SI , β 

0.6 

0.4 

0.2 

0 

CVX 

−0.2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

0.6 

0.4 

0.2 

0 

−0.2 

MMM 

−0.4 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

6 

4 

2 

0 

TXN 

−2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

β SI , β 

β SI , β 

β SI , β 

1.5 

1 

0.5 

HPQ 

0 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1.5 

1 

0.5 

PG 

0 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1.6 

1.4 

1.2 

1 

WAG 

0.8 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

Figure 3.46: ˆ βSI(k) calculated by the first condition: Y ≥ Y (k) ∩ X ≥ X(k) plotted in blue and by the second condition: Y ≥ Y (k) plotted in 

black for the upper tail calculated by least squares method applied on linear additive single factor model: X = β ·Y +ε in dependence of threshold 

k for k/N = 3% ... 6%. Reference ˆ β calculated for all data is given in green. Y denotes the index return vector of S&P 500 and Y denotes asset 

return vector of the nine assets for a time interval ranging from January 1991 to December 2000.

β SI , β 

β SI , β 

86 

β SI , β 

1.5 

1 

0.5 

BMY 

0 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

0.95 

0.9 

0.85 

0.8 

0.75 

0.7 

0.65 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1.5 

1 

0.5 

KO 

SGP 

0 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

β SI , β 

β SI , β 

β SI , β 

0.8 

0.6 

0.4 

0.2 

CVX 

0 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1.2 

1 

0.8 

0.6 

0.4 

MMM 

0.2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1 

0.5 

0 

−0.5 

−1 

TXN 

−1.5 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

β SI , β 

β SI , β 

β SI , β 

1.5 

1 

0.5 

0 

−0.5 

−1 

HPQ 

−1.5 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1.4 

1.2 

1 

0.8 

0.6 

PG 

0.4 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

2 

1 

0 

−1 

WAG 

−2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 


black for the lower tail calculated by least squares method applied on linear additive single factor model: X = β ·Y +ε in dependence of threshold 


return vector of the nine assets for a time interval ranging from January 1991 to December 2000.

β SI , β 

87 

β SI , β 

β SI , β 

1 

0.5 

0 

BMY 

−0.5 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

2.5 

2 

1.5 

1 

0.5 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

0.8 

0.6 

0.4 

0.2 

0 

KO 

SGP 

−0.2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

β SI , β 

β SI , β 

β SI , β 

1 

0.5 

0 

−0.5 

CVX 

−1 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

0.8 

0.6 

0.4 

0.2 

MMM 

0 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

2 

1.5 

1 

0.5 

0 

−0.5 

TXN 

−1 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

β SI , β 

β SI , β 

β SI , β 

1.4 

1.2 

1 

0.8 

0.6 

HPQ 

0.4 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

0.8 

0.6 

0.4 

0.2 

0 

PG 

−0.2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1 

0.8 

0.6 

0.4 

WAG 

0.2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 


black for the upper tail calculated by least squares method applied on linear additive single factor model: X = β ·Y +ε in dependence of threshold 


return vector of the nine assets for a time interval ranging from July 1985 to Mars 2008.

β SI , β 

88 

β SI , β 

β SI , β 

0.65 

0.6 

0.55 

0.5 

0.45 

0.4 

0.35 

BMY 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1.3 

1.2 

1.1 

1 

0.9 

0.8 

0.7 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1 

0.5 

0 

KO 

SGP 

−0.5 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

β SI , β 

β SI , β 

β SI , β 

0.65 

0.6 

0.55 

0.5 

0.45 

CVX 

0.4 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

0.8 

0.6 

0.4 

0.2 

MMM 

0 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

0.6 

0.4 

0.2 

0 

−0.2 

TXN 

−0.4 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

β SI , β 

β SI , β 

β SI , β 

1.4 

1.2 

1 

0.8 

0.6 

HPQ 

0.4 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1.5 

1 

0.5 

0 

PG 

−0.5 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 

1 

0.8 

0.6 

0.4 

WAG 

0.2 

0.03 0.035 0.04 0.045 

k/N 

0.05 0.055 0.06 


black for the lower tail calculated by least squares method applied on linear additive single factor model: X = β ·Y +ε in dependence of threshold 


return vector of the nine assets for a time interval ranging from July 1985 to Mars 2008.

As an additional information, ˆν was always taken constant for the whole intervals 

and estimated at k/N = 4%. Otherwise when ˆν was adapted to the threshold k, fluctuations 

of ˆ λ in k were much stronger and dominated by ˆν. In order to understand 

this, we reconsider figures (3.1) showing ˆν for the index S&P 500 and the nine assets 

for the positive tails and (3.3) showing ˆν for the index S&P 500 and the nine assets 

for the negative tails. In the positive tail of index S&P 500 we have a fluctuation 

spectrum ranging from around 3.2 for k/N = 3% to 2.2 for k/N = 10% for the bigger 

data sample, which is very high. In the negative tails the situation is similar. Fluctuations 

furthermore become stronger as k approaches zero and Hill’s estimator cannot be 

trusted anymore because of lack of data in that area. The situation for Gabaix’s b γ n is 

slightly better but as I wanted to observe the impact of β in k tail index α was assumed 

to be constant for the considered interval of threshold k calculated at k/N = 4%. 

We already know that adaptation of ˆ l to threshold k does not change ˆ λ significantly. 

This can also be observed in figures (3.50), (3.51), (3.52), and (3.53), where plots in 

green show ˆ l(k) and plots in red show ˆ l(k/N = 4%). In our area of interest k/N = 

3% . . .5% the two plots in red and green respectively stay close together showing us 

that ˆ l can be assumed more or less constant in the tails up to 10% of total data. For 

ˆ l(k) we used means that were not corrected for most extreme 0.5%: ˆ l(k) = 1 

k 

�k Xj,N 

j=1 Yj,N . 

Anyway the difference between ˆ λ calculated by uncorrected means and ˆ λ calculated by 

corrected means is small compared to differences between ˆ λ calculated by the different 

β-smile conditions and ˆ λ with ˆ β calculated by all data. Another approach would be 

to calculate ˆ l only by data that fulfills the ˆ βSI conditions. I also implemented this 

adapted ˆ lSI ’improvement’ for the first ˆ βSI condition. Plots are shown in figures (3.50) 

and (3.51) given in blue. Implementing adapted ˆ lSI for the second condition sometimes 

yielded negative ˆ l indicating that extreme positive return movements of assets happen 

simultaneously with extreme negative return movements of index S&P 500 or vice versa. 

This certainly yields meaningless results for ˆ λ. The differences between ˆ λ calculated by 

adapted ˆ lSI and ˆ λ calculated by ˆ l were also small. Another indication that ˆ λ is robust 

to the choice of ˆ l or that the assumption of constant ˆ l in the tails is appropriate. 

We can conclude that β seems to be the crucial parameter within these approaches. 

Sometimes the two or three estimates (for different β approaches) are consistent and 

for other cases they are not. These latter cases might diagnose the limits of the linear 

β model to describe the dependence structure between asset and index. 

89

90 

λ 

λ 

λ 

0.08 

0.06 

0.04 

0.02 

BMY 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.08 

0.06 

0.04 

0.02 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.14 

0.12 

0.1 

0.08 

0.06 

0.04 

KO 

SGP 

0.02 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

λ 

λ 

λ 

0.02 

0.015 

0.01 

0.005 

CVX 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.04 

0.03 

0.02 

0.01 

MMM 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

1 

0.8 

0.6 

0.4 

0.2 

TXN 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

λ 

λ 

λ 

0.1 

0.08 

0.06 

0.04 

0.02 

HPQ 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.5 

0.4 

0.3 

0.2 

0.1 

PG 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.3 

0.25 

0.2 

0.15 

0.1 

WAG 

0.05 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

Figure 3.50: ˆ λ + (k) for the upper tail of the nine assets and the index S&P 500 is plotted in black for ˆ β(k) calculated by least squares method 

applied to linear additive single factor model: X = β · Y + ε and adapted to the first condition: Y ≥ Y (k) ∩ X ≥ X(k), ˆν(k = 4%) and ˆ l(k) in 

dependence of threshold k for k/N = 3% ... 10%, in blue for ˆ β(k) and ˆ l(k) adapted to the first SI condition for comparison, in green for Reference 

ˆβ and ˆ l(k), and for Reference ˆ λ + (k = 4%) is given in red. Y denotes the index return vector of S&P 500 and X denotes asset return vector of the 

nine assets for a time interval ranging from January 1991 to December 2000.

91 

λ 

λ 

λ 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

BMY 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.05 

0.04 

0.03 

0.02 

0.01 

KO 

SGP 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

λ 

λ 

λ 

0.02 

0.015 

0.01 

0.005 

CVX 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

0.015 

MMM 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.01 

0.005 

TXN 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

λ 

λ 

λ 

0.08 

0.06 

0.04 

0.02 

HPQ 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.2 

0.15 

0.1 

0.05 

PG 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.08 

0.06 

0.04 

0.02 

WAG 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

Figure 3.51: ˆ λ + (k) for the lower tail of the nine assets and the index S&P 500 is plotted in black for ˆ β(k) calculated by least squares method 

applied to linear additive single factor model: X = β · Y + ε and adapted to the first condition: Y ≥ Y (k) ∩ X ≥ X(k), ˆν(k = 4%) and ˆ l(k) in 

dependence of threshold k for k/N = 3% ... 10%, in blue for ˆ β(k) and ˆ l(k) adapted to the first SI condition for comparison, in green for Reference 

ˆβ and ˆ l(k), and for Reference ˆ λ + (k = 4%) is given in red. Y denotes the index return vector of S&P 500 and X denotes asset return vector of the 

nine assets for a time interval ranging from January 1991 to December 2000.

92 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

BMY 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.08 

0.06 

0.04 

0.02 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

KO 

SGP 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

λ 

λ 

λ 

0.02 

0.015 

0.01 

0.005 

CVX 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.04 

0.03 

0.02 

0.01 

MMM 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

1 

0.5 

0 

−0.5 

TXN 

−1 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

0.025 

0.015 

0.005 

HPQ 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.03 

0.02 

0.01 

PG 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.5 

0.4 

0.3 

0.2 

0.1 

WAG 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

Figure 3.52: ˆ λ + (k) for the upper tail of the nine assets and the index S&P 500 is plotted in black for ˆ β(k) calculated by least squares method 

applied to linear additive single factor model: X = β · Y + ε and adapted to the second condition: Y ≥ Y (k), ˆν(k = 4%) and ˆ l(k) in dependence 

of threshold k for k/N = 3% ... 10%, in blue for ˆ β(k) and ˆ l(k) adapted to the first SI condition for comparison, in green for Reference ˆ β and ˆ l(k), 

and for Reference ˆ λ + (k = 4%) is given in red. Y denotes the index return vector of S&P 500 and X denotes asset return vector of the nine assets 

for a time interval ranging from January 1991 to December 2000.

93 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

BMY 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.2 

0.15 

0.1 

0.05 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.2 

0.15 

0.1 

0.05 

KO 

SGP 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

λ 

λ 

λ 

0.08 

0.06 

0.04 

0.02 

CVX 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.4 

0.3 

0.2 

0.1 

MMM 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

2.5 

1.5 

0.5 

x 10−7 

3 

2 

1 

TXN 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

λ 

λ 

λ 

0.14 

0.12 

0.1 

0.08 

0.06 

HPQ 

0.04 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.2 

0.15 

0.1 

0.05 

PG 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

0.2 

0.15 

0.1 

0.05 

WAG 

0 

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 

k/N 

Figure 3.53: ˆ λ + (k) for the lower tail of the nine assets and the index S&P 500 is plotted in black for ˆ β(k) calculated by least squares method 

applied to linear additive single factor model: X = β · Y + ε and adapted to the second condition: Y ≥ Y (k), ˆν(k = 4%) and ˆ l(k) in dependence 

of threshold k for k/N = 3% ... 10%, in blue for ˆ β(k) and ˆ l(k) adapted to the first SI condition for comparison, in green for Reference ˆ β and ˆ l(k), 

and for Reference ˆ λ + (k = 4%) is given in red. Y denotes the index return vector of S&P 500 and X denotes asset return vector of the nine assets 

for a time interval ranging from January 1991 to December 2000.


To establish uncertainty of upper and lower tail dependence coefficients I included 

bootstrap approximations 14 . The sampling has been performed by creating bootstrap 

samples of historical day-by-day return data of index S&P 500 and searching for corresponding 

asset return data. I always sampled the whole data sets because I think that 

this is more representative than just sampling of tails. Tables (3.20) and (3.21) show 

quantiles, extreme values, and standard deviations (’std bs’) of the results of bootstrap 

sampling of λ +,− for the smaller and bigger reference samples using ˆ βSI calculated by 

the first SI condition: {Y ≥ Y (k)} ∩ {X ≥ X(k)} and tables (3.22) and (3.23) show 

the results of bootstrap sampling of λ +,− for the smaller and bigger reference samples 

using ˆ βSI calculated by the second SI condition: Y ≥ Y (k). ˆ λ +,− were calculated using 

the mean values of ˆl = Xj,N/Yj,N for j = 1 . . .k with k = 100 for the smaller sample 

and with k = 229 for the bigger sample and ˆ λ +,− 

c were calculated using the mean values 

of ˆl from c . . .k with c = 13 and k = 100 for the smaller sample and for k = c . . .k with 

c = 29 and k = 229 for the bigger sample. 

Relative errors of average bootstrap tail dependence coefficients ˆ λ +,− 

bs compared to 

tail dependence coefficients ˆ λ +,− of the original sample were unusually big in most 

cases. If tail dependence coefficient estimates were very small (≈ 10 −3 ) relative errors 

could have become multiples of the estimates. Quantiles of the bigger data samples in 

some cases look very consistent and are approximately Gaussian distributed. In some 

other cases first of all if ˆ βSI are close to zero accuracy of the results become poor as 

it can be observed at the example of assets 3M Co. (MMM) and Texas-Instruments 

inc. (TXN) in the lower tails of the bigger data set applying the first SI condition 

and at the example of Schering & Plough Corp. In the positive tail also of the bigger 

data sample applying the first SI condition. 95% quantiles for example are around the 

size of the estimated total value of the respective tail dependence coefficient estimates 

provided ˆ λ is not too small. For the smaller data set it can be observed that for 

small estimates of the tail dependence coefficient by the first as well as the second 

condition, quantiles get relatively high. Accuracy here seems to be poor. Overall the 

situation concerning bootstrap quantiles of β-smile improvement estimates calculated 

by the non-parametric approach according to Sornette & Malevergne, applying the first 

or the second condition looks worse compared to the classic non-parametric approach 

with ˆ β calculated by all data. What makes the difference is that estimates calculated 

by SI conditions at times become very small and therefore relative error bars become 

very high. ˆ βSI therefore does not seem to have a negative stake on the error bars 

of ˆ λ. Another reason might be that because ˆ βSI shows strong fluctuations in k, it is 

a further source of uncertainty that distorts tail dependence estimates. Maybe if we 

use much bigger data sets with more data in the tails the estimates would become 

more stable. The problem is that this has not much practical application because our 

bigger data set already contains more than 5000 daily observations and for many assets 

there is not more data available. Looking at the second SI condition the situation is 

even worse. Relative errors are mostly within limits but tail dependence coefficient 

estimates are very inaccurate. Standard deviations are at least of the size of the total 

estimate. Anyways adaptation of β to the tails seems to add too much uncertainty to 

the estimates. 

14 The matlab m-file for the bootstrap sampling with replacement of λ +,− estimated by the non-parametric 

approach according to Sornette & Malevergne with ˆ β calculated by first and second SI conditions is denoted 

by: bootstr simple impr alldata.m and enclosed to the data CD 

94

95 

constr 1 m=2507 bs=1000 


λbs,mean 

λbs,mean 


BMY 0.06 0.42 0.93 0.12 0.08 0.08 0.01 1.79 0.52 0.03 0.02 0.02 

CVX 0.00 Inf 0.45 0.01 0.00 0.02 0.02 2.61 0.38 0.12 0.03 0.03 

HPQ 0.00 9.28 0.77 0.01 0.00 0.03 0.01 Inf 1.02 0.01 0.01 0.06 

KO 0.04 3.07 1.04 0.10 0.10 0.09 0.09 1.18 0.61 0.33 0.20 0.13 

MMM 0.04 Inf 1.00 0.23 0.04 0.17 0.04 0.55 1.00 0.12 0.06 0.07 

PG 0.37 0.21 0.63 0.58 0.55 0.40 0.05 2.58 0.56 0.17 0.10 0.08 

SGP 0.11 2.39 0.91 0.42 0.19 0.21 0.01 2.89 0.08 0.02 0.01 0.01 

TXN 0.79 0.24 0.84 0.82 0.78 0.40 0.04 Inf 1.01 0.10 0.04 0.21 

WAG 0.24 1.04 0.78 0.78 0.45 0.34 0.03 Inf 0.60 0.14 0.04 0.08 

λc 

BMY 0.06 0.43 0.92 0.12 0.10 0.09 0.02 1.86 0.51 0.05 0.03 0.03 

CVX 0.00 Inf 0.60 0.01 0.00 0.02 0.02 2.71 0.34 0.06 0.03 0.04 

HPQ 0.00 9.25 0.88 0.01 0.00 0.03 0.01 Inf 1.01 0.01 0.01 0.09 

KO 0.04 3.09 1.04 0.11 0.10 0.10 0.13 1.21 0.70 0.40 0.30 0.20 

MMM 0.04 Inf 0.96 0.20 0.04 0.19 0.05 0.60 1.00 0.11 0.07 0.08 

PG 0.37 0.24 0.61 0.59 0.58 0.40 0.09 2.48 0.66 0.30 0.20 0.10 

SGP 0.11 2.41 0.90 0.26 0.20 0.20 0.01 2.67 0.10 0.02 0.01 0.01 

TXN 0.79 0.23 0.83 0.81 0.74 0.42 0.04 Inf 1.04 0.10 0.04 0.20 

WAG 0.25 1.02 0.81 0.80 0.47 0.31 0.03 Inf 0.78 0.18 0.06 0.09 




interval from January 1991 to December 2000. ˆ βj have been calculated on the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k). ’inf’ denotes ·/0.

96 

constr 1 m=5736 bs=1000 


λbs,mean 

λbs,mean 


BMY 0.15 0.29 0.92 0.31 0.19 0.09 0.01 0.52 0.09 0.01 0.01 0.01 

CVX 0.18 1.01 0.78 0.60 0.30 0.21 0.03 0.10 0.29 0.03 0.03 0.03 

HPQ 0.04 0.32 0.49 0.12 0.11 0.12 0.01 0.01 0.00 0.01 0.01 0.01 

KO 0.65 0.41 0.59 0.58 0.58 0.38 0.21 0.08 0.31 0.20 0.19 0.12 

MMM 0.09 0.48 0.91 0.19 0.10 0.09 0.02 201.00 0.78 0.09 0.04 0.06 

PG 0.13 1.00 0.92 0.41 0.18 0.19 0.45 0.11 0.61 0.61 0.55 0.37 

SGP 0.02 9.03 0.49 0.09 0.05 0.05 0.05 3.02 0.92 0.20 0.08 0.12 

TXN 0.16 1.01 0.81 0.57 0.31 0.22 0.00 Inf 0.10 0.01 0.01 0.01 

WAG 0.12 0.32 0.90 0.23 0.16 0.10 0.02 6.96 0.57 0.09 0.02 0.07 

λc 

BMY 0.15 0.28 0.92 0.28 0.21 0.09 0.01 0.56 0.10 0.02 0.01 0.01 

CVX 0.18 1.00 0.79 0.57 0.30 0.21 0.03 0.10 0.31 0.04 0.03 0.03 

HPQ 0.04 0.29 0.47 0.13 0.11 0.10 0.01 0.03 0.03 0.01 0.01 0.01 

KO 0.65 0.40 0.59 0.57 0.56 0.39 0.28 0.09 0.33 0.32 0.20 0.10 

MMM 0.09 0.49 0.89 0.21 0.09 0.08 0.03 137.00 1.00 0.10 0.06 0.08 

PG 0.13 1.00 0.91 0.43 0.21 0.19 0.54 0.05 0.53 0.51 0.50 0.40 

SGP 0.02 9.02 0.52 0.10 0.04 0.05 0.07 3.02 0.91 0.30 0.11 0.19 

TXN 0.16 1.01 0.80 0.59 0.29 0.21 0.00 Inf 0.17 0.02 0.01 0.01 

WAG 0.12 0.31 0.91 0.26 0.21 0.09 0.03 6.01 0.79 0.11 0.03 0.08 




interval from July 1985 to Mars 2008. ˆ βj have been calculated on the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k). ’inf’ denotes ·/0.

97 

constr 2 m=2507 bs=1000 


λbs,mean 

λbs,mean 


BMY 0.14 0.95 0.88 0.29 0.21 0.22 0.19 0.09 0.82 0.21 0.19 0.10 

CVX 0.02 11.03 0.59 0.10 0.03 0.04 0.07 0.20 0.41 0.13 0.12 0.09 

HPQ 0.19 0.39 0.80 0.32 0.23 0.20 0.09 0.37 0.94 0.22 0.10 0.11 

KO 0.05 2.02 0.73 0.21 0.08 0.09 0.16 0.32 0.77 0.38 0.31 0.20 

MMM 0.01 2.96 0.21 0.05 0.03 0.03 0.27 0.03 0.71 0.39 0.18 0.18 

PG 0.05 37.95 0.87 0.18 0.09 0.11 0.11 0.48 0.89 0.22 0.19 0.13 

SGP 0.10 3.03 0.88 0.39 0.21 0.20 0.12 0.12 0.62 0.21 0.08 0.10 

TXN 0.00 Inf 0.81 0.00 0.00 0.03 0.00 Inf 0.04 0.00 0.00 0.00 

WAG 0.28 0.68 0.73 0.57 0.38 0.28 0.11 1.81 0.91 0.40 0.19 0.19 

λc 

BMY 0.15 0.96 0.90 0.40 0.20 0.21 0.27 0.10 0.69 0.32 0.20 0.16 

CVX 0.02 11.02 0.58 0.11 0.03 0.04 0.08 0.22 0.68 0.14 0.11 0.10 

HPQ 0.19 0.48 0.80 0.30 0.21 0.20 0.14 0.56 0.89 0.28 0.09 0.19 

KO 0.05 1.62 0.72 0.22 0.09 0.08 0.23 0.28 0.81 0.61 0.37 0.20 

MMM 0.01 2.73 0.19 0.05 0.03 0.03 0.33 0.04 0.70 0.48 0.26 0.23 

PG 0.05 40.04 0.89 0.26 0.08 0.12 0.18 0.41 0.79 0.40 0.20 0.21 

SGP 0.10 2.47 0.89 0.40 0.22 0.19 0.17 0.10 0.81 0.21 0.09 0.12 

TXN 0.00 Inf 0.86 0.00 0.00 0.03 0.00 Inf 0.13 0.00 0.00 0.00 

WAG 0.28 0.73 0.72 0.68 0.50 0.31 0.16 1.23 0.80 0.53 0.28 0.18 




interval from January 1991 to December 2000. ˆ βj have been calculated on the second SI condition: Y ≥ Y (k). ’inf’ denotes ·/0.

98 

constr 2 m=5736 bs=1000 


λbs,mean 

λbs,mean 


BMY 0.01 Inf 0.18 0.04 0.01 0.02 0.04 0.88 0.39 0.09 0.08 0.05 

CVX 0.01 Inf 0.30 0.04 0.01 0.03 0.08 0.21 0.47 0.08 0.07 0.10 

HPQ 0.06 2.03 0.68 0.20 0.10 0.09 0.04 0.20 0.21 0.04 0.03 0.02 

KO 0.19 1.02 0.79 0.79 0.38 0.30 0.34 0.07 0.70 0.31 0.21 0.10 

MMM 0.04 123.95 0.51 0.10 0.09 0.07 0.08 6.03 0.72 0.33 0.23 0.11 

PG 0.01 Inf 0.47 0.03 0.01 0.02 0.10 Inf 0.88 0.48 0.22 0.20 

SGP 0.02 Inf 1.02 0.07 0.02 0.08 0.04 Inf 0.82 0.21 0.10 0.09 

TXN 0.00 Inf 0.21 0.00 0.00 0.01 0.00 Inf 0.12 0.00 0.00 0.00 

WAG 0.04 2.01 0.47 0.13 0.07 0.07 0.05 2.01 0.87 0.13 0.05 0.09 

λc 

BMY 0.01 Inf 0.26 0.04 0.01 0.02 0.05 1.04 0.46 0.10 0.08 0.06 

CVX 0.01 Inf 0.31 0.04 0.01 0.03 0.09 0.19 0.47 0.09 0.07 0.06 

HPQ 0.06 2.02 0.68 0.20 0.11 0.09 0.05 1.00 0.30 0.07 0.04 0.03 

KO 0.19 1.02 0.81 0.81 0.40 0.29 0.45 0.06 0.61 0.32 0.28 0.18 

MMM 0.04 123.97 0.52 0.10 0.09 0.07 0.10 6.03 0.88 0.34 0.21 0.20 

PG 0.01 Inf 0.49 0.03 0.01 0.03 0.14 Inf 0.89 0.77 0.44 0.31 

SGP 0.02 Inf 1.01 0.07 0.02 0.08 0.06 Inf 0.90 0.21 0.20 0.10 

TXN 0.00 Inf 0.20 0.00 0.00 0.01 0.00 Inf 0.11 0.01 0.00 0.01 

WAG 0.04 1.98 0.49 0.13 0.07 0.07 0.06 2.02 0.92 0.20 0.06 0.08 




interval from July 1985 to Mars 2008. ˆ βj have been calculated on the second SI condition: Y ≥ Y (k). ’inf’ denotes ·/0.


I wanted to include a rolling time window observation of the β-smile improvement 

estimates just to see whether ˆ λ calculated by SI conditions performs similarly as ˆ λ 

calculated by ˆ β of the whole sample. Figures (3.54) and (3.55) show ˆ βSI for the window 

size of S = 2500 in dependence of time calculated by the first and by the second SI 

conditions and figures (3.56) and (3.57) show ˆ λ +,− calculated by ˆ βSI for the window 

size of S = 2500 in dependence of time by the first and by the second SI conditions. 

As we can see fluctuations of ˆ λ are quite strong and a window size of S = 2500 seems 

to be the minimum size to obtain more or less reasonable estimates. Choosing smaller 

window sizes yields instable results for both ˆ βSI and ˆ λ. It is interesting to observe that 

whereas ˆ βSI calculated by first condition seems to be high in the early period up to 

N = 3000 for most assets, ˆ βSI calculated by the second condition is mostly negative 

in that area. The spectron of fluctuations on the interval (−1, 1.5) seems to be similar 

for both ˆ βSI. Hewlett-Packard Co. (HPQ) is a special case because it shows for both 

conditions a significant difference between ˆ βSI calculated for the upper tail and ˆ βSI 

calculated for the lower tail. The case, where the linear dependences of asset Xj and 

index Yj are different in the positive and the negative tail, is an aspect that ˆ β calculated 

by all data is not able to display. 

What furthermore can be observed in all of the four figures is that the overall trend 

for my shown ˆ βSI and corresponding ˆ λ is the change in time of the positive tails 

dominating over negative tail to a tendency of negative tails to dominate the positive 

tails. This counts primarily for the second condition with assets Hewlett-Packard Co. 

(HPQ) and Texas Instruments Inc. (TXN) being exceptions that show a counter-trend. 

Looking at figure (3.57) showing ˆ λ calculated by the second SI condition, it seems that 

both upper and lower tail dependence have increased over time. 

I also applied bootstrap sampling over time to ˆ λ +,− calculated by first and second 

β-smile conditions in order to see whether my impression of unusual high quantiles is 

consistent over time 15 . This seems to be the case. Assets with more or less limited 

quantile deviations show this behavior over time and others remain inaccurate all the 

time. Furthermore it seems that there are windows were the quantiles become much 

bigger and then after some time get back to their normal extent. 

It is a pity that estimates of ˆ λ calculated by β-smile conditions are not more robust 

because the approach is very promising. Nevertheless the β-smile improvement conditions 

can be helpful as a complement to other approaches because they allow for new 

insights into the structure of extreme dependence of an asset and the index, representing 

the market factor, particularly in terms of showing differences between negative and 

positive tails. The classic non-parametric approach describes the dependence of asset 

and index by a linear relation that is assumed to be constant over the whole range of 

the distribution. This assumption might be to restrictive in some cases. In order to find 

and to describe these discrepancies by an advanced concept with the aspect of being 

more general in describing dependence structures, the β-smile improvement conditions 

can be applied. 

15 Results of bootstrap sampling of the rolling time window estimates for first and second SI conditions with 

window size S = 2500 data points can be observed on the data CD submitted with the thesis in the folder: 

window results 

99

100 

β 

β 

β 

1.5 

1 

0.5 

BMY 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

4 

3 

2 

1 

0 

−1 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2 

1.5 

1 

0.5 

0 

−0.5 

KO 

SGP 

−1 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

β 

β 

β 

2 

1.5 

1 

0.5 

0 

CVX 

−0.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1.5 

1 

0.5 

0 

MMM 

−0.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

6 

4 

2 

0 

−2 

TXN 

−4 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

β 

β 

β 

1.5 

1 

0.5 

0 

−0.5 

−1 

HPQ 

−1.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2 

1.5 

1 

0.5 

0 

PG 

−0.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2 

1 

0 

−1 

WAG 

−2 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 3.54: ˆ βSI(N) plotted in black for the upper tails and plotted in green for the lower tails for 9 assets given by X and index S&P 500 given 

by Y using the linear single factor model: X = β · Y + ε and the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k) 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.

101 

β 

β 

β 

1.5 

1 

0.5 

0 

−0.5 

BMY 

−1 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1.5 

1 

0.5 

0 

−0.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2 

1.5 

1 

0.5 

0 

−0.5 

KO 

SGP 

−1 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

β 

β 

β 

2 

1 

0 

−1 

CVX 

−2 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1.5 

1 

0.5 

0 

−0.5 

MMM 

−1 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2 

1 

0 

−1 

TXN 

−2 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

β 

β 

β 

2 

1.5 

1 

0.5 

0 

HPQ 

−0.5 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1.5 

1 

0.5 

0 

−0.5 

PG 

−1 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

2 

1.5 

1 

0.5 

WAG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 3.55: ˆ βSI(N) plotted in black for the upper tails and plotted in green for the lower tails for 9 assets given by X and index S&P 500 given 

by Y using the linear single factor model: X = β · Y + ε and the second SI condition: Y ≥ Y (k) for rolling time horizon windows of S = 2500 


102 

λ 

λ 

λ 

0.5 

0.4 

0.3 

0.2 

0.1 

BMY 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1 

0.8 

0.6 

0.4 

0.2 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.5 

0.4 

0.3 

0.2 

0.1 

KO 

SGP 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

1 

0.8 

0.6 

0.4 

0.2 

CVX 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.5 

0.4 

0.3 

0.2 

0.1 

MMM 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1 

0.8 

0.6 

0.4 

0.2 

TXN 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

HPQ 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

1 

0.8 

0.6 

0.4 

0.2 

PG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.5 

0.4 

0.3 

0.2 

0.1 

WAG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 3.56: λ + (N) for upper tails of index S&P 500 and the nine assets plotted in blue and λ − (N) for lower tails plotted in red using non- 

parametric approach given by equation ˆ λ +,− = 1/max 

� 

1, l 

ˆβ +,− 

SI 

� ˆν 

with coefficients ˆν, l, and ˆ β +,− 

SI 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. ˆ β +,− 

SI were 

calculated using the linear single factor model: X = β · Y + ε and the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k).

103 

λ 

λ 

λ 

0.2 

0.15 

0.1 

0.05 

BMY 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.5 

0.4 

0.3 

0.2 

0.1 

KO 

SGP 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.4 

0.3 

0.2 

0.1 

CVX 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.8 

0.6 

0.4 

0.2 

MMM 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.08 

0.06 

0.04 

0.02 

TXN 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

λ 

λ 

λ 

0.5 

0.4 

0.3 

0.2 

0.1 

HPQ 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.4 

0.3 

0.2 

0.1 

PG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

0.8 

0.6 

0.4 

0.2 

WAG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

N 

Figure 3.57: λ + (N) for upper tail of index S&P 500 and the nine assets plotted in blue and λ − (N) for lower tail plotted in red using non-parametric 

approach given by equation ˆ λ +,− = 1/max 

� 

l 1, ˆβ +,− 

�ˆν SI 

with coefficients ˆν, l, and ˆ β +,− 

SI 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. ˆ β +,− 

SI were calculated 

using the linear single factor model: X = β · Y + ε and the second SI condition: Y ≥ Y (k).

3.4 Non-Parametric Approaches according to 

Schmidt & Stadtmüller 

Within this section several concepts of non-parametric approaches for the estimation 

of tail dependence according to Schmidt & Stadtmüller published in [24] (2005) are 

discussed. In the first sub-section a presentation of the most important concepts and 

assumptions of the paper is provided. Then the second sub-section again is devoted to 

the implementation of the concepts and in the third sub-section results are analyzed. 


In [24] (2005) a set of non-parametric estimators is proposed for the upper and lower 

tail copula ΛU(x, y) and ΛL(x, y) (x, y) ′ ∈ R 2 +. Non-parametric estimation is used if 

no general finite-dimensional parametrization of tail copulas exists. 

Let Cm denote the empirical copula defined by: 

Cm(u, v) = Fm(G −1 

m (u), H −1 

m (v)), (u, v) ′ ∈ [0, 1] 2 

(3.77) 

with Fm, Gm, Hm being the empirical distribution functions corresponding to F, G, H. 

Analogously we define the ’empirical survival copula’ by: Cm(u, v) = F m(G −1 

m (u), H−1 m (v)), 

(u, v) ′ ∈ [0, 1] 2 with 

F m(x, y) = 1 

m 

m� 

j=1 

1 {X (j) >x,Y (j) >y} 

(3.78) 

and Gm = 1 − Gm, Hm = 1 − Hm. Let R (j) 

m1 and R (j) 

m2 denote the rank of X (j) and 

Y (j) , j = 1, . . .,m respectively. A set of estimators is given by: 

ˆΛL,m(x, y) := m 

k 

ˆΛU,m(x, y) := m 

k 

Cm( kx 

m 

Cm( kx 

m 

ky 1 

, ) ≈ 

m k 

ky 1 

, ) ≈ 

m k 

m� 

j=1 

m� 

j=1 

1 (j) 

{R m1≤kx and R(j) ms≤ky} 

1 (j) 

{R m1 >m−kx and R(j) ms>m−ky} 

(3.79) 

(3.80) 

with some parameter k ∈ {1, . . .,m} chosen by the statistician. The estimators 

ˆΛU,m(x, y) and ˆ ΛL,m(x, y) are referred to as ’empirical tail copulas’. For the asymptotic 

result we assume that k = k(m) → ∞ and k(m) → 0 as m → ∞. 

A related estimator was introduced by Huang in [25] (1992), [26] (1998), and [27] 

(1998). The relation between the upper tail copula and the stable tail dependence l is 

given by 

ΛU(x, y) = x + y − l(x, y). The Corresponding estimator for ΛL(x, y) on R2 + is: 

ˆΛ 

EV T 

L,m (x, y) = x + y − m 

� 

k 

≈ x + y − 1 

k 

Cm 

m� 

j=1 

� �� 

kx ky 

, 

m m 

1 (j) 

{R m1≤kx or R(j) 

m2≤ky}, (x, y) ∈ R2 + (3.81) 

104

and the corresponding estimator for ΛU(x, y) on R 2 + is: 

ˆΛ 

EV T 

U,m 

� � �� 

m kx ky 

(x, y) = x + y − Cm , 

k m m 

≈ x + y − 1 

k 

m� 

j=1 

1 (j) 

{R m1 >m−kx or R(j) 

m2 >m−ky}, (x, y) ∈ R2 + (3.82) 

with k = k(m) → ∞ and k(m) → 0 as m → ∞. An important practical problem 

again arises on the optimal choice of the threshold parameter k, which leads to the 

usual variance-bias problem. Based on the above estimates of the lower and upper tail 

copula, for the upper tail dependence coefficient 

ˆλU,m := ˆ ΛU,m(1, 1), and ÊV T 

λ 

EV T 

U,m := ˆ ΛU,m (1, 1) (3.83) 

are proposed as non-parametric estimators and analogous estimates for the lower tail 

dependence coefficient. 

3.4.2 Implementation of Non-Parametric Approaches 

It is the purpose of this sub-section to provide an insight into details concerning the 

implementation of the above introduced concepts into Matlab m-files and apply the 

function handles to our reference data sets of historical daily return data of the index 

S&P 500 and the nine assets for the calculation of upper and lower tail dependence coefficient 

estimates. Results can then be compared to further calculations, i.e. estimations 

of tail dependence λ +,− according to Sornette & Malevergne. 

Calculation of general Rank-Order Statistics 

For the calculation of tail copulas ˆ ΛU,m, ˆ ΛL,m, ÊV T ΛU,m , and ÊV T ΛL,m between the index S&P 

500 and an asset as described in (3.79), (3.80), (3.81), and (3.82) we need rank-ordered 

statistics of index returns X (j) and asset returns Y (j) denoted by R (j) 

m1 and R (j) 

m2 with 

j = 1, . . .,m. Then we define a count variable, which passes through the rank vectors 

checking for the required conditions given by Boolean ’and’ and ’or’. 

Given daily return data for any asset as from January 1991 to December 2000 

(m=2506) sorted in descending order raises the question, how to rank equal returns. 

The simplest way to calculate R (j) 

m,1 is to write an algorithm that first creates a returndata 

table sorted in descending order from which it top down picks out values and 

searches for equivalents in the historical return data table and then displaces the found 

equivalents by increasing rank numbers. Equal return values receive equal rank, described 

by natural numbers, whereas the step size always equals one. Looking at the 

resulting rank tables for different assets, we observe that maximal rank numbers vary. 

Therefore we would have to adapt threshold k to the different assets. 

Another way to calculate R (j) 

m1 and R (j) 

m2, which after my opinion is more reasonable, 

is to adapt the rank numbers to the weight of values, described by the number of equal 

return values. This means that equal return values still receive equal ranks but larger 

step sizes to preceding and successive ranks indicating the number (weight) of equal 

return values. To achieve this, I included the following relation to the above described 

code: 

105

R eq 

m,· = 

� n 

j=1 Rpre 

m,· + j 

, (3.84) 

n 

where R pre 

m,· denotes the rank previous to the set of n equal return-values ranked by R eq 

m,· 

This has the advantage that for each return table with equal amount of data measured 

simultaneously (i.e. m = 2506), maximal ranks are equal to m, which leaves threshold 

k consistent with all samples. To have equal k for different assets and index S&P 500 

is a big advantage in terms of the implementation because in all of the four equations 

(3.79), (3.80), (3.81), and (3.82) tail copulas are estimated by the summing up of 

indicator functions by certain conditions and then by division of the resulting sum by 

threshold k. In case of different k for asset and index S&P 500, we would have to divide 

by the mean of km,1 and km,2. This would lead to distortion of the estimate. 

To check for consistency of my results I added a small noise of amplitude (10 −6 ) 

to all my returns that none of them were equal anymore. In this way I removed the 

degeneracy evolved from the little modification described by equation (3.84). The 

results remained consistent up to the decimal places that were of interest. Therefore I 

can be pretty sure that there is no significant distortion caused by this modification. 

Table (3.24) shows the results of tail dependence coefficients ˆ λU,m, ˆ λL,m, ÊV T λU,m , and 

ˆλ EV T 

L,m of the index S&P 500 and the nine assets calculated for the shorter reference 

period and table (3.25) for the longer reference period, both at a threshold k equal to 

4% of total data m. Results achieved by adding some random noise are also shown for 

comparison 16 . As mentioned I analysed exactly the same data as for the calculations 

according to Sornette & Malevergne but decided not to present a version that neglects 

most extreme values because with the actual approaches quantitative return data does 

not affect the result, as by definition of the estimations based on the findings of Schmidt 

& Stadtmüller we only consider rank vectors representing relative magnitudes within 

historical return tables. Therefore extreme outliers don’t distort the results at all. 

To summarize for both upper and lower tails: upper and lower Tail dependence coefficients 

according to Schmidt & Stadtmüller were calculated by the following relations: 

and 

ˆλL,m = 1 

k 

ˆλU,m = 1 

k 

ˆλ 

EV T 

L,m 

ˆλ 

EV T 

U,m 

m� 

j=1 

m� 

j=1 

= 2 − 1 

k 

= 2 − 1 

k 

1 (j) 

R m1≤k and R(j) 

m2≤k 1 (j) 

R m1 >m−k and R(j) 

m2 >m−k 

m� 

j=1 

m� 

j=1 

1 (j) 

R m1≤k or R(j) 

m2≤k 1 (j) 

R m1 >m−k or R(j) 

m2 >m−k 

16 The matlab m-file for the implementation of λU,m, λL,m, λ EV T 

(3.85) 

(3.86) 

(3.87) 

(3.88) 

U,m , and λL,m estimated by approaches 

according to Schmidt & Stadtmüller is denoted by: Lambda Schmidt.m and enclosed to the appendix and to 

the data CD 

106 

EV T

ˆλU,m 

ˆλL,m 

ˆλ EV T 

U,m 

ˆλ EV T 

L,m 

m=2507 

k 100.28 100.28 100.28 100.28 

w. noise w. noise w. noise w. noise 

BMY 0.19 0.19 0.21 0.21 0.18 0.19 0.22 0.21 

CVX 0.18 0.17 0.20 0.20 0.17 0.17 0.21 0.20 

HPQ 0.25 0.25 0.27 0.27 0.23 0.25 0.27 0.27 

KO 0.27 0.26 0.23 0.23 0.25 0.26 0.23 0.23 

MMM 0.16 0.15 0.24 0.24 0.15 0.15 0.24 0.24 

PG 0.22 0.21 0.19 0.19 0.21 0.21 0.20 0.19 

SGP 0.14 0.13 0.23 0.23 0.13 0.13 0.23 0.23 

TXN 0.11 0.11 0.09 0.09 0.10 0.11 0.10 0.09 

WAG 0.19 0.19 0.27 0.27 0.18 0.19 0.27 0.27 

Table 3.24: Estimated values of upper and lower tail dependence for index S&P 500 and a set 

of nine major assets traded on the New York stock Exchange calculated by non-parametric 

approaches according to Schmidt & Stadtmüller. The tail represents the most extreme 4% of 

the return values during a time interval from January 1991 to December 2000 with a threshold 

value of k = 0.04 · m = 100.3. Columns denoted by ’w. noise’ correspond to columns on their 

right and are performed by the same approaches but with a small random noise of amplitude 

(10 −6 ) added to the returns that no two returns are equal anymore. 

whereas m denotes the size of the return sample i.e. number of daily observations, k 

again denotes the threshold typically assumed as 4% of m, and R (j) 

m1 and R (j) 

m2 with 

j = 1, . . .,m represent rank ordered statistics of index returns Y (j) and asset returns 

X (j) calculated by: R eq 

m,· = 

�n j=1 Rpre m,·+j 

denoting the ranks of equal returns and R pre 

m,· 

to the set of considered equal returns. 

3.4.3 Analysis of Coefficients 

n 

if we have n equal return values with R eq 

m,· 

denoting the rank of the return(s) previous 

Looking at equations (3.85), (3.86), (3.87), and (3.88) we can observe that ˆ λU,m, ˆ λL,m, 

ˆλ EV T 

U,m , and ÊV T λL,m all depend on threshold k, which is an indicator of what is assumed 

to represent the extreme tail of a certain data set. In our result tables, table (3.24) 

and table (3.25) we assumed the most extreme 4 % of data ordered by increasing or 

decreasing values to constitute the extreme tail. It might now be interesting to check for 

sensitivity of my tail dependence coefficient estimates to different choices of threshold 

k. For this purpose I plotted ˆ λL,m, ÊV T λL,m , ˆ λU,m, and ÊV T λU,m in dependence of k on an 

interval ranging from 0 % up to 6 % of total data m. Figure (3.58) shows the two plots 

at the example of the nine assets for the upper tails and figure (3.59) for the lower tails. 

It is interesting to observe that ˆ λL,m seems to be a lower bound of ÊV T λ 

L,m and ˆ λU,m seems 

to be a upper bound of ÊV T λU,m . Furthermore ˆ λL,m and ˆ λU,m seem to be less sensitive to 

small changes of threshold k than ÊV T λL,m and ÊV T λU,m . In fact when I choose step sizes 

equal to my available data (low data density), that means without scaling of my x-axis 

in terms of refining the k-density, then the curves of ÊV T λU,m and ˆ λU,m become nearly 

identical. This holds also for negative tails and might be an indication for the Boolean 

107

ˆλU,m 

ˆλL,m 

ˆλ EV T 

U,m 

ˆλ EV T 

L,m 

m=5736 

k 229.40 229.40 229.40 229.40 

w. noise w. noise w. noise w. noise 

BMY 0.22 0.22 0.28 0.28 0.22 0.22 0.28 0.28 

CVX 0.17 0.17 0.27 0.27 0.17 0.17 0.27 0.27 

HPQ 0.28 0.28 0.29 0.29 0.28 0.28 0.29 0.29 

KO 0.27 0.27 0.28 0.28 0.26 0.27 0.29 0.28 

MMM 0.25 0.25 0.27 0.27 0.25 0.25 0.27 0.27 

PG 0.19 0.19 0.20 0.20 0.19 0.19 0.20 0.20 

SGP 0.17 0.17 0.26 0.26 0.17 0.17 0.27 0.26 

TXN 0.17 0.17 0.16 0.16 0.17 0.17 0.17 0.16 

WAG 0.21 0.21 0.29 0.29 0.20 0.21 0.29 0.29 

Table 3.25: Estimated values of upper and lower tail dependence for index S&P 500 and a set 

of nine major assets traded on the New York stock Exchange calculated by non-parametric 

approaches according to Schmidt & Stadtmüller. The tail represents the most extreme 4% 

of the return values during a time interval from July 1985 to April 2008 with a threshold 

value of k = 0.04 · m = 229.4. Columns denoted by ’w. noise’ correspond to columns on their 

right and are performed by the same approaches but with a small random noise of amplitude 

(10 −6 ) added to the returns that no two returns are equal anymore. 

’or’ criterion applied to real financial data to exhibit higher fluctuations compared to 

the Boolean ’and’ criterion. 

108

EVT 

λ , λ 

U,m U,m 

EVT 

λ , λ 

U,m U,m 

109 

EVT 

λ , λ 

U,m U,m 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

BMY 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

KO 

SGP 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

EVT 

λ , λ 

U,m U,m 

EVT 

λ , λ 

U,m U,m 

EVT 

λ , λ 

U,m U,m 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

CVX 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

MMM 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

EVT 

λ , λ 

U,m U,m 

EVT 

λ , λ 

U,m U,m 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 

k/m 

EV T 

Figure 3.58: ˆ λU,m for upper tails of index S&P 500 and the nine assets plotted in black and ˆ λ 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

TXN 

U,m 

EVT 

λ , λ 

U,m U,m 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

HPQ 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

PG 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

WAG 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

for upper tails plotted in red, both in dependence 

of threshold k on an interval from k/m = 0% ... 6% for m = 2507 daily return values during a time interval ranging from January 1991 to December 

2000.

110 

EVT 

λ , λ 

L,m L,m 

EVT 

λ , λ 

L,m L,m 

EVT 

λ , λ 

L,m L,m 

0.4 

0.3 

0.2 

0.1 

BMY 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.4 

0.3 

0.2 

0.1 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.4 

0.3 

0.2 

0.1 

KO 

SGP 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

EVT 

λ , λ 

L,m L,m 

EVT 

λ , λ 

L,m L,m 

EVT 

λ , λ 

L,m L,m 

0.4 

0.3 

0.2 

0.1 

CVX 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.4 

0.3 

0.2 

0.1 

MMM 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.4 

0.3 

0.2 

0.1 

TXN 

0 

0 0.01 0.02 0.03 0.04 0.05 0.06 

k/m 

EV T 

L,m 

Figure 3.59: ˆ λL,m for lower tails of index S&P 500 and the nine assets plotted in black and ˆ λ 

EVT 

λ , λ 

L,m L,m 

EVT 

λ , λ 

L,m L,m 

EVT 

λ , λ 

L,m L,m 

0.4 

0.3 

0.2 

0.1 

HPQ 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.4 

0.3 

0.2 

0.1 

PG 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

0.4 

0.3 

0.2 

0.1 

WAG 

0 

0 0.01 0.02 0.03 

k/m 

0.04 0.05 0.06 

for lower tails plotted in red, both in dependence 

of threshold k on an interval from k/m = 0% ... 6% for m = 2507 daily return values during a time interval ranging from January 1991 to December 

2000.

Estimation of optimal Threshold k 

To estimate the optimal threshold k for my tail dependence estimates according to 

Schmidt & Stadtmüller it is proposed in chapter seven of paper [24] (2005) to use a 

simple plateau finding algorithm after smoothing the plots of ˆ λL,m(k), ÊV T λL,m (k), ˆ λU,m(k), 

and ÊV T λU,m (k) by some box kernel. To make the results comparable to my further 

estimates, I wanted to choose a threshold value k around 4% of total data. Therefore I 

first wanted to smooth out my function of estimated tail dependence in dependence of 

threshold k by using a kernel with a bell-type shape and unit integral (i.e. the function 

is normalized). Basically, I wanted to replace a data point by the function centered on 

its abscissa times an amplitude equal to the ordinate of that data point by choosing a 

Gaussian or just a window function (zero outside of an interval and constant inside the 

window). The general formula of the approach is as follows: 

λsmoothed(k) = ˆ f(k) · λ(k) (3.89) 

n� 

� � 

with f(k) ˆ 

1 k − kj 

= K 

(3.90) 

nh h 

The kernel estimator ˆ f with kernel K is a sum of boxes or ’bumps’ placed at the 

observations. If K satisfies the condition: � ∞ 

−∞ K(k)dk = 1, ˆ f represents a probability 

density function of my estimates that weights the observations with window width h 

and n denoting the number of observations. 

After some tries I realized that whatever width of the filter I used, the supposedly 

smooth function remained sharp toothed. The problem is that the local density of data 

in my case is constant over the whole plot interval. ˆ f by definition only considers values 

of the x-axis for the weighting. Because these values in my case are equally distributed, 

the only solution of ˆ f, if the parameter of window width is adapted properly, could be 

equal to 1 and therefore λsmoothed becomes equal to λ indicating that this smoothing 

approach has no effect. 

Therefore I think that we should choose another smoothing method: A locally 

weighted scatter plot smooth using least squares quadratic polynomial fitting yielded 

the best results. It has a strong smoothing effect and still leaves the plots accurate 

enough. The regression weights for each data point in the span d(k) are given by the 

� � � 

� � 

tricube function: wi = 1− � 3�3 , whereas a second degree least squares regression 

was performed: 

S = 

n� 

j=1 

r 2 j = 

� k−ki 

d(k) 

j=1 

n� 

wj(yj − ˆyj) 2 with ˆyj = p1 · k 2 j + p2 · kj + p3 (3.91) 

j=1 

Figure (3.60) shows smoothing of ˆ λL,m(k) and ˆ λU,m(k) for all assets and the smaller 

interval and figure (3.61) for the bigger interval by setting d(k) = 0.125 of the considered 

interval k/m = 0% . . .8% or in my case of constant local data density a span of 

25 values (out of 2507) for the smaller sample and a span of 57 values (out of 5736) 

for the bigger sample. k = 4% seems to be a good choice for the thresholds also for 

estimates according to Schmidt & Stadtmüller. Most of the plots show stiff behaviour 

betwenn k/m = 3% . . .5%. Therefore and to be able to compare the results estimated 

by methods according to Sornette & Malevergne I decided to stick with k = 4%. 

111

λ BMY 

λ KO 

112 

λ SGP 

0.4 

0.3 

0.2 

0.1 

0 

0 0.02 0.04 

k/m 

0.06 0.08 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

0 0.02 0.04 

k/m 

0.06 0.08 

0.4 

0.3 

0.2 

0.1 

0 

0 0.02 0.04 0.06 0.08 

k/m 

λ CVX 

λ MMM 

λ TXN 

0.3 

0.2 

0.1 

0 

−0.1 

0 0.02 0.04 

k/m 

0.06 0.08 

0.3 

0.2 

0.1 

0 

−0.1 

0 0.02 0.04 

k/m 

0.06 0.08 

0.15 

0.1 

0.05 

0 

−0.05 

0 0.02 0.04 0.06 0.08 

k/m 

λ HPQ 

λ PG 

λ WAG 

0.4 

0.3 

0.2 

0.1 

0 

−0.1 

0 0.02 0.04 

k/m 

0.06 0.08 

0.4 

0.3 

0.2 

0.1 

0 

0 0.02 0.04 

k/m 

0.06 0.08 

0.8 

0.6 

0.4 

0.2 

0 

0 0.02 0.04 0.06 0.08 

Figure 3.60: Smoothed ˆ λL,m plotted in black and smoothed ˆ λU,m plotted in green for the nine assets with the index S&P 500 given for the most 

extreme 8% during a time interval from January 1991 to December 2000. Smoothing is performed by locally weighted scatter plot smooth using 

least squares quadratic polynomial fitting filters with 25 considered data points by step or [0.125 · 0.08 · m] values with a total of m = 2507 data 

points and [·] denoting integer numbers. 

k/m

λ BMY 

λ KO 

113 

λ SGP 

0.4 

0.3 

0.2 

0.1 

0 

0 0.02 0.04 

k/m 

0.06 0.08 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

0 0.02 0.04 

k/m 

0.06 0.08 

0.4 

0.3 

0.2 

0.1 

0 

−0.1 

0 0.02 0.04 0.06 0.08 

k/m 

λ CVX 

λ MMM 

λ TXN 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

0 0.02 0.04 

k/m 

0.06 0.08 

0.4 

0.3 

0.2 

0.1 

0 

−0.1 

0 0.02 0.04 

k/m 

0.06 0.08 

0.2 

0.15 

0.1 

0.05 

0 

−0.05 

0 0.02 0.04 0.06 0.08 

k/m 

λ HPQ 

λ PG 

λ WAG 

0.4 

0.3 

0.2 

0.1 

0 

0 0.02 0.04 

k/m 

0.06 0.08 

0.3 

0.2 

0.1 

0 

−0.1 

0 0.02 0.04 

k/m 

0.06 0.08 

0.4 

0.3 

0.2 

0.1 

0 

0 0.02 0.04 0.06 0.08 

Figure 3.61: Smoothed ˆ λL,m plotted in black and smoothed ˆ λU,m plotted in green for the nine assets with the index S&P 500 given for the most 

extreme 8% during a time interval from July 1985 to April 2008. Smoothing is performed by locally weighted scatter plot smooth using least 

squares quadratic polynomial fitting filters with 57 considered data points by step or [0.125 · 0.08 · m] values with a total of m = 5736 data points 

and [·] denoting integer numbers. 

k/m

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

0 1 2 3 4 5 6 7 8 

θ 

λ L 

0.2 

0.18 

0.16 

0.14 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

0 

0 1 2 3 4 5 6 7 8 

Figure 3.62: Lower tail dependence λL(θ) = 2−1/θ and corresponding asymptotic variance 

σ2 L (θ) = 2−1/θ − 3 

24−1/θ + 1 

28−1/θ for the Pareto copula. 


Schmidt & Stadtmüller proposed in chapter (5) of paper [24] (2005) a method to approximate 

the unknown asymptotic variance of tail dependence estimates obtained by 

their methods. They choose the Pareto copula as a simple but flexible parametric copula, 

calculated its tail copula, and utilized the corresponding variance functional σ 2 L (θ) 

as an approximation of the unknown asymptotic variance. The tail copula of the Pareto 

copula is given by: 

C(u, v) = max � [u −θ + v −θ − 1] −1/θ , 0 � , θ ∈ [−1, ∞) \ 0. 

Further the lower tail copula exists for θ > 0 and can be expressed by: 

The asymptotic variance was given by: 

ΛL(x, y) = (x −θ + y −θ ) −1/θ . 

σ 2 L(θ) = 2 −1/θ − 3 

2 4−1/θ + 1 

2 8−1/θ , (3.92) 

where θ was replaced by the maximum likelihood estimator ˆ θ. 

It can be shown that the Pareto copula is lower tail dependent with lower tail 

dependence coefficient λL = ΛL(1, 1) = 2−1/θ for θ > 0, (i.e. page 77 of paper [28] 

(2008)). Given our estimated results of ˆ λU,m and ÊV T λU,m , σ2 L (ˆ θ) can be calculated. Figure 

(3.62) shows plots of λL(θ) and σ2 L (θ). 

Looking at the result table for my tail dependence estimates according to Schmidt & 

Stadtmüller in tables (3.24) and (3.25) we observe that all of the estimates are between 

0.1 and 0.3, which yields ˆ log(2) 

θ = −log{λ∈(0.1,0.3)} 

∈ (0.3, 0.6) and finally σ2 L (ˆ θ) ∈ (0.08, 0.18). 

114 

θ 

2 

σ 

L

The right hand side of figure (3.62) shows that this is exactly the area where σ 2 L (θ) 

peaks. To achieve results of higher accuracy, meaning that the variance functional 

σ 2 L (ˆ θ) becomes lower, we would need very high estimates of lower tail dependence 

coefficients ˆ λL,m. Even if our estimates were around 0.7 we would still calculate standard 

deviations of around 50% of the estimates by the variance functional. This, as we will 

see shortly, would still be about twice the amount of bootstrap sampling standard 

deviation estimates. Therefore I don’t think that this approximation approach has 

meaningful appliance to my data. 

Now I come to the estimation of error bars by the bootstrap method. I built bootstrap 

samples of S&P 500 index return data Y1,m, Y2,m, . . .,Ym,m, where m denotes the 

total number of values, and searched for the asset return data corresponding in time. 

For the smaller sample consisting of m = 2507 data points I built 1000 bootstrap samples 

and for the bigger sample consisting of m = 5736 data points I built 800 bootstrap 

samples because this already required considerable computing power. Table (3.26) 

shows results of bootstrap sampling for the smaller data set and table (3.27) shows the 

results for the bigger data set 17 . 

Relative errors of the mean values of bootstrap sample estimates compared to original 

sample estimates denoted by ’rel err’ were small for each asset by both approaches ˆ λ 

and λ ˆ EV T : the biggest relative error was calculated for λbs,mean of the upper tail of 

asset Minnesota Mining & MFG (MMM) shown in table (3.26) for smaller sample 

bootstrap estimates. It showed 12 % relative deviation from the real value. Performing 

this calculation again yielded similar values. Here again the assumption of errors to 

be approximately Gaussian distributed seems to hold as calculated standard deviations 

within and between the different bootstrap samples are very close of being equal to 

half of 95% quantiles. This holds for both sample sizes. The standard deviations 

between the bootstrap sample estimates of the smaller sample were found to be around 

18% up to maximally 30% of total tail dependence estimates. The lower the tail 

dependence estimates the less accurate were the quantiles and standard deviations. For 

the bigger sample relative deviations were significantly lower. Here standard deviations 

were around 10% to 15% of total tail dependence estimates. 

The results obtained by approaches according to Schmidt & Stadtmüller seem to be 

very consistent. Tables (3.24) and (3.25) also show that, compared to other methods, 

estimates are close to each other, whereas estimates for lower tails tend to be higher 

than estimates for upper tails like it used to be the case for the non-parametric and 

the parametric approach according to Sornette & Malevergne applied to the smaller 

data set and using corrected ˆl and Ĉ given by ˆlc and Ĉc. This also holds for the nonparametric 

approach according to Poon, Rockinger, and Tawn using corrected ˆ L given 

by ˆ Lc. 

17 The matlab m-file for the bootstrap sampling with replacement of λU,m, λL,m, λ EV T 

U,m , and λL,m estimated 

by approaches according to Schmidt & Stadtmüller is denoted by: bootstr Schmidt.m and enclosed to the data 

CD 

115 

EV T

116 

m=2507 bs=1000 


λbs,mean 

λbs,mean 

λ·,m k=100.28 rel err max dev 95% q 90% q std bs k=100.28 rel err max dev 95% q 90% q std bs 

BMY 0.20 0.03 0.11 0.07 0.06 0.04 0.21 0.00 0.19 0.08 0.07 0.04 

CVX 0.18 0.01 0.09 0.08 0.06 0.04 0.19 0.02 0.20 0.07 0.06 0.04 

HPQ 0.25 0.01 0.10 0.08 0.07 0.04 0.27 0.00 0.09 0.08 0.07 0.04 

KO 0.26 0.05 0.19 0.08 0.07 0.04 0.22 0.02 0.10 0.08 0.07 0.04 

MMM 0.17 0.08 0.08 0.08 0.07 0.04 0.24 0.00 0.11 0.08 0.06 0.04 

PG 0.21 0.03 0.11 0.08 0.06 0.04 0.19 0.00 0.12 0.07 0.06 0.04 

SGP 0.14 0.02 0.10 0.07 0.06 0.04 0.23 0.02 0.09 0.08 0.06 0.04 

TXN 0.11 0.01 0.07 0.06 0.05 0.03 0.09 0.04 0.08 0.05 0.05 0.03 

WAG 0.18 0.04 0.10 0.08 0.06 0.04 0.27 0.01 0.12 0.08 0.07 0.04 

λEV T 

·,m 

BMY 0.19 0.06 0.12 0.07 0.06 0.04 0.22 0.02 0.11 0.08 0.07 0.04 

CVX 0.17 0.02 0.10 0.08 0.07 0.04 0.20 0.00 0.09 0.07 0.06 0.04 

HPQ 0.24 0.02 0.09 0.08 0.07 0.04 0.28 0.01 0.10 0.08 0.07 0.04 

KO 0.25 0.04 0.20 0.08 0.07 0.04 0.23 0.00 0.11 0.08 0.07 0.04 

MMM 0.16 0.12 0.11 0.08 0.07 0.04 0.25 0.02 0.12 0.07 0.06 0.04 

PG 0.20 0.01 0.12 0.08 0.06 0.04 0.20 0.02 0.08 0.07 0.06 0.04 

SGP 0.13 0.06 0.10 0.07 0.06 0.04 0.24 0.04 0.17 0.08 0.06 0.04 

TXN 0.10 0.03 0.09 0.06 0.05 0.03 0.10 0.08 0.10 0.06 0.05 0.03 

WAG 0.17 0.01 0.10 0.08 0.07 0.04 0.28 0.02 0.13 0.09 0.07 0.04 

Table 3.26: Establishing the uncertainty of non-parametrically estimated upper tail dependence coefficients ˆ λU,m and ˆ λEV T 

U,m and lower tail dependence 

coefficients ˆ λL,m and ˆ λEV T 

L,m 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% of the return 

values during a time interval from January 1991 to December 2000.

117 

m=5736 bs=800 


λbs,mean 

λbs,mean 

λ·,m k=229.40 rel err max dev 95% q 90% q std bs k=229.40 rel err max dev 95% q 90% q std bs 

BMY 0.22 0.01 0.07 0.05 0.04 0.02 0.28 0.00 0.10 0.06 0.05 0.03 

CVX 0.18 0.03 0.09 0.05 0.04 0.02 0.27 0.01 0.09 0.06 0.05 0.03 

HPQ 0.28 0.01 0.10 0.05 0.04 0.03 0.28 0.02 0.10 0.05 0.04 0.03 

KO 0.27 0.03 0.11 0.06 0.05 0.03 0.29 0.02 0.08 0.05 0.04 0.03 

MMM 0.25 0.02 0.09 0.05 0.04 0.03 0.26 0.01 0.09 0.05 0.04 0.03 

PG 0.20 0.02 0.09 0.05 0.04 0.03 0.20 0.01 0.09 0.05 0.04 0.02 

SGP 0.17 0.02 0.08 0.05 0.04 0.02 0.26 0.00 0.09 0.05 0.05 0.03 

TXN 0.17 0.00 0.07 0.05 0.04 0.02 0.16 0.02 0.07 0.05 0.04 0.02 

WAG 0.21 0.01 0.08 0.05 0.04 0.03 0.28 0.03 0.10 0.06 0.05 0.03 

λEV T 

·,m 

BMY 0.22 0.00 0.08 0.05 0.04 0.03 0.28 0.01 0.10 0.06 0.05 0.03 

CVX 0.17 0.04 0.08 0.05 0.04 0.02 0.27 0.00 0.09 0.06 0.05 0.03 

HPQ 0.28 0.01 0.09 0.05 0.05 0.03 0.29 0.01 0.11 0.05 0.04 0.03 

KO 0.27 0.04 0.10 0.06 0.05 0.03 0.29 0.02 0.08 0.05 0.05 0.03 

MMM 0.24 0.01 0.09 0.05 0.04 0.03 0.27 0.00 0.09 0.05 0.04 0.03 

PG 0.19 0.03 0.09 0.05 0.04 0.03 0.20 0.00 0.09 0.05 0.04 0.03 

SGP 0.17 0.01 0.08 0.05 0.04 0.02 0.27 0.00 0.09 0.05 0.05 0.03 

TXN 0.17 0.01 0.07 0.05 0.04 0.02 0.16 0.00 0.07 0.05 0.04 0.02 

WAG 0.21 0.02 0.08 0.05 0.04 0.03 0.29 0.02 0.09 0.06 0.05 0.03 

Table 3.27: Establishing the uncertainty of non-parametrically estimated upper tail dependence coefficients ˆ λU,m and ˆ λEV T 

U,m and lower tail dependence 

coefficients ˆ λL,m and ˆ λEV T 

L,m by creating 800 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% of the return 

values during a time interval from July 1985 to April 2008.


I also implemented the rolling horizon time window to the Schmidt & Stadtmüller 

estimator for the three window sizes S = 2500, S = 1600, and S = 800. Figures (3.63), 

(3.64), and (3.65) show the results. We can see that for most of the time left-tail 

dependence ˆ λL,m plotted in black dominates right-tail dependence ˆ λU,m given in green. 

The bigger the time window, the more obvious this becomes. In terms of trends it is 

difficult to draw conclusion because plots for different window sizes look a little different 

from each other. Assuming that plots with window size S = 2500 given in figure (3.63) 

represent something like a smoothed version of the other plots with smaller window 

sizes we get the impression that tail dependence for both positive and negative tails 

have become remarkably constant recently. Comparing the right part of the plots, 

representing estimates from 1985 to 1995, to the left part of the plots, representing 

estimates from 2000 to 2008, the left side is more fluctuating than the right side. For 

most assets it is also this area, where ˆ λL,m grows clearly beyond ˆ λU,m. It is not clear 

whether there is in general a decreasing or an increasing tendency of ˆ λL,m and ˆ λU,m. 

Also here asset Texas Instruments inc. (TXN) looks different from the others as it 

shows a clear increasing trend over time. Overall it is remarkable how consistent most 

of the assets remain over time. Except asset TXN, all assets are fluctuating around a 

constant value. Even if we look at figure (3.65), which shows estimates for the smallest 

windows of size S = 800 fluctuations remain limited. This was not the case for i.e. first 

and second β-smile conditions. 

Error Bars in Dependence of Time 

To establish uncertainty of estimated upper and lower tail dependence estimates in 

dependence of time I performed bootstrap sampling for the rolling time windows 18 . 

Therefore I used a window size of S = 1600. The Schmidt and Stadtmüller approaches 

request the highest computing power of all implemented concepts. 

Because bootstrap estimates showed good accuracy I think that S = 1600 daily observations 

per estimate is enough for this approach. Results provided on the data CD 

show that bootstrap quantiles remain stable and low enough for the chosen window 

size. 95% quantiles are around 40% to 50% of total estimates for upper and lower tails 

of most assets. This yields standard deviations around 20% to 25% of the estimates, 

which is comparably low. ÊV T λL,m and ÊV T λU,m perform similarly to ˆ λL,m and ˆ λU,m in terms 

of bootstrap quantile estimations. 

18 Results of bootstrap sampling of the rolling time window estimates for λU,m, λL,m, λ EV T 

U,m , and λL,m and 

window size of S = 1600 data points can be observed on the data CD submitted with the thesis in the folder: 

window results 

118 

EV T

119 

λ 

λ 

λ 

0.35 

0.3 

0.25 

0.2 

0.15 

BMY 

0.1 

2500 3000 3500 4000 4500 5000 5500 6000 

0.5 

0.4 

0.3 

0.2 

m 

0.1 

2500 3000 3500 4000 4500 5000 5500 6000 

0.35 

0.3 

0.25 

0.2 

0.15 

KO 

m 

SGP 

0.1 

2500 3000 3500 4000 4500 5000 5500 6000 

λ 

λ 

λ 

0.35 

0.3 

0.25 

0.2 

0.15 

CVX 

0.1 

2500 3000 3500 4000 4500 5000 5500 6000 

0.5 

0.4 

0.3 

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MMM 

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2500 3000 3500 4000 4500 5000 5500 6000 

0.2 

0.15 

0.1 

0.05 

m 

TXN 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

λ 

λ 

λ 

0.32 

0.3 

0.28 

0.26 

0.24 

0.22 

HPQ 

0.2 

2500 3000 3500 4000 4500 5000 5500 6000 

0.35 

0.3 

0.25 

0.2 

0.15 

m 

PG 

0.1 

2500 3000 3500 4000 4500 5000 5500 6000 

0.5 

0.4 

0.3 

0.2 

m 

WAG 

0.1 

2500 3000 3500 4000 4500 5000 5500 6000 

m 

m 

m 

Figure 3.63: λU,m(m) for upper tails of index S&P 500 and the nine assets plotted in green and λL,m(m) for lower tails plotted in black using a 

non-parametric approach given by equation ˆ λL,m = 1 �m k j=1 1 R (j) 


m2≤k and ˆ λU,m = 1 �m k j=1 1 R (j) 

for rolling time horizon 


m2 >m−k 

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.

120 

λ 

λ 

λ 

0.5 

0.4 

0.3 

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BMY 

0.1 

1000 2000 3000 4000 5000 6000 

0.5 

0.4 

0.3 

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0.35 

0.3 

0.25 

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KO 

m 

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λ 

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m 

m 

m 







m2 >m−k 


Mars 2008.

121 

λ 

λ 

λ 

0.5 

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0.1 

BMY 

0 

0 1000 2000 3000 4000 5000 6000 

0.8 

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0.5 

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m 

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λ 

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Mars 2008.

122 

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0.4 

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0 

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0.5 

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0.5 

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m 

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0 

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λ 

λ 

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0 

2500 3000 3500 4000 4500 5000 5500 6000 

0.4 

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0.4 

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0 

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λ 

λ 

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0 

2500 3000 3500 4000 4500 5000 5500 6000 

0.5 

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m 

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0 

2500 3000 3500 4000 4500 5000 5500 6000 

0.4 

0.3 

0.2 

0.1 

m 

WAG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

m 

m 

m 

Figure 3.66: λL,m(N) for lower tails of index S&P 500 and the nine assets for rolling time horizon windows of size S = 2500 plotted in blue, 

S = 1600 plotted in green, and S = 800 plotted in red using a non-parametric approach given by equation ˆ λU,m = 1 �m k j=1 1 R (j) 


m2 >m−k 

from N = (2501 − S ...S),(2502... S + 1),... ,(5736 − S + 1... 5736).

123 

λ 

λ 

λ 

0.5 

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BMY 

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m 

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m 

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SPG 

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m 

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λ 

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m 

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m 

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λ 

λ 

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0 

2500 3000 3500 4000 4500 5000 5500 6000 

m 

0.4 

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0 

2500 3000 3500 4000 4500 5000 5500 6000 

m 

0.5 

0.4 

0.3 

0.2 

0.1 

WAG 

0 

2500 3000 3500 4000 4500 5000 5500 6000 

m 

Figure 3.67: λU,m(N) for lower tails of index S&P 500 and the nine assets for rolling time horizon windows of size S = 2500 plotted in blue, 

S = 1600 plotted in green, and S = 800 plotted in red a using non-parametric approach given by equation ˆ λL,m = 1 

k 

N = (2501 − S ...S),(2502... S + 1),... ,(5736 − S + 1...5736). 

� m 

j=1 1 R (j) from 


m2≤k

3.5 Conclusions 

Within this section I want to compare the results of the different concepts and conclude 

about which approaches have more practical application and which approaches are more 

of theoretical or supplemental interest. 

It is interesting to observe that by the three main concepts: approaches according to 

Poon, Rockinger, and Tawn with respective non-parametric upper and lower tail dependence 

estimator denoted by ˆχ, approaches according to Sornette & Malevergne with 

respective parametric and non-parametric tail dependence estimators λ +,− including 

β-smile improvement conditions, and approaches according to Schmidt & Stadtmüller 

with respective non-parametric tail dependence estimators denoted by λL,m and λU,m 

presented in this chapter all differ significantly in their results. 

The estimator ˆχ that was covered in the first section (3.1) yielded results reported 

in tables (3.1) and (3.2), that were all close to one. This would mean that all of our 

considered assets show a nearly perfect extremal dependence with the index S&P 500 

for corrected upper and lower tails. Furthermore, most of the estimates were close 

together or even identical. Therefore results are not of much use. The only column 

that is different from the others reports estimates for ˆχ yielded by approximations 

of univariate survival functions that were estimated without correction for the most 

extreme outliers for tail values with rank (1 . . .k) and k denoting the rank number of 

least extreme tail value, which is somewhat questionable. 

Results according to Sornette & Malevergne calculated by ˆ β of the whole data set 

reported in tables (3.8) and (3.10) using Hill’s estimator and reported in tables (3.9) 

and (3.11) using Gabaix’s estimator are consistent, whereas parametric estimators tend 

to be somewhat higher than non-parametric estimators. The only estimates where the 

performance of the parametric and the non-parametric approach was significantly different 

from each other was yielded applying the parametric approach to negative tails by 

approximation of corresponding tail distributions by univariate survival functions without 

correction for the most extreme outliers, namely by tail values with rank (1 . . .k), 

where k denotes the rank number of the least extreme tail values. Anyway plots of 

the parametrized survival function compared to the empirical complementary cumulative 

distribution function for the smaller data set plotted in figure (3.38) show big 

differences between the two distribution functions and again call the parametrization 

of survival copulas with uncorrected average values for the estimation of scale factors 

ĈY and Ĉε into question. Using the bigger data set we can overcome these difficulties 

by some extent, which can be observed in figures (3.31) and (3.33) and also in 

tables (3.10) and (3.11) showing estimates for the bigger data sets. In case of both 

non-parametric and parametric estimators we observe that estimates with corrected l 

and corrected C, calculated for tail values with rank c . . .k excluding the most extreme 

0.5%, are in most cases higher than estimates for the positive tails. Using the Gabaix 

estimator shown in tables (3.11) and (3.9) differences between upper and lower tail 

dependence estimates are more distinctive. This finding agrees well with previous literature. 

Comparing differences between tail dependence estimates for index S&P 500 

and the nine assets of the smaller and the bigger data sets the estimates are consistent 

in most cases. Overall, estimates between the different assets and the index are not as 

close together as for ˆχ, although the biggest values were estimated for assets Hewlett- 

Packard Co. (HPQ) and Coca-Cola Co (KO) around 12% to 14%, which intuitively 

seems very small. 

β-smile estimates for the first and second SI conditions given in table (3.18) and 

(3.19) sometimes were similar to the approaches using ˆ β calculated by all data and 

124

sometimes were significantly different. The problem here is that ˆ βSI are fluctuating 

and show large error bars. Mostly the β-smile estimators tend to be more radical than 

estimates calculated by classic ˆ β in terms of yielding more extreme results (close to 

one or close to zero). It might be good to have the β-smile improvement approaches 

as additional indication for very low or high tail dependence coefficient estimates and 

complex dependence structures but I would not trust in these estimates because of the 

high error bars obtained by bootstrap sampling. 

Coming to concepts according to Schmidt & Stadtmüller with results reported in 

tables (3.25) and (3.24) we observe that all the estimates lie in-between the two other 

concepts. They are bigger than estimates according to Sornette & Malevergne and 

smaller than estimates according to Poon, Rockinger, and Tawn. Also here, lower tail 

dependence estimates tend to be generally higher than tail dependence estimates of 

the upper tails. Estimates are consistent for the different time windows and they are 

rather close together for all assets. Studying the definition of the estimators proposed 

by Schmidt & Stadtmüller we find that the dependence structure in the tails is fully 

explained by the rank order of the input data. Absolute return values have no impact 

on the results. That means that highly inconvenient outliers are just weighted by their 

ranks and not in terms of their absolute value relative to the rest of data. This is 

different to the other estimators where we always have a certain impact of the extent 

of extreme outliers in terms of a proportion i.e. by ˆl in the non-parametric estimator or 

in terms of a proportion of scale factors Ĉε/ ĈY in the parametric approach proposed 

by Sornette & Malevergne. The χ estimator uses Z, which is the minimum value of tail 

distribution approximations by unit Fréchet marginals S and T of assets and index. 

Now we come to the error bar estimations of the different concepts: bootstrap estimates 

for error bars of ˆχ are given in tables (3.3) and (3.4). Looking at the smaller 

sample, quantiles are much bigger in the negative tails than in the positive tails for both 

uncorrected tails and tails corrected for most extreme outliers. For the bigger sample 

this only counts for uncorrected tails. Anyway quantiles for corrected tails seem to be 

low indicating that estimates are accurate. For uncorrected tails quantiles were very 

� 

Zk,N 

high for some assets. Using the proposed relation ˆσ = 

2k(N−k) N3 for the calculation 

of error bars is only recommended for tails that are corrected for outliers because these 

outliers seem to have a strong impact on the accuracy of the estimate for concepts 

according to Poon, Rockinger, and Tawn. 

Looking at the non-parametric and parametric approaches proposed by Sornette & 

Malevergne with bootstrap estimates of error bars for smaller and bigger data sets 

provided in tables (3.12) and (3.13) for the non-parametric approach and provided in 

tables (3.15) and (3.16) for the parametric approach the situation looks very similarly. 

Errors seem to be approximately Gaussian distributed for both parametric and nonparametric 

estimators. The only thing we have to keep an eye on is that the parametric 

approach seems to perform poorly with uncorrected negative tails primarily for the 

smaller data set. Correcting the tails for outliers helps to achieve stable estimates. The 

non-parametric approach seems to be less sensitive to outliers. 

Tail dependence estimates calculated by β-smile improvement conditions perform 

very poorly in terms of error bars that were estimated by bootstrap sampling with 

replacement shown in tables (3.20) and (3.21), for the non-parametric approach with 

ˆβSI calculated by the first SI condition applied to the smaller and the bigger data sets, 

and in tables (3.22) and (3.23), for the non-parametric approach with ˆ βSI calculated 

by the second SI condition applied to the smaller and the bigger data sets. Standard 

deviations ’std bs’ are mostly around the size of the estimates for both conditions. 

125

Now we come to bootstrap estimates performed for the Schmidt & Stadtmüller 

approaches shown in tables (3.24) and (3.25) for the smaller and the bigger data 

sets. Errors seem to be approximately Gaussian distributed and consistent without 

exception. Furthermore ’std bs’ seem to be comparably low for both smaller and bigger 

data sets. 

As a last point in this chapter I wanted to compare the rolling time window estimates 

of the different methods: it is remarkable how similar the estimators ˆ λU,m and 

ˆλL,m provided by Schmidt & Stadtmüller compared to the estimators ˆ λ +,− provided 

by Sornette & Malevergne look for some assets if we just consider the shapes of their 

respective plots in dependence of time. Also the extent of movements is very similar. 

The main difference seems to be that ˆ λU,m and ˆ λL,m generally perform on a higher 

estimation level. Dynamics in dependence of time seem to be similar to ˆ λ +,− . ˆ λ +,− 

estimated by SI conditions shows much stronger fluctuations than the other methods. 

For the implementation part of the thesis presented in the next chapter I will provide 

results estimated by all the different methods to observe the impact on their 

performance applying the concepts to different data sets. Anyway from this point it 

seems that the non-parametric estimator by Sornette & Malevergne and the also nonparametric 

estimator by Schmidt & Stadtmüller have more practical application than 

the rest of the estimators because of their accuracy in terms of error bars and their 

consistent performance for different time windows. 

126

Chapter 4 

Application of Concepts to various 

Data and Purposes 

After implementation of the different methods I wanted to extend the scope of this work 

by applying the concepts to various data and purposes and also show some interesting 

applications. 

In a first step described in section (4.1), I applied the concepts to indexes from 

major finance centers in the world and respective assets in a comparable way as on the 

example of S&P 500 and the nine assets to gain further experience by comparing the 

results. In section (4.2) I wanted to calculate tail dependence coefficient estimates for 

market indexes from different countries, to find out to what extent extreme measures 

depend on geographical distances, and then compare the results with tail dependence 

coefficient estimates of exchange rates to check whether they accord well with each 

other. Finally in section (4.3) I will implement the different approaches to synthetic 

time series to find out what the best result can be. Furthermore it will show me biases 

and give me the lower bounds for the errors of the different concepts. 

4.1 Application to major Financial Centers in the World 

I downloaded a lot of asset and index data for the implementation of the different 

concepts to the major financial centers in the world. Table (4.1) shows the indexes 

including abbreviations adopted from 1 and tables (4.2), (4.3), (4.4), and (4.5) give 

respective assets included in the indexes plus abbreviations also adopted from Yahoo 

finance. Most of the assets used for the calculations are among the stocks with the 

largest capitalization within their respective indexes, whereas their weight in the indexes 

should not be too high in order that dependence does not stem from their overlap with 

the market factor. Anyway I tried to choose a certain spectrum of different weights 

to observe the impact on the estimates 2 . A problem was that for most indexes I did 

not find exact index component weights. Only exceptions were Dow Jones Industrial 

Average and S&P 500 3 . One has to remember that even if I knew the exact component 

weights of the indexes, they could be altering in time as I sometimes consider time 

intervals covering several decades. Anyway this is primarily of high interest for my 

approaches according to Sornette & Malevergne because these approaches depend on 

1 http://finance.yahoo.com 

2 Asset and index data is provided on the data CD 

3 Information about index component weights for assets of S&P 500 and most important assets of Dow Jones 

Industrial Average index is provided on: http://www.indexarb.com/indexComponentWtSwitch.html updated 

on September-16-2008. 

127

the regression coefficient β of the linear additive single factor model X = β · Y + ε, 

which above has been found to be the crucial parameter for this type of models. If 

β is small, tail dependence estimators ˆ λ +,− (β) are low as well and vice versa. This 

has been recognized as a weakness of the model, which lead to the introduction of the 

β-smile improvement conditions, where βSI(k) was calculated only by tail return data 

up to threshold number k but unfortunately yielded highly volatile estimates. Using 

the classic β that is calculated by all data, though yields stable solutions, focuses on 

linear dependences over the whole return data range, whereas moderate return data is 

weighted equally like extreme data. 

First of all I defined a time interval for which I had complete data for all assets 

and indexes. The biggest ’common’ time interval ranges from begin of February 2003 

to end of August 2008 and contains, depending on the index of the financial center, 

from 1372 data points for Mercado de Valores (MERVAL) to 1433 data points for SSE 

Composite index (SSEC). Data for index Tel Aviv 100 (TA 100) and included assets 

was available not till end of August 2003 but the impact on tail dependence coefficients 

of the absent of these early data should be marginal. Anyway for some indexes there is 

much more data available i.e. Dow Jones Industrial Average (DOW) price data starting 

from January 1962 up to now consisting of more than 11’700 observations for assets 

Alcoa Inc. (AA), Boeing Co. (BA), and Caterpillar Inc. (CAT) or S%P 500 price data 

for assets Hewlett-Packard Co. (HPQ) and Coca-Cola Co. (KO) also starting from 

January 1962 up to now consisting of more than 11’600 observations. These huge data 

sets can be used to check for the consistency of the estimators over time. Anyway the 

target is to be able to draw conclusions with far less data as for example by our above 

described five and a half year interval from early 2003 up to now. 

First I implemented the parametric and the non-parametric approaches according to 

Sornette & Malevergne and already made a very unexpected finding: tail dependence 

estimates for all assets and indexes outside of the U.S. were significantly smaller than 

0.001. That means that their extreme dependence of the market factor would be close 

to zero, what intuitively seems to be highly questionable. All the U.S. indexes instead 

exhibited more or less reasonable estimates. My first idea was that maybe the market 

capitalization of the chosen assets might be too small but then, performing the estimates 

on assets with different known weights (assets of S&P 500 and some assets of Dow 

Jones Industrial Average), I realized that the only cause of vanishing tail dependence 

estimates for all but the U.S. indexes and corresponding assets were differences in 

estimated β because the estimated tail dependence coefficients are very sensitive to 

β calculated by the linear single factor model. If β < 0.2 ˆ λ +,− → 0. That means 

that when we observe a low dependence of moderate changes between asset and index, 

extreme dependence vanishes as well. This can be observed on table (4.6), which shows 

results for parametric and non-parametric ˆ λ +,− for a data set consisting of N = 2000 

daily returns of index S%P 500 and assets with different component weights denoted 

by ’C.W.’. 

I also observed on index S&P 500 that assets, which have a low weight in the index 

(< 0.5%) tend to have smaller β and therefore low ˆ λ +,− . What is interesting now 

is that this is no longer the case if we calculate βSI(k) by SI conditions: estimates 

of λ +,− that were close to zero when we use β of the whole sample could become 

significantly higher than zero when calculated using βSI calculated by first or second 

β-smile conditions. This can be observed in table (4.7) on the same data set as above 

consisting of N = 2000 data points for index S&P 500 and corresponding assets and 

would imply that assets that overall have a very low tendency to move simultaneously 

128

with the index can have much stronger dependence in their extreme tails. 

Additionally, I performed bootstrap sampling with replacement of my biggest data 

sets up to 11’640 data points to find out whether the estimates become more accurate if 

we boost the data sets. The relative errors between the results of the original sample and 

the means of all bootstrap samples became small but the standard deviations between 

the samples for all my estimates were still precisely of the size of the estimates for both 

the first and the second SI conditions. For comparison I added the results of the tail 

dependence estimates of the other U.S indexes Dow Jones Industrial Average (DOW) 

and respective assets included in table (4.8) for the parametric and non-parametric 

approaches with β calculated by all data and in table (4.9) for the non-parametric 

approach with βSI only calculated for tail data. NASDAQ Composite (NASDAQ) and 

respective assets are included in table (4.10) for both, β calculated by all data and βSI. 

The data sets also consists of N = 2000 data points, which has been identified to be 

the smallest size to provide stable estimates for the approaches according to Sornette & 

Malevergne. Assets with biggest weights in the NASDAQ Composite Index are i.e. Microsoft 

Corp. (MSFT) with around 9%, Intel Corp. (INTC) with around 6%, and Cisco 

Systems Inc. (CSCO) with around 5.4% but as mentioned, unfortunately I couldn’t 

access more precise data. Error bar results of bootstrap sampling with replacement for 

the indexes Dow Jones Industrial Average (DOW) and NASDAQ Composite (NAS- 

DAQ) were fully consistent with my S%P 500 results presented in subsections (3.2.3) 

and (3.2.5). 

Now I come to the other approaches according to Poon, Rockinger, and Tawn and 

according to Schmidt & Stadtmüller. In table (4.11) I show results at the example of 

Australian Securities Exchange index (AORD) and corresponding assets for the common 

data samples ranging from begin of February 2003 to end of August 2008 and 

consisting of N = 1398 daily return observations. That was the maximum common 

interval with data available for assets and index. Table (4.12) shows estimates by the 

same data but by the non-parametric approach according to Sornette & Malevergne 

with first and second β-smile conditions. 95% and 90% bootstrap error quantiles show 

on table (4.11) that for approaches according to Poon, Rockinger, and Tawn and according 

to Schmidt & Stadtmüller errors are mostly lower than the size of the estimates 

but still for many assets uncertainty in terms of error bars seems to be very high. This 

is similar for all samples with less than N = 2000 data points showing us that we need 

an interval of at least eight years to obtain accurate results by any of the methods 

presented. The problem is that the data that is available for assets traded outside 

the U.S. mostly does not go back that far in time. Anyway in most cases estimates 

yielded by smaller data sets are close to estimates yielded by bigger ones but there are 

some exceptions. Considering the limit of N = 2000 there are only a few assets of my 

gathered sample data remaining that are traded outside the U.S.: 

�Australian Securities Exchange index (AORD): Billabong International Limited 

(BBG) and Commonwealth Bank of Australia (CBA) with N = 2059 daily observations 

up to now 

�Financial Times Stock Exchange index (FTSE 100): United Utilities Gr. (UU) 

with N = 2117 daily observations up to now 

�MIBTEL: Tiscali (TIS) with N = 2123 daily observations up to now 

Applying Schmidt & Stadtmüller approaches to these bigger data sets yields lower 

bootstrap quantiles than to the smaller data sets consisting of N = 1398 data points. 

129

Index Abbrev. Country 

Australian Securities Exchange (ASX) AORD Australia 

CAC 40 CAC 40 France 

Deutscher Aktien Index DAX Germany 

Dow Jones Industrial Average DOW U.S. 

Financial Times Stock Exchange Index FTSE 100 U.K 

Jakarta Composite Index JKSE Indonesia 

Mercado de Valores MERVAL Argentina 

MIBTEL MIBTEL Italy 

NASDAQ Composite NASDAQ U.S. 

Swiss Market Index SMI Swiss 

SSE Composite Index SSEC China 

Tel Aviv 100 TA 100 Israel 

Table 4.1: Indexes that were used for the implementation of the different concepts given by 

their name, further used abbreviation, and country of origin. 

This is fully consistent with my bootstrap sampling estimates for the two reference data 

sets of S&P 500 presented in subsection (3.4.3). 

Applying approaches according to Poon, Rockinger and Tawn with tail dependence 

coefficient given by χ +,− , estimates sometimes yield lower bootstrap quantiles but for 

several assets i.e. Commonwealth Bank of Australia (CBA) errors remain high. This is 

also consistent with my further results presented in subsection (3.1.2), where bootstrap 

error bars for the different assets included in the index S&P 500 were also inconsistent. 

130

Index Asset Abbrev. 

AORD Australian Agricultural Company Limited AAC 

Abacus Property Group ABP 

AGL Energy Limited AGK 

Asciano Group AIO 

Axa Asia Pacific Holdings Limited AXA 

Billabong International Limited BBG 

Bioral Limited BLD 

Commonwealth Bank of Australia CBA 

Centennial Coal Company Limited CEY 

CSR Limited CSR 

Futuris Corporation Limited FCL 

Ing Industrial Fund IIF 

Macquarie Office Trust MOF 

Newcrest Mining Limited NCM 

Nido Petroleum Limited NDO 

CAC Alstom ALO 

Michelin ML 

EDF EDF 

Peugeot UG 

L’Oreal OR 

France Telecom FTE 

Sanofi-Aventis SAN 

Alcatel-Lucent ALU 

PPR PP 

Renault RNO 

DAX Adidas ADS 

Bayer BAY 

Deutsche Bank N DBK 

Hypo Real Estate HRX 

Linde LIN 

Metro MEO 

Merck MRK 

Munich Re Group MUV2 

SAP SAP 

Siemens SIE 

Tui TUI1 

Table 4.2: Assets included in indexes ’AORD’ (ASX), ’CAC 40’, and ’DAX’ that were used for 

the implementation of the different concepts given by their name, further used abbreviation, 

and country of origin. 

131

Index Asset Abbrev. Comp. Weight 

DOW Alcoa Inc. AA 1.95% 

American Int’l. Group AIG 0.28% 

Boeing Co. BA 4.54% 

Caterpillar Inc. CAT 4.77% 

General Motors GM 0.80% 

The Home Depot Inc. HD 2.03% 

Honeywell International Inc. HON - 

Intel Corporation INTC 1.43% 

Johnson and Johnson JNJ 5.14% 

Mc Donald’s MCD 4.73% 

Wal Mart Stores Inc. WMT 4.58% 

Exxon Mobil Corp. XOM 5.63% 

FTSE 100 Anglo American AAL 

Aviva AV 

Bae Systems BA 

Barclays BARC 

British American Tobacco BATS 

BG Group BG 

Cable and Wireless CW 

Glaxo Smith Klein GSK 

International Power IPR 

Lonmin LMI 

Tui Travel TT 

United Utilities Gr. UU 

JKSE Astra Agro Lestari Tbk AALI 

Tiga Pilar Sejahtera Food Tbk AISA 

AKR Corporindo Tbk AKRA 

Aneka Tambam Tbk ANTM 

Bank Rakyat Indonesia BBRI 

Bank Mandiri Tbk BMRI 

Central Proteinaprima Tbk CPRO 

Gozco Plantations Tbk GZCO 

Hexindo Adiperkasa Tbk HEXA 

Laguna Cipta Griya Tbk LCGP 

Sampoerna Agro Tbk SGRO 

Sorini Agro Asia Corporindo SOBI 

Table 4.3: Assets included in indexes ’DOW’ (Dow Jones), ’FTSE 100’, and ’JKSE’ that 

were used for the implementation of the different concepts given by their name, further used 

abbreviation, and country of origin. 

132

Index Asset Abbrev. 

MERVAL Acindar-Escriturales ACIN 

Petrobras Ordinarias APBR 

Cresud CRES 

Siderar ’A’ Voto Escri ERAR 

Irsa Investments and Representations Inc. IRSA 

Mirgor-Ord.Esc MIRG 

MIBTEL Ascopiave ASC 

Alitalia AZA 

Benetton Group BEN 

Buzzi Unicem BZU 

Banca Carige CRG 

Ducati Motor Hold. DMH 

Eni ENI 

Mediobanca MB 

Milano Ass. MI 

Risanamento RN 

ST Microelectronics STM 

Tiscali TIS 

Terna TRN 

NASDAQ Composite Apple Inc. AAPL 

Abraxis BioScience Inc. ABII 

Alliance Bankshare Corporation ABVA 

Cephalon Inc. CEPH 

Cico Systems Inc. CSCO 

Henry Schein Inc. HSIC 

Intel Corporation INTC 

Microsoft Corporation MSFT 

On Semiconductor Corp. ONNN 

Pacific Ethanol Inc. PEIX 

Perry Ellis International Inc. PERY 

Power Shares QQQ QQQQ 

SMI Julius Baer Holding N BAER 

Adecco N ADEN 

Ciba Holding N CIBN 

The Swatch Group UHR 

Zurich finl. Services ZURN 

Syngenta N SYNN 

Swiss Re N RUKN 

Holcim N HOLN 

Richemont Units A CFR 

Table 4.4: Assets included in indexes ’MERVAL’, ’MIBTEL’, ’NASDAQ’, and ’SMI’ that 

were used for the implementation of the different concepts given by their name, further used 

abbreviation, and country of origin. 

133

Index Asset Abbrev. Comp. Weight 

S&P 500 Bristol-Myers Squibb Co. BMY 0.40% 

Chevron CVX 1.59% 

Hewlett-Packard Co. HPQ 1.12% 

Coca-Cola Co. KO 1.03% 

3M MMM 0.46% 

Procter and Gamble Co. PG 2.07% 

Schering-Plough Corp. SGP 1.12% 

Texas Instruments Inc. TXN 0.28% 

Walgreen Co. WAG 0.31% 

new assets: 

Citygroup C 0.81% 

McDonalds MCD 0.69% 

Occidental Petroleum OXY 0.54% 

Dell DE 0.30% 

PNC Financial Services PNC 0.25% 

Noble Energy Inc. NBL 0.10% 

Parker-Hannifin PH 0.09% 

SSEC Northeast Express 600003 

Anhui Expressway 600012 

Henan Zhongyuan Ex 600020 

Shanghai Electricity Power 600021 

Jinan Iron and Steel 600022 

Huadian Power International 600027 

Hubai Chutian Expr. 600035 

CNTIC Trading 600056 

Minmetals Development 600058 

Zhengzhou Yutong 600066 

TA 100 AFR-ISR INV ILS AFIL 

Alony Hetz PTY and I ALHE 

Audiocodes AUDC 

Cellcom Israel CEL 

Global Industries Ltd. GLOB 

HSG and Constr. Holding HUCN 

Nice Systems NICE 

Scailex Corp. SCIX 

Table 4.5: Assets included in indexes ’S&P 500’, ’SSEC’, and ’TA 100’ that were used for 

the implementation of the different concepts given by their name, further used abbreviation, 

and country of origin. Additional assets included in index S&P 500 that were not part of the 

reference samples presented in chapter (3) are listed below ’new assets’. 

134

C. W. β Non-Par Par 


ˆν 3.16 3.69 3.16 3.69 3.16 3.69 3.16 3.69 

tail 1... k 1... k c... k c... k 1...k 1... k c...k c... k 

ˆλ 

BMY 0.40% 0.52 0.03 0.01 0.03 0.01 0.03 0.01 0.03 0.01 

CVX 1.59% 0.46 0.06 0.01 0.06 0.02 0.07 0.00 0.07 0.03 

HPQ 1.12% 1.07 0.07 0.04 0.07 0.05 0.10 0.02 0.10 0.07 

KO 1.03% 0.33 0.02 0.01 0.02 0.01 0.02 0.01 0.02 0.01 

MMM 0.46% 0.58 0.08 0.03 0.08 0.05 0.10 0.00 0.11 0.08 

PG 2.07% 0.25 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 

SGP 1.12% 0.51 0.02 0.00 0.02 0.00 0.02 0.00 0.02 0.01 

TXN 0.28% 0.87 0.01 0.01 0.02 0.01 0.02 0.00 0.02 0.01 

WAG 0.31% 0.48 0.03 0.01 0.03 0.01 0.03 0.01 0.03 0.02 

MCD 0.69% 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

C 0.81% 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

OXY 0.54% 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

DE 0.30% -0.11 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 

PNC 0.25% 0.16 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

NBL 0.10% 0.09 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

PH 0.09% 0.22 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

Table 4.6: Estimated upper and lower tail dependence ˆ λ +,− applying the non-parametric and 

parametric approaches according to Sornette & Malevergne to index S&P 500 and 16 assets 

included in dependence of their component weights in the index denoted by ’C.W.’ and β. 

The data samples contain N = 2000 daily price observations on a range from 12.04.2000 

to 31.03.2008. Tail index ˆν was calculated using Hill’s estimator, k = 0.04 · N = 80, c = 

0.005 · N = 10, and ∗ denotes negative ˆ β. 

135

136 

C. W. β β + 

SI1 Cond. 1 

up up 

β+ 

SI2 Cond. 2 

up up 

β− 

SI1 Cond. 1 

lo lo 

β+ 

SI2 Cond. 2 

lo lo 

ˆν 3.16 3.16 3.16 3.16 3.69 3.69 3.69 3.69 

tail 

ˆλ 

1... k c... k 1... k c... k 1... k c...k 1... k c...k 

BMY 0.40% 0.52 0.19 0.00 0.00 -0.34 0.00* 0.00* -0.26 0.00* 0.00* 0.92 0.06 0.08 

CVX 1.59% 0.46 0.35 0.03 0.02 -0.20 0.00* 0.00* -0.23 0.00* 0.00* 0.79 0.11 0.16 

HPQ 1.12% 1.07 1.25 0.11 0.11 2.04 0.51 0.54 -0.16 0.00* 0.00* 0.13 0.00 0.00 

KO 1.03% 0.33 0.51 0.07 0.07 0.42 0.04 0.04 0.11 0.00 0.00 0.60 0.07 0.07 

MMM 0.46% 0.58 0.81 0.23 0.24 0.78 0.21 0.22 0.43 0.01 0.02 1.19 0.46 0.70 

PG 2.07% 0.25 0.12 0.00 0.00 0.04 0.00 0.00 1.59 1.00 1.00 1.06 0.44 0.77 

SGP 1.12% 0.51 -0.20 0.00* 0.00* -0.01 0.00* 0.00* 0.09 0.00 0.00 0.84 0.02 0.03 

TXN 0.28% 0.87 1.75 0.14 0.14 -0.28 0.00* 0.00* 0.51 0.00 0.00 0.24 0.00 0.00 

WAG 0.31% 0.48 0.66 0.07 0.08 -0.03 0.00* 0.00* 0.30 0.00 0.00 1.06 0.21 0.24 

MCD 0.69% 0.01 1.84 1.00 1.00 -0.62 0.00* 0.00* -0.69 0.00* 0.00* 0.14 0.00 0.00 

C 0.81% 0.04 -0.06 0.00* 0.00* -0.53 0.00* 0.00* 5.45 1.00 1.00 0.13 0.00 0.00 

OXY 0.54% 0.06 -0.72 0.00* 0.00* -0.60 0.00* 0.00* -0.51 0.00* 0.00* 0.65 0.03 0.03 

DE 0.30% -0.11 -0.22 0.00* 0.00* -1.09 0.00* 0.00* -0.42 0.00* 0.00* -0.01 0.00* 0.00* 

PNC 0.25% 0.16 -0.02 0.00* 0.00* 0.11 0.00 0.00 0.26 0.00 0.00 -0.01 0.00* 0.00* 

NBL 0.10% 0.09 -0.80 0.00* 0.00* -0.82 0.00* 0.00* 0.77 0.04 0.04 1.15 0.16 0.16 

PH 0.09% 0.22 1.20 0.30 0.31 -0.43 0.00* 0.00* 3.02 1.00 1.00 0.42 0.01 0.01 

Table 4.7: Estimated upper and lower tail dependence ˆ λ +,− applying the non-parametric approach according to Sornette & Malevergne to index 

S&P 500 and 16 assets included using βSI only calculated for the extreme tails by first and second β-smile conditions. Component weights of assets 

within the index are denoted by ’C.W.’, βSI for the first and second conditions and β calculated for all data are listed to observe their impact on 

the estimates. The data samples contain N = 2000 daily price observations on a time interval from 12.04.2000 to 31.03.2008. Tail index ˆν was 

calculated using Hill’s estimator. ’Cond 1’ denotes: Y ≥ Y (k) ∩ X ≥ X(k), ’Cond. 2’ denotes Y ≥ Y (k), k = 0.04 · N = 80, c = 0.005 · N = 10, 

and ∗ denotes negative ˆ β.

C. W. β N-Par Par 


ˆν 3.15 3.84 3.15 3.84 3.15 3.84 3.15 3.84 

tail 1... k 1...k c... k c...k 1... k 1... k c... k c... k 

ˆλ 

AA 1.95% 1.35 0.27 0.19 0.26 0.19 0.37 0.31 0.37 0.31 

BA 4.54% 1.08 0.23 0.13 0.22 0.14 0.29 0.21 0.29 0.23 

CAT 4.77% 1.18 0.29 0.26 0.28 0.27 0.47 0.40 0.47 0.47 

GM 0.80% 1.38 0.17 0.12 0.17 0.13 0.24 0.20 0.25 0.21 

HON - 1.37 0.28 0.23 0.29 0.27 0.43 0.36 0.46 0.44 

JNJ 5.14% 0.51 0.08 0.05 0.07 0.05 0.08 0.05 0.08 0.07 

MCD 4.73% 0.66 0.07 0.04 0.07 0.05 0.08 0.05 0.08 0.05 

WMT 4.58% 0.88 0.18 0.17 0.17 0.17 0.25 0.24 0.25 0.25 

XOM 5.63% 0.79 0.24 0.11 0.24 0.11 0.29 0.19 0.28 0.20 

HD 2.03% 1.31 0.25 0.18 0.26 0.20 0.40 0.19 0.41 0.33 

INTC 1.43% 1.59 0.21 0.12 0.21 0.14 0.33 0.17 0.34 0.23 

AIG 0.28% 0.35 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 

Table 4.8: Estimated upper and lower tail dependence ˆ λ +,− applying the parametric and the 

non-parametric approach according to Sornette & Malevergne to index Dow Jones Industrial 

Average and 12 assets included using β calculated by all data. Component weights of assets 

within the index are denoted by ’C.W.’, and β calculated for all data are listed to observe 

their impact on the estimates. The data samples contain N = 2000 daily price observations 

on a time interval from 16.08.2000 to 31.07.2008. Tail index ˆν was calculated using Hill’s 

estimator, k = 0.04 · N = 80, and c = 0.005 · N = 10. 

137

138 

C. W. β β + 

SI1 Cond. 1 

up up 

β+ 

SI2 Cond. 2 

up up 

β− 

SI1 Cond. 1 

lo lo 

β+ 

SI2 Cond. 2 

lo lo 

ˆν 3.15 3.15 3.15 3.15 3.84 3.84 3.84 3.84 

tail 

ˆλ 

1... k c... k 1... k c... k 1... k c...k 1... k c...k 

AA 1.95% 1.35 0.23 0.00 0.00 1.20 0.19 0.19 0.83 0.03 0.03 1.50 0.27 0.27 

BA 4.54% 1.08 0.38 0.01 0.01 0.56 0.03 0.03 1.88 1.00 1.00 2.30 1.00 1.00 

CAT 4.77% 1.18 0.54 0.03 0.02 1.20 0.27 0.26 0.48 0.01 0.01 0.81 0.06 0.06 

GM 0.80% 1.38 0.24 0.00 0.00 1.20 0.10 0.10 1.04 0.04 0.04 1.70 0.27 0.29 

HON - 1.37 0.91 0.08 0.08 1.50 0.35 0.36 2.21 1.00 1.00 2.40 1.00 1.00 

JNJ 5.14% 0.51 0.69 0.20 0.19 0.93 0.51 0.49 2.12 1.00 1.00 0.80 0.28 0.29 

MCD 4.73% 0.66 0.08 0.00 0.00 -0.09 0.00 0.00 -0.45 0.00 0.00 0.42 0.01 0.01 

WMT 4.58% 0.88 0.48 0.03 0.03 0.89 0.18 0.18 0.11 0.00 0.00 0.60 0.04 0.04 

XOM 5.63% 0.79 0.88 0.34 0.33 0.83 0.27 0.27 0.01 0.00 0.00 0.66 0.06 0.05 

HD 2.03% 1.31 0.76 0.05 0.05 1.30 0.26 0.26 1.86 0.68 0.79 2.10 1.00 1.00 

INTC 1.43% 1.59 0.67 0.01 0.01 1.80 0.28 0.28 -0.08 0.00 0.00 0.49 0.00 0.00 

AIG 0.28% 0.35 0.44 0.01 0.01 -0.51 0.00 0.00 -0.24 0.00 0.00 -0.17 0.00 0.00 

Table 4.9: Estimated upper and lower tail dependence ˆ λ +,− applying the non-parametric approach according to Sornette & Malevergne to index 

Dow Jones Industrial Average and 12 assets included using βSI only calculated for the extreme tails by first and second β-smile conditions. 

Component weights of assets within the index denoted by ’C.W.’, βSI for the first and second conditions and β calculated for all data are listed 

to observe their impact on the estimates. The data samples contain N = 2000 daily price observations on a time interval from 16.08.2000 to 

31.07.2008. Tail index ˆν was calculated using Hill’s estimator, k = 0.04 · N = 80, and c = 0.005 · N = 10.

C. W. β N-Par Par 


ˆν 2.59 3.83 2.59 3.83 2.59 3.83 2.59 3.83 

tail 1... k 1... k c... k c... k 1... k 1... k c...k c... k 

ˆλ 

AAPL - 1.01 0.26 0.12 0.25 0.16 0.28 0.04 0.28 0.28 

INTC - 1.25 0.52 0.26 0.49 0.31 0.65 0.43 0.64 0.62 

MSFT - 0.85 0.41 0.26 0.40 0.29 0.50 0.49 0.50 0.57 

CSCO - -0.08 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 0.00* 

CEPH - 0.34 0.02 0.00 0.02 0.00 0.02 0.00 0.02 0.00 

PERY - 0.21 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

HSIC - 0.26 0.02 0.00 0.02 0.00 0.02 0.00 0.02 0.00 

QQQQ - 1.12 0.87 0.63 0.86 0.62 0.96 0.99 0.96 0.99 

ONNN - 0.57 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 

ˆλ β + 

SI1 β− 

SI1 

AAPL 0.28 -0.60 0.01 0.00* 0.01 0.00* - - - - 

INTC 0.68 0.69 0.11 0.03 0.10 0.03 - - - - 

MSFT 0.69 0.62 0.23 0.07 0.23 0.08 - - - - 

CSCO -0.10 0.09 0.00* 0.00 0.00* 0.00 - - - - 

CEPH 0.20 0.21 0.00 0.00 0.00 0.00 - - - - 

PERY 1.55 0.69 0.50 0.02 0.50 0.03 - - - - 

HSIC 0.58 1.46 0.15 1.00 0.14 1.00 - - - - 

QQQQ 0.99 0.89 0.64 0.26 0.63 0.26 - - - - 

ONNN -3.24 0.53 0.00* 0.00 0.00* 0.00 - - - - 

ˆλ β + 

SI2 β− 

SI2 

AAPL 0.38 -0.43 0.02 0.00* 0.02 0.00* - - - - 

INTC 0.82 0.19 0.18 0.00 0.17 0.00 - - - - 

MSFT 0.82 0.39 0.37 0.01 0.37 0.02 - - - - 

CSCO -0.11 -1.00 0.00* 0.00* 0.00* 0.00* - - - - 

CEPH -0.66 -0.57 0.00* 0.00* 0.00* 0.00* - - - - 

PERY -0.25 -1.67 0.00* 0.00* 0.00* 0.00* - - - - 

HSIC -0.15 0.34 0.00* 0.01 0.00* 0.01 - - - - 

QQQQ 1.04 1.04 0.73 0.47 0.72 0.47 - - - - 

ONNN -1.11 -1.84 0.00* 0.00* 0.00* 0.00* - - - - 

Table 4.10: Estimated upper and lower tail dependence ˆ λ +,− applying the parametric and the 

non-parametric approach according to Sornette & Malevergne to index NASDAQ Composite 

and 9 assets included using β calculated for all data and βSI only calculated for the extreme 

tails by first and second β-smile conditions. Component weights of assets within the index 

denoted by ’C.W.’, βSI for the first and second conditions and β calculated for all data 

are listed to observe their impact on the estimates. The data samples contain N = 2000 

daily price observations on a time interval from 15.08.2000 to 31.07.2008. Tail index ˆν was 

calculated using Hill’s estimator, k = 0.04·N = 80, c = 0.005·N = 10, and ∗ denotes negative 

ˆβ. 

139

140 

ˆχ + 95% Q 90% Q ˆχ + c 95% Q 90% Q ˆχ − 95% Q 90% Q ˆχ − c 95% Q 90% Q 

CBA 0.91 0.82 0.10 0.87 0.05 0.04 0.30 0.49 0.47 0.88 0.09 0.07 

BBG 0.91 0.83 0.09 0.92 0.09 0.07 0.91 0.06 0.05 0.88 0.06 0.05 

AAC 0.92 0.05 0.05 0.91 0.05 0.04 0.91 0.79 0.79 0.88 0.07 0.06 

IIF 0.92 0.03 0.03 0.92 0.05 0.04 0.91 0.84 0.08 0.88 0.05 0.04 

MOF 0.91 0.79 0.79 0.89 0.05 0.04 0.91 0.05 0.04 0.88 0.05 0.04 

NCM 0.91 0.79 0.79 0.89 0.05 0.04 0.91 0.05 0.04 0.88 0.05 0.04 

ABP 0.92 0.05 0.04 0.92 0.05 0.04 0.91 0.06 0.06 0.88 0.06 0.05 

AGK 0.91 0.84 0.07 0.88 0.05 0.04 0.66 0.53 0.53 0.88 0.07 0.06 

BLD 0.92 0.04 0.04 0.90 0.04 0.04 0.91 0.04 0.04 0.88 0.05 0.04 

NDO 0.84 0.46 0.46 0.78 0.65 0.65 0.84 0.50 0.50 0.79 0.59 0.59 

CEY 0.91 0.75 0.75 0.86 0.06 0.05 0.91 0.77 0.77 0.88 0.06 0.05 

CSR 0.85 0.29 0.13 0.92 0.06 0.05 0.31 0.51 0.50 0.88 0.10 0.08 

FCL 0.83 0.71 0.71 0.92 0.75 0.75 0.88 0.79 0.13 0.88 0.06 0.05 

AXA 0.91 0.83 0.08 0.92 0.07 0.06 0.91 0.83 0.08 0.88 0.07 0.05 

ˆλU,m 95% Q 90% Q ˆ λEV T 

U,m 95% Q 90% Q ˆ λL,m 95% Q 90% Q ˆ λEV T 

L,m 95% Q 90% Q 

CBA 0.23 0.10 0.08 0.23 0.11 0.09 0.16 0.11 0.07 0.19 0.10 0.08 

BBG 0.13 0.10 0.08 0.12 0.09 0.08 0.14 0.09 0.08 0.17 0.10 0.08 

AAC 0.07 0.08 0.06 0.06 0.09 0.07 0.07 0.07 0.06 0.10 0.08 0.06 

IIF 0.20 0.10 0.09 0.19 0.11 0.09 0.22 0.11 0.09 0.24 0.11 0.08 

MOF 0.23 0.12 0.10 0.23 0.11 0.09 0.25 0.11 0.10 0.28 0.11 0.10 

NCM 0.23 0.12 0.10 0.23 0.11 0.09 0.25 0.11 0.10 0.28 0.11 0.10 

ABP 0.13 0.09 0.07 0.12 0.09 0.09 0.18 0.09 0.07 0.21 0.10 0.08 

AGK 0.05 0.05 0.05 0.05 0.06 0.05 0.13 0.09 0.07 0.15 0.09 0.08 

BLD 0.09 0.07 0.06 0.08 0.08 0.06 0.04 0.05 0.04 0.06 0.05 0.05 

NDO 0.00 0.03 0.01 0.03 0.09 0.07 0.02 0.05 0.04 0.01 0.08 0.06 

CEY 0.05 0.05 0.05 0.05 0.06 0.06 0.07 0.07 0.06 0.10 0.08 0.07 

CSR 0.09 0.08 0.06 0.08 0.07 0.07 0.07 0.07 0.06 0.10 0.08 0.06 

FCL 0.02 0.03 0.03 0.01 0.05 0.04 0.07 0.07 0.07 0.10 0.08 0.06 

AXA 0.13 0.09 0.07 0.12 0.09 0.08 0.11 0.09 0.07 0.14 0.08 0.08 

Table 4.11: Estimated upper and lower tail dependence according to Poon, Rockinger, and Tawn (ˆχ + · , ˆχ− · 

Schmidt & Stadtmüller in the lower part ( ˆ λ·,m, ˆ λEV T 

·,m 

) in the upper part and according to 

) for index AORD (ASX) and 14 assets included including error bars estimated by bootstrap 

sampling with replacement for bs = 1000 bootstrap samples shown next to the estimates. The data samples contain N = 1398 daily price 

observations on a time interval from begin of February 2003 to end of August 2008.

141 

ˆλ + 

SI1 95% Q 90% Q ˆ λ + 

SI1,c 95% Q 90% Q ˆ λ − 

SI1 95% Q 90% Q ˆ λ − 

SI1,c 95% Q 90% Q 

CBA 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

BBG 0.03 * * 0.03 * * 0.00 * * 0.00 * * 

AAC 0.00 * * 0.00 * * 0.02 * * 0.02 * * 

IIF 0.10 * * 0.10 * * 0.10 * * 0.10 * * 

MOF 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

NCM 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

ABP 0.06 0.06 0.02 0.06 * * 0.00 * * 0.00 * * 

AGK 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

BLD 0.44 * * 0.42 * * 0.00 * * 0.00 * * 

NDO 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

CEY 0.05 * * 0.05 * * 0.00 * * 0.00 * * 

CSR 0.91 0.58 0.58 0.90 * * 0.40 * * 0.70 * * 

FCL 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

AXA 0.03 * * 0.03 * * 0.04 * * 0.04 * * 

ˆλ + 

SI2 95% Q 90% Q ˆ λ + 

SI2,c 95% Q 90% Q ˆ λ − 

SI2 95% Q 90% Q ˆ λ − 

SI2,c 95% Q 90% Q 

CBA 0.00 * * 0.00 * * 0.02 * * 0.05 * * 

BBG 0.01 * * 0.01 * * 0.00 * * 0.00 * * 

AAC 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

IIF 0.28 * * 0.29 * * 0.11 * * 0.11 * * 

MOF 0.12 * * 0.11 * * 0.06 * * 0.07 * * 

NCM 0.12 * * 0.11 * * 0.06 * * 0.07 * * 

ABP 0.02 * * 0.02 * * 0.00 * * 0.00 * * 

AGK 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

BLD 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

NDO 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

CEY 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

CSR 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

FCL 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

AXA 0.00 * * 0.00 * * 0.00 * * 0.00 * * 

Table 4.12: Estimated ˆ λ +,− applying the non-parametric approach according to Sornette & Malevergne to index AORD (ASX) and 14 assets 

included using βSI only calculated for the extreme tails by first and second β-smile conditions. Error bars estimated by bootstrap sampling with 

replacement for bs = 1000 bootstrap samples were large and therefore mostly denoted by *. The data samples contain N = 1398 data points from 

February 2003 to end of August 2008. Tail index ˆν was calculated using Hill’s estimator.

Abbreviation Currency exchange rate 

A USD-Australian Dollar 

Eur USD-Euro 

GBP USD-British Pound 

BrR USD-Brazilian Real 

CHFR USD-Swiss Frank 

CHY USD-Chinese Yuan 

indoRU USD-Indonesian Rupiah 

indRu USD-Indian Rupee 

JPY USD-Japanese Yen 

Table 4.13: Further used abbreviations of currency exchange rates that were used for the 

estimation of tail dependence. 

4.2 Application to Exchange Rates 

I wanted to apply the different concepts to index data of different markets. Here the 

advantage is that in terms of historical index prices there is much more data available 

than it is the case for individual asset prices. But the problem is that the data provided 

does not accord in time. Choosing a sample size of N = 2000 data points for the different 

indexes, which has been shown to be the minimum sample size in order to obtain 

stable estimates, already shows shifts of two months time between different indexes 4 . 

Because the data is not synchronous it makes no sense to apply tail dependence. 

Now I come to the application of the concepts for the estimation of tail dependence to 

exchange rates of various foreign currencies with the US$. An explanation of abbreviations 

of different currency exchange rates is provided in table (4.13) 5 . The dependence 

between different currency exchange rates is an indicator for the dependence between 

different economies. The existence of a dependence between foreign exchange markets 

could affect firms’ investment decisions. Knowledge of short- and long-term extreme 

relations could furthermore have important implications for risk management. 

For the implementation I wanted to use relatively small, preferably up-to-date data 

sets. Fortunately the provided data is consistent and perfectly corresponding in time. 

I choose sample sizes of N = 2000 daily observations on a time interval ranging from 

20.10.2002 to 10.04.2008 for several currency exchange rates between the US$ and 

major foreign currencies. Table (4.14) shows tail dependence coefficients estimated by 

the non-parametric approach according to Sornette & Malevergne for the chosen time 

interval. The results look quite interesting because the spectrum is large. Sometimes 

dependences seem very strong and in other cases we estimate zero tail dependence. 

Regarding the results intuitively, they look reasonable to me. It is interesting that 

extreme changes of exchange rates of the US$ to any of the chosen currencies seem to 

have no significant influence on the exchange rate of the US$ to the Chinese Yuan so 

far. It looks as if the Chinese Yuan has been approximately independent of the other 

currencies in terms of extreme changes up to now. The Indian Rupee shows a similar 

situation. It looks as if these huge countries remained somewhat autonomic of any 

other economy. Another interesting finding is that by far the strongest tail dependence 

is estimated between the currency exchange rate of the US$ and the Euro and the 

4 

This is shown in Excel file: Index data.xls enclosed on the data CD, where several indexes are listed in 

parallel 

5 

Data tables of daily currency exchange rates can be downloaded at FXHistory: 

http://www.oanda.com/convert/fxhistory 

142

exchange rate of the US$ and the Swiss Franc. This seems to be reasonable. 

I also performed the estimations by the other methods for comparison. This can be 

seen in table (4.15) for the χ approach according to Poon, Rockinger, and Tawn [13] 

(2004) and in table (4.16) for the λ approaches according to Schmidt & Stadtmüller [24] 

(2005). 

Comparing the results achieved by the different methods, at first sight the results 

seem very different, but again, if we take into account relative differences, table (4.14) 

showing results according to Sornette & Malevergne and table (4.16) showing results 

according to Schmidt & Stadtmüller make similar statements i.e. tail dependence for 

currencies within Europe is larger than outside of Europe. Looking for example at 

tail dependence coefficient estimates of currency exchange rates in US$ between Euro 

and Swiss Frank, absolute results are nearly identical and in both tables represent 

the maximum estimates achieved. Another example is the Australian $: Comparing 

currency exchange rates of the Australian $ and the Euro with currency exchange rates 

of the Australian $ and the British Pound and currency exchange rates of the Australian 

$ and the Swiss Frank we notice that in both tables tail dependence estimates for all of 

the three exchange rates are relatively high whereas estimates for the Euro are slightly 

higher than for the Pound and estimates for the Pound are slightly higher than for 

the Swiss Frank. The rest of the currency exchange rate estimates with the Australian 

$ are significantly lower with estimates for Japanese Yen being higher than the rest. 

Many more of these equivalences can be found. 

Comparing the χ-approach shown in table (4.15) with results achieved by the other 

two methods it is much more difficult to draw inferences because also applied to currency 

exchange rates results of the χ-approach are very close together, which makes it hard 

to assess differences between the estimates because differences are mostly smaller than 

estimated error bars. What we notice is that for all cases ˆχ = 1 could not be rejected 

and therefore it can be concluded that asymptotic dependence exists between all of the 

chosen currency exchange rates. For further details we refer to section (3.1). 

Although the Sornette-Malevergne factor model is not so natural for the application 

to currency exchange rates because we do not expect a ’common factor’ to appear, 

the results look reasonable also compared to the results yielded by the Schmidt & 

Stadtmüller approaches. 

143

A Eur GBP BrR CHFR CHY indoRu indRu JPY 

A λ + * 0.29 0.23 0.01 0.20 0.00 0.01 0.00 0.06 

λ− 0.13 0.13 0.00 0.07 0.00 0.00 0.00 0.02 

λ + c 

λ 

0.28 0.22 0.01 0.19 0.00 0.01 0.00 0.06 

− Eur 

c 

λ 

0.13 0.14 0.00 0.07 0.00 0.00 0.00 0.02 

+ * 0.33 0.00 0.77 0.00 0.00 0.00 0.11 

λ− 0.46 0.00 0.71 0.00 0.00 0.00 0.10 

λ + c 

λ 

0.31 0.00 0.76 0.00 0.00 0.00 0.11 

− GBP 

c 

λ 

0.46 0.00 0.71 0.00 0.00 0.00 0.11 

+ * 0.00 0.33 0.00 0.00 0.00 0.06 

λ− 0.00 0.18 0.00 0.00 0.00 0.02 

λ + c 

λ 

0.00 0.34 0.00 0.00 0.00 0.06 

− BrR 

c 

λ 

0.00 0.18 0.00 0.00 0.00 0.02 

+ * 0.00 0.00 0.00 0.00 0.00 

λ− 0.00 0.00 0.01 0.01 0.00 

λ + c 

λ 

0.00 0.00 0.00 0.00 0.00 

− CHFR 

c 

λ 

0.00 0.00 0.01 0.01 0.00 

+ * 0.00 0.00 0.00 0.19 

λ− 0.00 0.00 0.00 0.12 

λ + c 

λ 

0.00 0.00 0.00 0.19 

− CHY 

c 

λ 

0.00 0.00 0.00 0.12 

+ * 0.00 0.00 0.00 

λ− 0.00 0.00 0.00 

λ + c 

λ 

0.00 0.00 0.00 

− indoRu 

c 

λ 

0.00 0.00 0.00 

+ * 0.00 0.01 

λ− 0.00 0.01 

λ + c 

λ 

0.00 0.01 

− indRu 

c 

λ 

0.00 0.01 

+ * 0.01 

λ− 0.00 

λ + c 

0.01 

0.00 

λ − c 

Table 4.14: Estimated ˆ λ +,− applying the non-parametric approach according to Sornette & 

Malevergne given by: ˆ λ +,− � 

= 1/max 1, l 

�ν β to exchange rates of various foreign currencies 

with the US$. The tail represents the most extreme 4% of the return values during a time 

interval ranging from 20.10.2002 to 10.04.2008 consisting of N = 2000 daily observations. ’c’ 

means that tails were corrected for 0.5% of most extreme data for the estimation of l. Tail 

index ˆν was calculated using Hill’s estimator. 

144


A χ + * 0.92 0.91 0.92 0.91 0.92 0.92 0.92 0.92 

χ− 0.91 0.91 0.91 0.91 0.84 0.91 0.91 0.91 

χ + c 

χ 

0.88 0.86 0.87 0.86 0.89 0.89 0.87 0.89 

− Eur 

c 

χ 

0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 

+ * 0.91 0.92 0.91 0.92 0.92 0.92 0.92 

χ− 0.91 0.91 0.91 0.84 0.91 0.91 0.91 

χ + c 

χ 

0.86 0.87 0.86 0.88 0.88 0.87 0.88 

− GBP 

c 

χ 

0.86 0.86 0.86 0.86 0.86 0.86 0.86 

+ * 0.91 0.91 0.91 0.91 0.91 0.91 

χ− 0.93 0.92 0.84 0.93 0.92 0.93 

χ + c 

χ 

0.86 0.86 0.86 0.86 0.86 0.86 

− BrR 

c 

χ 

0.90 0.89 0.90 0.90 0.88 0.90 

+ * 0.91 0.92 0.92 0.92 0.92 

χ− 0.92 0.84 0.93 0.92 0.93 

χ + c 

χ 

0.86 0.87 0.87 0.87 0.87 

− CHFR 

c 

χ 

0.89 0.91 0.90 0.88 0.90 

+ * 0.91 0.91 0.91 0.91 

χ− 0.84 0.92 0.92 0.92 

χ + c 

χ 

0.86 0.86 0.86 0.86 

− CHY 

c 

χ 

0.89 0.89 0.88 0.89 

+ * 0.93 0.92 0.93 

χ− 0.84 0.84 0.84 

χ + c 

χ 

0.91 0.87 0.89 

− indoRu 

c 

χ 

0.90 0.88 0.90 

+ * 0.92 0.93 

χ− 0.92 0.93 

χ + c 

χ 

0.87 0.89 

− indRu 

c 

χ 

0.88 0.90 

+ * 0.92 

χ− 0.92 

χ + c 

0.87 

0.88 

χ − c 

Table 4.15: Estimated ˆχ +,− applying the non-parametric approach according to Poon, 

Rockinger, and Tawn given by: ˆχ +,− = Zk,N ·k 

N to exchange rates of various foreign currencies 

with the US$. The tail represents the most extreme 4% of the return values during a time 

interval ranging from 20.10.2002 to 10.04.2008 consisting of N = 2000 daily observations. ’c’ 

means that tails were corrected for 0.5% of most extreme data for the estimation of L. Tail 

indexes ˆν were calculated using Hill’s estimator. 

145


A λU,m * 0.40 0.36 0.14 0.34 0.08 0.13 0.05 0.29 

λL,m 0.41 0.33 0.10 0.43 0.04 0.13 0.15 0.26 

λEV T 

U,m 

λEV T 

L,m 

0.40 

0.44 

0.36 

0.35 

0.14 

0.12 

0.34 

0.45 

0.07 

0.06 

0.12 

0.15 

0.05 

0.17 

0.29 

0.29 

Eur λU,m * 0.53 0.13 0.73 0.09 0.08 0.04 0.29 

λL,m 0.43 0.09 0.69 0.08 0.14 0.09 0.24 

λEV T 

U,m 

λEV T 

L,m 

0.52 

0.45 

0.12 

0.11 

0.72 

0.71 

0.09 

0.10 

0.07 

0.16 

0.04 

0.11 

0.29 

0.26 

GBP λU,m * 0.09 0.44 0.06 0.06 0.08 0.25 

λL,m 0.08 0.43 0.06 0.09 0.09 0.26 

λEV T 

U,m 

λEV T 

L,m 

0.09 

0.10 

0.44 

0.45 

0.06 

0.09 

0.06 

0.11 

0.07 

0.11 

0.25 

0.29 

BrR λU,m * 0.08 0.06 0.09 0.10 0.10 

λL,m 0.06 0.04 0.09 0.06 0.06 

λEV T 

U,m 

λEV T 

L,m 

0.07 

0.09 

0.06 

0.06 

0.09 

0.11 

0.10 

0.09 

0.10 

0.09 

CHFR λU,m * 0.08 0.10 0.05 0.36 

λL,m 0.08 0.11 0.08 0.31 

λEV T 

U,m 

λEV T 

L,m 

0.07 

0.10 

0.10 

0.14 

0.05 

0.10 

0.36 

0.34 

CHY λU,m * 0.04 0.10 0.05 

λL,m 0.04 0.05 0.09 

λEV T 

U,m 

λEV T 

L,m 

0.04 

0.06 

0.10 

0.07 

0.05 

0.11 

indoRu λU,m * 0.10 0.10 

λL,m 0.08 0.06 

λEV T 

U,m 

λEV T 

L,m 

0.10 

0.10 

0.10 

0.09 

indRu λU,m * 0.10 

λL,m 

0.11 

λEV T 

U,m 

λEV T 

0.10 

0.14 

L,m 

Table 4.16: Estimated ˆ λL,m, ˆ λU,m, ˆ λEV T 

L,m , and ˆ λU,m applying the non-parametric approaches 

according to Schmidt & Stadtmüller explained in section (3.4) to exchange rates of various 

foreign currencies with the US$. The tail represents the most extreme 4% of the return values 

during a time interval ranging from 20.10.2002 to 10.04.2008 consisting of N = 2000 daily 

observations. 

EV T 

146

4.3 Application to Synthetic Time Series 

To check for the bias of the parameters ˆ β and tail index ˆα and to get a sense of what 

the best result could be by confining resulting errors on the estimators to statistical 

errors in order to find lower error bounds, I implemented the different concepts to 

synthetic time series. The difficulty creating a synthetic sample on the one hand is 

that it should be as simple as possible enabling me to control everything and ensuring 

exact knowledge about the structure of the data and on the other hand not to simplify 

the model too much because we want it to become realistic. 

I approximated the real return time series by a distribution consisting of two parts: 

the distribution body was modeled Gaussian and the tails up to a certain threshold were 

modeled by heavy tailed distribution functions. The probability density distribution 

function is given by: 

� 

N(µ, σ) if |y| < Yk,N 

p(y) = α 

y1+α if |y| ≥ 

(4.1) 

Yk,N 

with Yk,N denoting the threshold value or the least extreme return value still counted 

to the tails. 

Now I will explain the implementation of the distribution 6 : First I created two 

samples of N uniformly distributed random values on the interval (0, 1) denoted by 

’u1’ and ’u2’, whereas N denotes the sample size. Then I sorted the two samples of 

uniformly distributed random values in descending order (’u1desc’ and ’u2desc’) and 

applied the inverse error function for transformation to standard normal distributed 

values N(0,1) with zero mean and variance equal to one given by equation: 

z1 = √ 2 · erfinv(2 · u1desc − 1) (4.2) 

z2 = √ 2 · erfinv(2 · u2desc − 1) (4.3) 

Now I built the tail samples by first creating four samples of Z uniformly distributed 

random values on the interval (0, 1) with Z = [0.04 · N] and then transforming the 

uniformly distributed tail values U(0,1) to heavy tail distributed values by applying 

relation: � � 

1 1/α 

to get positive tails and by multiplying the same relation by factor 

1−x 

(−1) for negative tails with α denoting the tail index of the heavy tails. The heavy tailed 

parts where then sorted by decreasing order and scaled in a way that the least extreme 

values still counted to the tails were equal to the most extreme values counted to the 

body of the respective distribution. Then the tails of the standard normal distributed 

samples ’u1desc’ and ’u2desc’ were replaced by the heavy tails, scaled to typical size 

in order to obtain synthetic return series that are comparable to original return series, 

and named Y and ε denoting vectors of index returns and idiosyncratic noise. Error 

terms ε calculated for real time series were typically of similar size as respective return 

values. Therefore the scaling of the two vectors Y and ε was performed equally. As 

a next step asset return vector X had to be built by Y , ε, and a predefined factor 

β applying the linear single factor model: X = β · Y + ε, whereas firstly the vectors 

were brought back into a random sequence by uniform sampling without replacement 

to avoid violation of the i.i.d (independent and identically distributed) criterion. Then 

after comparison of the synthetic samples with real historic return series, concepts to 

6 the m-file for the estimation of tail dependence coefficients on synthetic samples is enclosed to the appendix 

and to the data CD and denoted by: synthdata all.m 

147

estimate tail dependence were applied. Figure (4.1) shows non-parametric probability 

density distribution functions of the real index S&P 500 and asset CVX (Chevron Corp.) 

return series on a time interval ranging from July 1985 to Mars 2008 representing our 

smaller reference samples and a synthetic sample created by the m-file described within 

this section. Both were estimated by a Gaussian box kernel described in subsection 

(3.4.3) and the data sets consist of N = 2507 daily observations. We can see that the 

tails of the synthetic distribution were chosen similar to the tails of index S&P 500, 

whereas some assets show a slightly different tail behavior. As we are interested in the 

tails of the distributions I also plotted the distribution on a semi-log scale to the base 

ten shown in figure (4.2). First of all we notice the bumps caused by the Gaussian 

box kernel. The three bumps on the left that can be observed on the non-parametric 

probability density distribution function of asset CVX represent the most extreme 

outliers and therefore are reasonable. Looking at the non-parametric probability density 

distribution function of the sample of the synthetic time series plotted in red we observe 

that it lies in-between the two real data distributions. This is exactly what we intended. 

The transition from the Gaussian part to the heavy tailed part of the distribution seems 

to be as smooth as on the example of real time series and should not generate problems 

for the calculation of tail dependence coefficient estimates. Random returns Xi and 

random errors εi have the same distribution and tails modeled by the same tail index 

α as it is requested for the concepts according to Sornette & Malevergne. 

Implementing first the concept of non-parametric tail dependence, estimated by β 

calculated for all data, to synthetic time series we determine that there is no significant 

bias created by the estimation of β. This can be seen on table (4.17) showing that 

the maximum bias of the β-estimator, calculated as an average value performing 1000 

samples with identical inputs in dependence of the size of the original β that was predefined 

for the synthetic samples, was always smaller than 0.5% of the original value 

βorig. Additionally to show the effect on ˆ λ +,− of the bias created by ˆ β, tail dependence 

estimators calculated by the original β denoted by ˆ λ +,− (βorig) were compared 

� 

to tail 

ˆβalldata 

dependence estimators in dependence of estimated β denoted by ˆ λ +,− 

� 

. Only 

for small ˆ λ +,− 

� � 

ˆβalldata the bias is significantly different from zero and reaches a maximum 

of two percent of the estimate. It has to be added that the bias is independent 

of the size of the synthetic sample and that tail index α was always set equal to three 

to avoid a second source of errors. An estimation of error bars is also provided in table 

(4.17) by quoting standard deviations and 95% quantiles within the 1000 samples. 

Standard deviations of ˆ βalldata and ˆ λ +,− 

� 

resulting from ˆ β compaired with the 

� ˆβall data 

95% quantiles show that errors seem to be approximately Gaussian distributed. 

Using βSI calculated by β-smile conditions yields surprising results: as we can observe 

in table (4.18) there is no bias in the estimation of β when we apply the second 

β-smile condition given by: � Y ≥ Y (k) � to 1000 synthetic samples with identical inputs 

in dependence of the size of the original β. This is surprising insofar as it only 

counts for the second β-smile condition. The first and the third β-smile conditions both 

yield β-estimates that are strongly biased. As discussed in section (3.3) we still face 

the problem of wide error bars for tail dependence coefficient estimates performed by 

β-smile conditions. Even if βSI 2 is not biased itself, table (4.18) shows that primarily 

for small βorig we obtain a significant bias in ˆ λ +,− with ˆ β estimated by the second βsmile 

condition. Also here tail index α was always set equal to three to avoid a second 

source of errors. 

148

Density 

Density 

60 

50 

40 

30 

20 

10 

Fitted synthetic time series 

Fitted S&P 500 

Fitted asset CVX 

0 

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 

Data 

Figure 4.1: Non-parametric probability density distribution functions of index S&P 500, assets 

CVX (Chevron Corp.), and a synthetic sample estimated using a Gaussian box kernel for a 

time interval of N = 2507 data points ranging from January 1991 to December 2000. 

10 0 

10 −5 

10 −10 

10 −15 

10 −20 

10 −25 

10 −30 

Fitted synthetic time series 

Fitted S&P 500 

Fitted asset CVX 

−0.8 −0.6 −0.4 −0.2 0 0.2 

Data 

Figure 4.2: Non-parametric probability density distribution functions of index S&P 500, assets 

CVX (Chevron Corp.), and a synthetic sample plotted on a semi-log scale and estimated using 

a Gaussian box kernel for a time interval of N = 2507 data points ranging from January 1991 

to December 2000. 

149

150 

βorig 

ˆβall data bias rel. bias std 95% quantile 

+,− 

λ ˆ (βorig) 

+,− 

λ ˆ ( ˆ α = 3 

βall data) bias rel. bias std 95% quantile 

0.05 0.05 0.00 0.00 0.02 0.04 0.00 0.01 0.01 inf 0.10 0.01 

0.10 0.10 0.00 0.00 0.02 0.03 0.00 0.00 0.00 0.00 0.00 0.00 

0.20 0.20 0.00 0.00 0.02 0.03 0.01 0.01 0.00 0.01 0.00 0.00 

0.30 0.30 0.00 0.00 0.01 0.01 0.02 0.02 0.00 0.02 0.01 0.01 

0.40 0.40 0.00 0.00 0.02 0.03 0.05 0.05 0.00 0.00 0.01 0.02 

0.50 0.50 0.00 0.00 0.02 0.03 0.09 0.09 0.00 0.00 0.01 0.03 

0.60 0.60 0.00 0.00 0.02 0.03 0.13 0.13 0.00 0.00 0.02 0.04 

0.70 0.70 0.00 0.00 0.02 0.03 0.18 0.18 0.00 0.00 0.03 0.04 

0.80 0.80 0.00 0.00 0.02 0.04 0.23 0.23 0.00 0.00 0.03 0.05 

0.90 0.90 0.00 0.00 0.02 0.03 0.28 0.28 0.00 0.00 0.03 0.06 

1.00 1.00 0.00 0.01 0.02 0.03 0.33 0.33 0.00 0.00 0.04 0.06 

1.10 1.10 0.00 0.00 0.02 0.03 0.38 0.38 0.00 0.00 0.04 0.06 

1.20 1.20 0.00 0.00 0.02 0.03 0.42 0.42 0.00 0.00 0.04 0.07 

1.30 1.30 0.00 0.00 0.02 0.03 0.46 0.46 0.00 0.00 0.04 0.07 

Table 4.17: Establishing bias and error bars of ˆ β calculated by all data and effect on ˆ λ +,− estimated by applying the non-parametric approach 

according to Sornette & Malevergne to 1000 synthetic samples consisting of N = 2507 data points. The bias is calculated by comparing mean 

values ˆ βall data and ˆ λ +,− 

( ˆ βall data) to original values denoted by βorig and corresponding tail dependence estimator ˆ λ +,− (βorig) and for the estimation 

of error bars standard deviation ’std’ and 95% quantiles of the estimates are provided. Relative bias (’rel.bias’) is calculated as percentage of 

ˆλ +,− 

(βorig). Tail indexes α were set equal to three to avoid a second source of error and ’inf’ means ·/0.

151 

βorig 

ˆβSI 2 bias rel. bias std 95% quantile 

+,− 

λ ˆ (βorig) 

+,− 

λ ˆ ( ˆ α = 3 

βSI 2) bias rel. bias std 95% quantile 

0.05 0.05 0.00 0.00 0.11 0.17 0.00 0.28 0.28 * 0.45 0.71 

0.10 0.10 0.00 0.00 0.11 0.15 0.00 0.12 0.12 * 0.32 0.87 

0.20 0.20 0.00 0.00 0.10 0.16 0.01 0.04 0.03 * 0.17 0.02 

0.30 0.30 0.00 0.00 0.10 0.16 0.02 0.04 0.02 0.62 0.08 0.05 

0.40 0.40 0.00 0.00 0.10 0.16 0.05 0.06 0.01 0.20 0.06 0.07 

0.50 0.50 0.00 0.00 0.10 0.16 0.08 0.09 0.01 0.13 0.06 0.10 

0.60 0.06 0.00 0.00 0.11 0.17 0.13 0.14 0.01 0.08 0.08 0.12 

0.70 0.70 0.00 0.00 0.11 0.16 0.18 0.19 0.01 0.06 0.08 0.14 

0.80 0.80 0.00 0.00 0.11 0.17 0.23 0.23 0.00 0.00 0.10 0.17 

0.90 0.90 0.00 0.00 0.11 0.17 0.28 0.28 0.00 0.00 0.10 0.17 

1.00 1.00 0.00 0.00 0.11 0.16 0.33 0.33 0.00 0.00 0.11 0.19 

1.10 1.10 0.00 0.00 0.10 0.16 0.38 0.39 0.01 0.03 0.11 0.19 

1.20 1.20 0.00 0.00 0.10 0.16 0.42 0.43 0.01 0.02 0.12 0.19 

1.30 1.30 0.00 0.00 0.10 0.17 0.46 0.47 0.01 0.02 0.11 0.20 

Table 4.18: Establishing bias and error bars of ˆ βSI 2 calculated by second β-smile condition: � Y ≥ Y (k) � and effect on ˆ λ +,− estimated by applying 

the non-parametric approach according to Sornette & Malevergne to 1000 synthetic samples consisting of N = 2507 data points. The bias is 

calculated by comparing mean values ˆ βalldata and ˆ λ +,− 

( ˆ βall data) to original values denoted by βorig and corresponding tail dependence estimator 

ˆλ +,− (βorig) and for the estimation of error bars standard deviation ’std’ and 95% quantiles of the estimates are provided. Tail indexes α were set 

equal to three to avoid a second source of error and ∗ means that relative bias (’rel.bias’) calculated as percentage of ˆ λ +,− 

(βorig) ≥ 100%.

152 

α ˆν bias rel. bias std 95% quantile ˆ b γ n bias rel. bias std 95% quantile 

1.50 1.53 0.03 0.02 0.16 0.27 1.50 0.00 0.00 0.21 0.35 

2.00 2.05 0.05 0.03 0.21 0.34 2.02 0.02 0.01 0.29 0.51 

2.50 2.56 0.06 0.02 0.26 0.44 2.53 0.03 0.01 0.36 0.65 

3.00 3.07 0.07 0.02 0.31 0.52 2.99 0.01 0.03 0.41 0.76 

3.50 3.57 0.07 0.02 0.38 0.62 3.50 0.00 0.00 0.49 0.81 

4.00 4.07 0.07 0.02 0.46 0.79 3.98 0.02 0.01 0.59 1.05 

4.50 4.59 0.09 0.02 0.45 0.80 4.50 0.00 0.00 0.60 1.00 

5.00 5.12 0.12 0.02 0.53 0.95 5.04 0.04 0.01 0.73 1.29 

Table 4.19: Establishing bias and error bars of ˆν and ˆb γ n, estimated for 1000 synthetic samples consisting of N = 2507 data points. The bias is 

calculated by comparing mean values ˆν and ˆb γ 

n to original values denoted by α and for the estimation of error bars standard deviation ’std’ and 

95% quantiles of the estimates are provided. Relative bias (’rel.bias’) is calculated as percentage of α.

153 

βorig 

ˆλ +,− 

(α,βorig) 

+,− 

λ ˆ (ˆν, ˆ βall data) bias rel. bias std 95% quantile 

+,− 

λ ˆ (ˆν, ˆ α = 3 

βSI 2) bias rel. bias std 95% quantile 

0.05 0.00 0.01 0.01 * 0.08 0.01 0.27 0.27 * 0.44 0.73 

0.10 0.00 0.00 0.00 0.30 0.00 0.00 0.12 0.12 * 0.32 0.88 

0.20 0.01 0.01 0.00 0.05 0.01 0.01 0.05 0.04 * 0.18 0.03 

0.30 0.02 0.02 0.00 0.01 0.01 0.02 0.03 0.01 0.39 0.05 0.05 

0.40 0.05 0.05 0.00 0.01 0.02 0.04 0.06 0.01 0.21 0.05 0.08 

0.50 0.09 0.08 0.00 0.00 0.03 0.06 0.09 0.01 0.11 0.07 0.11 

0.60 0.13 0.12 0.00 0.02 0.04 0.07 0.14 0.01 0.08 0.07 0.14 

0.70 0.18 0.17 0.00 0.01 0.04 0.07 0.19 0.01 0.06 0.09 0.16 

0.80 0.23 0.23 0.00 0.00 0.05 0.08 0.24 0.01 0.06 0.10 0.16 

0.90 0.28 0.27 0.00 0.00 0.06 0.10 0.28 0.01 0.02 0.11 0.19 

1.00 0.33 0.32 0.01 0.02 0.06 0.10 0.33 0.01 0.02 0.12 0.20 

1.10 0.38 0.37 0.00 0.00 0.06 0.10 0.38 0.00 0.01 0.12 0.18 

1.20 0.42 0.42 0.01 0.01 0.06 0.11 0.42 0.00 0.00 0.12 0.20 

1.30 0.46 0.46 0.01 0.01 0.06 0.10 0.47 0.01 0.01 0.13 0.21 

Table 4.20: Establishing bias and error bars of ˆ λ +,− (ˆν, ˆ βall data) with ˆ β calculated by all data and ˆ λ +,− (ˆν, ˆ βSI 2) with ˆ β calculated by second 

β-smile condition: � Y ≥ Y (k) � by applying the non-parametric approach according to Sornette & Malevergne to 1000 synthetic samples consisting 

of N = 2507 data points. The bias is calculated by comparing mean values ˆ λ +,− 

(ˆν, ˆ βall data) and ˆ λ +,− 

(ˆν, ˆ βSI 2) to original values denoted by 

ˆλ +,− 

(α,βorig) and for the estimation of error bars standard deviation ’std’ and 95% quantiles of the estimates are provided. ∗ means that relative 

bias (’rel.bias’) calculated as percentage of ˆ λ +,− 

(βorig) ≥ 100%.

We also find a bias of Hill’s estimator denoted by ˆν reported in table (4.19). Com- 

pared to Gabaix’s estimator denoted by ˆb γ n the bias of the Hill estimator is significantly 

bigger but ˆb γ n instead has a wider error bar. 

In table (4.20) results of ˆ λ +,− 

(ˆν, ˆ βalldata) and ˆ λ +,− 

(ˆν, ˆ βSI2) estimated as mean values 

of 1000 synthetic samples with identical inputs are shown. We notice that the bias of the 

tail dependence estimators does not significantly change when we calculate the tail index 

by the Hill estimator compared to results reported in tables (4.17) and (4.18), where we 

used correct α. Error bars instead become larger because as shown in (4.19) tail index 

estimators add another source of uncertainty to tail dependence estimators. Anyway 

the calculation of Hill’s estimator is faster and more simple compared to Gabaix’s 

estimator and as tail dependence estimators are not sensitive to moderate changes of 

tail index α but are very sensitive to changes in β, Hill’s estimator might be accurate 

enough here because on synthetic time series bias resulting from the Hill estimator has 

negligible effect on the tail dependence coefficient estimates. 

We conclude that, applied to our synthetic time series, which are generally built of 

Gaussian distributions with tails modeled by heavy tailed distribution parts, concepts 

according to Sornette & Malevergne with ˆ β calculated by all data allow to estimate 

λ +,− in a accurate way showing us the best solution that we can expect. Furthermore 

we found that the second β-smile condition performs well for βorig ≥ 0.5 with respect 

to relative bias as well as error bars. 

Now, for comparison, I implemented concepts according to Poon, Rockinger, and 

Tawn and concepts according to Schmidt & Stadtmüller to synthetic samples. For 

these concepts there is no need to calculate any linear factor, but for example we 

can compare the results to the tail dependence estimators according to Sornette & 

Malevergne applying their non-parametric approach and compare the resulting error 

bars to error bars estimated by bootstrap sampling with replacement. Table (4.21) 

shows tail dependence coefficients estimated by the different concepts including error 

bars calculated for 1000 synthetic samples compared to their average values. Tail 

indexes were calculated by Hill’s estimator. For comparison I added βorig and ˆ λ +,− (βorig) 

with tail indexes also calculated by Hill’s estimator. Respective standard deviations 

and 95% quantiles, also given in table (4.21), show us that error bars resulting from 

both factors, ˆν and ˆ β are around double size of errors bars just resulting from ˆ β given 

in table (4.17) and of same extent as error bars estimated by bootstrap sampling with 

replacement in subsection (3.2.3). For the methods according to Schmidt & Stadtmüller 

a relation for the calculation of the estimates’ variance σ2 L (θ) was proposed in their 

paper [24] (2005) that was already discussed in subsection (3.4.3). Applied to real 

historical data of index S&P 500 and corresponding assets the error bar estimates given 

by relation (3.92) were high compared to bootstrap quantiles. Compared to the error 

bars calculated for the 1000 synthetic samples the proposed estimator works better. 

σL(θ) are close to standard deviations ’std’ calculated for the 1000 samples compared 

to their average values. Another standard deviation estimator ˆσ, which was introduced 

in the paper [13] (2004) to provide error bar estimates for ˆχ +,− seems significantly lower 

than bootstrap estimates of real historical data provided in subsection (3.1.2) as well as 

standard deviations calculated for the 1000 samples compared to their average values 

for synthetic samples provided in table (4.21). Looking at total values of the estimates, 

the different estimators still yield mostly different results. In terms of error bars the 

three presented methods all yield reasonable results that seem to be approximately 

Gaussian distributed and agree well with results presented in chapter (3). 

154

155 

βorig 

ˆλ +,− EV T 

(ˆν,βall data) std 95% quantile λ·,m 

ˆ std 95% quantile λˆ ·,m std 95% quantile ˆχ ˆσ std 95% quantile 

0.05 0.01 0.08 0.01 0.05 0.02 0.04 0.05 0.02 0.04 0.87 0.09 0.13 0.06 

0.1 0.00 0.00 0.00 0.06 0.02 0.04 0.06 0.02 0.04 0.87 0.09 0.14 0.07 

0.2 0.01 0.01 0.01 0.10 0.03 0.05 0.10 0.03 0.05 0.87 0.09 0.14 0.07 

0.3 0.02 0.01 0.02 0.15 0.03 0.05 0.15 0.03 0.05 0.87 0.08 0.14 0.06 

0.4 0.05 0.02 0.04 0.19 0.04 0.06 0.19 0.06 0.05 0.87 0.09 0.14 0.07 

0.5 0.08 0.03 0.06 0.24 0.04 0.06 0.25 0.04 0.06 0.88 0.09 0.15 0.07 

0.6 0.12 0.04 0.07 0.28 0.04 0.06 0.28 0.04 0.06 0.88 0.09 0.14 0.06 

0.7 0.17 0.04 0.07 0.32 0.04 0.07 0.32 0.04 0.07 0.88 0.09 0.13 0.06 

0.8 0.23 0.05 0.08 0.36 0.04 0.07 0.36 0.04 0.07 0.89 0.09 0.12 0.06 

0.9 0.27 0.06 0.10 0.40 0.04 0.07 0.40 0.04 0.07 0.89 0.09 0.13 0.06 

1 0.32 0.06 0.10 0.43 0.04 0.07 0.43 0.04 0.07 0.89 0.09 0.12 0.05 

1.1 0.37 0.06 0.10 0.46 0.04 0.07 0.46 0.04 0.07 0.89 0.09 0.12 0.05 

1.2 0.42 0.06 0.11 0.49 0.04 0.07 0.49 0.04 0.07 0.89 0.09 0.12 0.05 

1.3 0.46 0.06 0.10 0.51 0.04 0.07 0.52 0.04 0.07 0.89 0.09 0.12 0.05 

Table 4.21: Comparison of estimates by non-parametric estimators ˆ λ·,m, ˆ EV T 

λ·,m according to Schmidt & Stadtmüller, and ˆχ+,− according to Poon, 

Rockinger, and Tawn with ˆ λ +,− (βall data) according to Sornette & Malevergne by applying the different concepts to 1000 synthetic samples consisting 

of N = 2507 data points created by different βorig. For the estimation of error bars standard deviation ’std’ and 95% quantiles of the estimates 

are calculated and for ˆχ +,− the proposed standard deviation ˆσ is given for comparison. Tail indexes α were estimated by the Hill estimator.

Chapter 5 

Concluding Remarks 

Now we summarize the findings of chapter (4) and finally draw some conclusions for 

practical use of the different concepts. 

In section (4.1) we have seen that all tail dependence coefficients ˆ λ +,− estimated by 

the non-parametric approach and by the parametric approach according to Sornette 

& Malevergne for assets and indexes traded outside the U.S. were vanishing while all 

estimates of the crucial parameter ˆ β were comparably small. This seems implausible. 

Furthermore we found on U.S. assets and indexes that for ˆ β < 0.2 tail dependence 

coefficient estimates ˆ λ → 0. However, for index AORD (ASX) and included assets 

results estimated by concepts according to Schmidt & Stadtmüller reported in table 

(4.11) look reasonable. To achieve diversification of extreme financial risks within a 

portfolio we are looking for small or vanishing tail dependence coefficients, which we 

expect to be rare. 

In contrast, as shown in section (4.2), applied to exchange rates of various foreign 

currencies with the US$ results by the methods according to Sornette & Malevergne 

were more reasonable. Concepts according to Sornette & Malevergne yielded mostly 

vanishing results while results by concepts according to Schmidt & Stadtmüller were 

all close together and around 0.9. Approaches according to Poon, Rockinger, and Tawn 

again yielded nearly identical results for all cases. 

In section (4.3) we found that the non-parametric approach according to Sornette 

& Malevergne with ˆ β estimated for all data applied to synthetic time series succeeded 

in providing estimates that were unbiased and of satisfying accuracy. This at least 

provides us with lower error bounds (the optimum solution we can expect) and is a 

promising result for application of the concepts to real data. Estimation of tail index 

α by Hill’s estimator has shown a small bias on our synthetic sample surveys reported 

in table (4.19). But the impact on the final estimators, as shown in table (4.20), was 

marginal. Furthermore, for real data we can apply the Gabaix estimator to calculate 

tail index ˆα for comparison because in spite of wider error bars the Gabaix estimator 

should perform better than Hill’s estimator with respect to bias facing heteroscedastic 

time series. 

Anyway, at the example of synthetic time series we were assuming that assets and index 

have a common factor that is constant over the whole distribution interval, whereas 

by the β-smile improvement conditions we are generally assuming that dependence in 

the extreme parts of the distributions might be different compared to dependence in 

the body of the distributions. As we were able to calculate the common factors β only 

by tail data applying the second β-smile condition reported in table (4.18), we should 

always check on real data whether βalldata is close to βSI· to get a feeling about dependences 

between assets and index. Applying the β-smile improvement conditions to 

156

synthetic time series we detected a strong bias in the first and in the third SI conditions. 

The second SI condition has shown to perform well for (β > 0.5) with respect to bias. 

Tables (4.6), (4.8), and (4.10) show ˆ β-values calculated for all data on historical time 

series that are much bigger than all of our synthetic sample estimates ˆ β reported in 

table (4.17), which all fulfill ˆ β < 0.5. This might indicate that the bias of historic time 

series could be different from the bias of synthetic time series. Now we come to the final 

comparison of the different concepts on synthetic time series: in table (4.21) we see 

all the different concepts including error bar estimates applied to synthetic time series. 

Estimates by the different methods are mostly inconsistent as it was also the case for 

historic time series. Whereas ˆχ is always big and within a narrow range, the two other 

estimators ˆ λ +,− and ˆ λ·,m are on a wider range but still different from each other. What 

all methods have in common is that they are increasing in increasing β, whereas ˆ λ·,m 

tends to be slightly higher than ˆ λ +,− primarily for small ˆ β. This is consistent with 

real data results i.e. reported in section (4.1) where the different concepts were applied 

to major financial centers in the world. Comparing estimates of λ +,− to estimates of 

λ·,m on the historical reference data series of the index S&P 500 and the nine assets 

provided in tables (3.8) and (3.10) for the Sornette & Malevergne estimators and in 

tables (3.24) and (3.25) for the Schmidt & Stadtmüller estimators we detected that 

within different ranges the relative proportions agree in many cases. Asset Texas Instruments 

Inc. (TXN) is the smallest observation in positive and negative tails observed 

by both methods and sample sizes and Hewlett-Packard Co. (HPQ) and Coca-Cola Co. 

(KO) both belong to the biggest observations in the positive and negative tails also for 

both reference data sets. There are many of these conformities showing us that the 

two estimators qualitatively point into the same direction. As it is our purpose to find 

differences between the tail dependences of the different assets and the index for small 

ˆβ, we should always calculate λ·,m according to Schmidt & Stadtmüller for comparison 

because λ +,− according to Sornette & Malevergne mostly yield zero tail dependence in 

that case. 

It cannot be concluded that one of the presented concepts for the estimation of 

tail dependence outperforms the others by all criteria. Anyway, having the methods 

on our hands that were discussed within this report, I would recommend to apply 

all of the available concepts for the estimation of the tail dependence coefficient to 

various data representing alternative choices out of the field of interest and then draw 

conclusions considering the differences between the estimates of alternative portfolio 

choices. Having a pool of alternatives evaluated by the different concepts might allow 

the investor to pick out the method that fits best to his respective purpose and helps 

him to get an intuition about dependences between assets and indexes over the whole 

range of the distribution. Finally it might enable him to choose between the considered 

alternatives. 

The coefficient of tail dependence provides the investor with a tool to minimize 

extreme correlations within his portfolio, which first of all should help him to endure 

bearish markets by optimized diversification criteria. Globalization might have lead 

to increasing extreme dependences across and within economies making extreme lefttail 

events more and more likely to happen. To be responsive to this movements the 

coefficient of tail dependence represents a simple and concrete extension to financial 

risk management. 

157

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159

Appendix 

M-Files 

In the first part of the appendix Matlab m-files are provided. Comments are parenthesized 

in: %...%, which is also read as a comment by the Matlab solver. Implementation 

works by copy and paste, whereas data sets should be provided in Excel format (xls or 

xlsx). 

Import of Data 

clear all; 

exl = actxserver(’excel.application’); 

exlWkbk = exl.Workbooks; 

exlFile = exlWkbk.Open([’%set path for opening%’]); 

nn=%number of rows%; 

n=%number of columns%; 

exlSheet=exlFile.Sheets.Item(’table1’); %choice of table%; 

dat_range = [’B3:e’ num2str(nn)]; %range in Excel file%; 

rngObj=exlSheet.Range(dat_range); 

exlData=rngObj.Value; %end of read-in%; 

Prices(:,:)=exlData; 

Prices=cell2mat(Prices); %define Prices as matrix%; 

for j=1:n; 

%for i=nn-2:-1:1; 

for i=nn-3:-1:1; 

Returns(i,j)=log(Prices(i,j))-log(Prices(i+1,j)); 

end 

end %transform daily prices to returns% 

a=1; 

b=nn-3; %data range% 

Data can also be imported by: 

Prices=xlsread(’testdata.xls’, 1, ’A4:B5’) %1 denotes table1% 

But then the Excel-data sheet has to be copied into the folder where the m-file is safed. 

160

Approach according to Poon, Rockinger, and Tawn 

retdesc=sort(Returns(a:b,1:n),’descend’); 

retasc=sort(Returns(a:b,1:n)); %sort Returns% 

Z=roundn((b-a)*0.04,0); %define ’k’% 

ZZ=roundn(0.005*(b-a),0); %define ’c’% 

%calculate tail indexes by the Hill estimator%; 

for j=1:n; 

vk(j)=(1/(Z)*sum(log(retdesc(1:Z,j)))-log(retdesc(Z,j)))^-1; 

vka(j)=(1/(Z)*sum(log(-retasc(1:Z,j)))-log(-retasc(Z,j)))^-1; 

end 

%calculate l% 

t=linspace(1,Z,Z); 

p=size(t); 

p=p(2); 

for j=1:n; 

for i=1:p; 

f=t(i); 

d=@(f)(f/(b-a)*(retdesc(f,j))^vk(j)); 

dd=@(f)(f/(b-a)*(-retasc(f,j))^vka(j)); 

lpos(i,j)=d(f); 

lneg(i,j)=dd(f); 

end 

end 

lmk_pos=mean(lpos(1:Z,:)); 

lmy_pos=mean(lpos(ZZ:Z,:)); 

lmk_neg=mean(lneg(1:Z,:)); 

lmy_neg=mean(lneg(ZZ:Z,:)); 

%calculate S and T% 

for j=1:n-1; 

for i=1:Z; 

S(i,1)=-1/log(1-lmk_pos(1)*retdesc(i,1)^(-vk(1))); 

T(i,j)=-1/log(1-lmk_pos(j+1)*retdesc(i,j+1)^(-vk(j+1))); 

S_y(i,1)=-1/log(1-lmy_pos(1)*retdesc(i,1)^(-vk(1))); 

T_y(i,j)=-1/log(1-lmy_pos(j+1)*retdesc(i,j+1)^(-vk(j+1))); 

Sn(i,1)=-1/log(1-lmk_neg(1)*(-retasc(i,1))^(-vka(1))); 

Tn(i,j)=-1/log(1-lmk_neg(j+1)*(-retasc(i,j+1))^(-vka(j+1))); 

Sn_y(i,1)=-1/log(1-lmy_neg(1)*(-retasc(i,1))^(-vka(1))); 

Tn_y(i,j)=-1/log(1-lmy_neg(j+1)*(-retasc(i,j+1))^(-vka(j+1))); 

end 

end 

%calculate Z% 

for j=1:n-1; 

for i=1:Z; 

z(i,j)=min(S(i,1),T(i,j)); 

z_y(i,j)=min(S_y(i,1),T_y(i,j)); 

zn(i,j)=min(Sn(i,1),Tn(i,j)); 

zn_y(i,j)=min(Sn_y(i,1),Tn_y(i,j)); 

end 

end 

161

%estimate overline_chi and corresponding sigma% 

for j=1:n-1; 

ovchi(j)=2/Z*(sum(log(z(:,j)./z(Z,j))))-1; 

sigovchi(j)=(ovchi(j)+1)/sqrt(Z); 

ovchi_y(j)=2/Z*(sum(log(z_y(:,j)./z_y(Z,j))))-1; 

sigovchi_y(j)=(ovchi_y(j)+1)/sqrt(Z); 

ovchin(j)=2/Z*(sum(log(zn(:,j)./zn(Z,j))))-1; 

sigovchin(j)=(ovchin(j)+1)/sqrt(Z); 

ovchin_y(j)=2/Z*(sum(log(zn_y(:,j)./zn_y(Z,j))))-1; 

sigovchin_y(j)=(ovchin_y(j)+1)/sqrt(Z); 

end 

%output% 

ovchi 

sigovchi 

%estimate chi and corresponding sigma if overline_chi=1% 

for j=1:n-1; 

if abs(ovchi(j)-1)

Non-Parametric Approach according to Sornette & Malevergne 




Y=roundn(0.005*(b-a),0); %define ’c’% 

kretdesc=retdesc(1:Z,:); %from 1 to ’k’; 

yretdesc=retdesc(Y:Z,:); %from ’c’ to ’k’; 

kretasc=retasc(1:Z,:); 

yretasc=retasc(Y:Z,:); 

%calculate beta for the whole data set% 

for q=1:n-1; 

coeff=polyfit(Returns(a:b,1),Returns(a:b,q+1),1); 

bbeta(q)=coeff(1); 

end 

%estimate tail indexes by the Hill estimator% 

vk=(1/(Z)*sum(log(kretdesc(:,1)))-log(kretdesc(Z,1)))^-1; 

vka=(1/(Z)*sum(log(-kretasc(:,1)))-log(-kretasc(Z,1)))^-1; 

%estimate ’l’ and lambda% 

for i=1:n-1; 

ly(i)=mean(yretdesc(:,i+1)./(yretdesc(:,1))); 

lk(i)=mean(kretdesc(:,i+1)./(kretdesc(:,1))); 

lya(i)=mean(yretasc(:,i+1)./(yretasc(:,1))); 

lka(i)=mean(kretasc(:,i+1)./(kretasc(:,1))); 

if(bbeta(i)>0) 

L_nonpar_y_up(i)=1/(max(1,ly(i)/bbeta(i)))^vk; 

L_nonpar_k_up(i)=1/(max(1,lk(i)/bbeta(i)))^vk; 

L_nonpar_y_lo(i)=1/(max(1,lya(i)/bbeta(i)))^vka; 

L_nonpar_k_lo(i)=1/(max(1,lka(i)/bbeta(i)))^vka; 

elseif(bbeta(i)

Parametric Approach according to Sornette & Malevergne 




ZZ=roundn(0.005*(b-a),0); %define ’c’% 

%estimate tail indexes by the Hill estimator% 

vk=(1/(Z)*sum(log(retdesc(1:Z,1)))-log(retdesc(Z,1)))^-1 

vka=(1/(Z)*sum(log(-retasc(1:Z,1)))-log(-retasc(Z,1)))^-1 

%calculate beta for the whole data set% 

for q=1:n-1; 

coeff=polyfit(Returns(a:b,1),Returns(a:b,q+1),1); 

bbeta(q)=coeff(1); 

end 

%calculate error epsilon% 

for i=1:n-1; 

eps=Returns(:,i+1)-(bbeta(1,i)*Returns(:,1)); 

Eps(:,i)=eps; 

end 

%estimate Cy and mean value of Cy for most extr. 4 per cent% 

t=linspace(1,Z,Z); 

p=size(t); 

p=p(2); 

for i=1:p; 

f=t(i); 

d=@(f)(f/(b-a)*(retdesc(f,1))^vk); 

dd=@(f)(f/(b-a)*(-(retasc(f,1)))^vka); 

cypos(i,1)=d(f); 

cyneg(i,1)=dd(f); 

end; 

cymk_pos=mean(cypos(1:Z)); 

cymy_pos=mean(cypos(ZZ:Z)); 

cymk_neg=mean(cyneg(1:Z)); 

cymy_neg=mean(cyneg(ZZ:Z)); 

%estimate Ceps and mean value of Ceps for most extr. 4 per cent% 

epsdesc=sort(Eps(a:b,:),’descend’); 

epsasc=sort(Eps(a:b,:)); 

for i=1:n-1; 

for j=1:p; 

f=t(j); 

d=@(f)(f/(b-a)*(epsdesc(f,i))^vk); 

dd=@(f)(f/(b-a)*(-(epsasc(f,i)))^vka); 

cepos(j,i)=d(f); 

ceneg(j,i)=dd(f); 

end 

end 

cemk_pos=mean(cepos(1:Z,:)); 

cemy_pos=mean(cepos(ZZ:Z,:)); 

cemk_neg=mean(ceneg(1:Z,:)); 

cemy_neg=mean(ceneg(ZZ:Z,:)); 

164

%estimate lambda% 

for j=1:n-1; 

if(bbeta(j)>0) 

L_par_k_up(1,j)=(1+(bbeta(1,j)^-vk)*cemk_pos(1,j)/cymk_pos)^-1; 

L_par_y_up(1,j)=(1+(bbeta(1,j)^-vk)*cemy_pos(1,j)/cymy_pos)^-1; 

L_par_k_lo(1,j)=(1+(bbeta(1,j)^-vka)*cemk_neg(1,j)/cymk_neg)^-1; 

L_par_y_lo(1,j)=(1+(bbeta(1,j)^-vka)*cemy_neg(1,j)/cymy_neg)^-1; 

elseif(bbeta(j)

β-Smile Improvement Conditions 

A=Returns(a:b,1:n); 

retdesc=sort(A,’descend’); 

retasc=sort(A); %sort Returns% 

Z=roundn((b-a+1)*0.04,0); %define ’k’% 

Y=roundn(0.005*(b-a+1),0); %define ’c’% 

%Implementation of chosen condition and calculation of beta_SI% 

for u=1:n-1; 

h=1; 

ha=1; 

for i=1:b-a+1; 

if (A(i,1)>=retdesc(Z,1))&&(A(i,u+1)>=retdesc(Z,u+1)); 

%if (A(i,1)>=retdesc(Z,1)); 

%if (A(i,u+1)>=retdesc(Z,u+1)); 

D(h,1)=A(i,1); 

D(h,2)=A(i,u+1); 

h=h+1; 

end 

if (A(i,1)

Approaches according to Schmidt & Stadtmüller 


p=size(A); 

B=sort(A); 

C=zeros(p(1,1),p(1,2)); 

for k=1:p(1,2); along columns% 

pp=1; 

for i=1:p(1,1); %along rows% 

v=find(A(:,k)==B(i,k)); 

t=size(v); 

t=t(1,1); 

if t>1; 

for z=1:t; 

C(v(z),k)=0.5-t/2+i; 

end 

else C(v,k)=i; 

v=[]; 

end 

end 

end 

k=0.04*(b-a+1); 

sum_L(1,1:n-1)=0; 

sum_U(1,1:n-1)=0; 

sum_L_EVT(1,1:n-1)=0; 

sum_U_EVT(1,1:n-1)=0;%set sums to zero% 

%conditions for the different sums% 

for j=1:n-1; 

for i=1:b-a+1; 

if(C(i,1)(b-a+1)-k); 

sum_U(1,j)=sum_U(1,j)+1; 

end 

if(C(i,1)>(b-a+1)-k) || (C(i,j+1)>(b-a+1)-k); 

sum_U_EVT(1,j)=sum_U_EVT(1,j)+1; 

end 

if(C(i,1)

Tail Indexes by the Gabaix Estimator 


AA=sort(A,’descend’); 

B=sort(-A,’descend’); %sort Returns% 

p=size(A); 

C=NaN(p(1,1),p(1,2)); 

c=NaN(p(1,1),p(1,2)); 

for k=1:p(1,2); %along columns% 

for i=1:p(1,1); %along rows% 

v=find(-A(:,k)==B(i,k));%negative tail% 

vv=find(A(:,k)==AA(i,k));%positive tail% 

t=size(v); 

t=t(1,1); 

tt=size(vv); 

tt=tt(1,1); 

if t>1; 

for z=1:t; 

C(v(z),k)=0.5-t/2+i;%negative tail% 

end 

else C(v,k)=i; 

end 

if tt>1; 

for z=1:tt; 

c(vv(z),k)=0.5-tt/2+i;%positive tail% 

end 

else c(vv,k)=i; 

end 

end 

end 

Z=round(0.04*(b-a+1)); %define ’k’% 

Y=round(0.005*(b-a)); %define ’c’% 

AAA=log(AA(1:Z,:)); 

BB=log(B(1:Z,:)); 

CC=log(sort(C)-0.5); 

cc=log(sort(c)-0.5); 

for i=1:n; 

p=polyfit(AAA(:,i),cc(1:Z,i),1);%positive tail 

pp=polyfit(BB(:,i),CC(1:Z,i),1);%negative tail 

bnn(i)=-(pp(1,1)); 

bn(i)=-(p(1,1)); 

end 

%output% 

bn %upper tail indexes% 

bnn %lower tail indexes% 

168

Composition of Synthetic Data Set 

clear all 

for jj=1:1000; 

p=2507; %sample size% 

vvvv=3; %tail index% 

bbeta=0.9; %predefined beta% 

u1=rand(p,1); 

u2=rand(p,1); 

Z=round(0.04*p); 

u1desc=sort(u1,’descend’); 

u2desc=sort(u2,’descend’); 

%Box-Muller trafo; 

z1(:,1)=sqrt(2) *erfinv (2*u1desc - 1); 

z2(:,1)=sqrt(2) *erfinv (2*u2desc - 1); 

%here we have the standard normal distributed part for eps and x% 

u1desc_up(:,1)=rand(Z,1); 

u2desc_up(:,1)=rand(Z,1); 

u1desc_lo(:,1)=rand(Z,1); 

u2desc_lo(:,1)=rand(Z,1); 

for i=1:Z; 

z1_up(i,1)=(1/(1-u1desc_up(i,1)))^(1/vvvv); 

z2_up(i,1)=(1/(1-u2desc_up(i,1)))^(1/vvvv); 

z1_lo(i,1)=-((1/(1-u1desc_lo(i,1)))^(1/vvvv)); 

z2_lo(i,1)=-((1/(1-u2desc_lo(i,1)))^(1/vvvv)); 

end 

%heavy tail parts are created% 

%set the different distribution parts together into a single total% 

%distribution% 

z1_up=sort(z1_up,’descend’); 

z2_up=sort(z2_up,’descend’); 

z1_lo=sort(z1_lo,’descend’); 

z2_lo=sort(z2_lo,’descend’); 

z1_up=z1(101,1)/z1_up(Z,1)*z1_up; 

z2_up=z2(101,1)/z2_up(Z,1)*z2_up; 

z1_lo=z1(p-Z)/z1_lo(1,1)*z1_lo; 

z2_lo=z2(p-Z)/z2_lo(1,1)*z2_lo; 

z1(1:Z,1)=z1_up; 

z2(1:Z,1)=z2_up; 

z1(p-Z+1:p,1)=z1_lo; 

z2(p-Z+1:p,1)=z2_lo; 

y=z1; 

eps=z2; 

169

%define mu and sigma if requested% 

y_s=(0.05*10^(-3)+y(:).*0.02)*0.5; 

eps_s=(0.05*10^(-3)+eps(:).*0.02)*0.5; 

%now we have to build the y values by x, eps, and beta% 

%The problem now is that both x_s and eps_s are in descending order, which% 

%means that high return-values correspond to high eps-values. This is a% 

%violation of i.i.d. Therefore we bring back the random sequence to our% 

%samples by uniform sampling w.o. replacement% 

o1=randperm(p)’; 

o2=randperm(p)’; 

for i=1:p 

ys(i,1)=y_s(o1(i),1); 

epss(i,1)=eps_s(o2(i),1); 

x_s(i,1)=ys(i,1)*bbeta+epss(i,1); 

end 

xdesc=sort(x_s,’descend’); 

epsdesc=sort(epss,’descend’); 

ydesc=sort(ys,’descend’); 

xasc=sort(x_s); 

epsasc=sort(epss); 

yasc=sort(ys); 

170

Descriptive Statistics 

The kurtosis is a measure of the extent to which data is concentrated in the peak versus 

the tail. A positive value (leptokurtic) indicates that data is concentrated in the peak; 

a negative value (platykurtic) indicates that data is concentrated in the tail. For all 

assets and indexes we observe a kurtosis greater than zero, which is inconsistent with the 

assumption of Gaussian distributed returns. The negative skewness for all assets means 

that there is an elongated tail on the left of the probability density distribution function 

meaning that we find more data in the left tail compared to Gaussian distributed 

returns. 

N Mean Std Skew. Kurt. 

Statistics Statistics Statistics Statistics Std. error Statistik Std. error 

ret SP500 2506 2.43E-04 4.05E-03 -3.21E-01 4.90E-02 5.42E+00 9.80E-02 

ret BMY 2506 1.40E-05 1.20E-02 -1.41E+01 4.90E-02 3.61E+02 9.80E-02 

ret CVX 2506 2.30E-05 8.73E-03 -1.60E+01 4.90E-02 5.46E+02 9.80E-02 

ret HPQ 2506 1.00E-05 1.38E-02 -9.81E+00 4.90E-02 1.96E+02 9.80E-02 

ret KO 2506 5.10E-05 1.06E-02 -1.61E+01 4.90E-02 4.40E+02 9.80E-02 

ret MMM 2506 2.70E-05 8.65E-03 -1.54E+01 4.90E-02 5.18E+02 9.80E-02 

ret PG 2506 -2.40E-05 1.17E-02 -1.49E+01 4.90E-02 3.62E+02 9.80E-02 

ret SGP 2506 4.60E-05 1.38E-02 -1.28E+01 4.90E-02 2.83E+02 9.80E-02 

ret TXN 2506 3.00E-05 1.81E-02 -7.49E+00 4.90E-02 1.26E+02 9.80E-02 

ret WAG 2506 -2.70E-05 1.44E-02 -1.38E+01 4.90E-02 2.82E+02 9.80E-02 

Table 5.1: Descriptive statistics of historical return series of S&P 500 and nine assets included 

during a time interval from January 1991 to December 2000 (smaller reference sample of 

chapter (3)) 


Statistic Statistic Statistic Statistic Std. Error Statistic Std. Error 

ret SP500 5736 1.46E-04 4.61E-03 -1.94E+00 3.20E-02 4.15E+01 6.50E-02 

ret BMY 5736 -7.90E-05 1.08E-02 -1.27E+01 3.20E-02 3.54E+02 6.50E-02 

ret CVX 5736 6.30E-05 8.71E-03 -1.43E+01 3.20E-02 4.87E+02 6.50E-02 

ret HPQ 5736 2.10E-05 1.20E-02 -6.49E+00 3.20E-02 1.51E+02 6.50E-02 

ret KO 5736 -1.00E-05 1.10E-02 -2.02E+01 3.20E-02 6.87E+02 6.50E-02 

ret MMM 5736 0.00E+00 9.32E-03 -1.73E+01 3.20E-02 5.53E+02 6.50E-02 

ret PG 5736 1.60E-05 1.06E-02 -1.69E+01 3.20E-02 4.61E+02 6.50E-02 

ret SGP 5736 -8.80E-05 1.30E-02 -1.16E+01 3.20E-02 2.66E+02 6.50E-02 

ret TXN 5736 -8.00E-05 1.66E-02 -8.69E+00 3.20E-02 2.09E+02 6.50E-02 

ret WAG 5736 2.40E-05 1.12E-02 -1.29E+01 3.20E-02 3.38E+02 6.50E-02 


during a time interval from July 1985 to April 2008 (bigger reference sample of chapter (3)) 

171



ret SP500 2000 -2.85E-05 4.82E-03 5.34E-02 5.47E-02 2.48E+00 1.09E-01 

ret BMY 2000 -2.38E-04 9.03E-03 -2.08E+00 5.47E-02 2.44E+01 1.09E-01 

ret CVX 2000 -1.09E-05 9.13E-03 -1.78E+01 5.47E-02 5.78E+02 1.09E-01 

ret HPQ 2000 -2.51E-04 1.32E-02 -4.37E+00 5.47E-02 9.46E+01 1.09E-01 

ret KO 2000 5.27E-05 5.74E-03 -1.83E-01 5.47E-02 2.78E+00 1.09E-01 

ret MMM 2000 -4.79E-05 9.33E-03 -1.86E+01 5.47E-02 6.20E+02 1.09E-01 

ret PG 2000 5.12E-06 8.73E-03 -2.07E+01 5.47E-02 7.15E+02 1.09E-01 

ret SGP 2000 -2.36E-04 1.03E-02 -2.10E+00 5.47E-02 2.40E+01 1.09E-01 

ret TXN 2000 -3.56E-04 1.62E-02 -3.17E+00 5.47E-02 6.61E+01 1.09E-01 

ret WAG 2000 6.89E-05 7.69E-03 -4.58E-01 5.47E-02 6.81E+00 1.09E-01 

ret MCD 2000 1.24E-04 7.43E-03 -1.92E-01 5.47E-02 5.15E+00 1.09E-01 

ret C 2000 -1.06E-04 8.58E-03 -2.01E-01 5.47E-02 6.97E+00 1.09E-01 

ret OXY 2000 4.61E-04 7.88E-03 -1.21E-01 5.47E-02 8.12E-01 1.09E-01 

ret DE 2000 -2.06E-04 1.15E-02 -1.18E-01 5.47E-02 6.47E+00 1.09E-01 

ret PNC 2000 1.42E-04 7.33E-03 -4.90E-01 5.47E-02 7.33E+00 1.09E-01 

ret NBL 2000 3.44E-04 9.01E-03 -2.36E-01 5.47E-02 1.77E+00 1.09E-01 

ret PH 2000 2.19E-04 8.36E-03 -7.08E-02 5.47E-02 3.94E+00 1.09E-01 


during a time interval from 12.04.2000 to 31.03.2008 (sample consisting of N = 2000 

observations used in chapter (4)) 



ret DOW 2000 6.02E-06 4.63E-03 -3.83E-02 5.47E-02 3.55E+00 1.09E-01 

ret AA 2000 3.09E-05 9.80E-03 -2.96E-03 5.47E-02 2.09E+00 1.09E-01 

ret BA 2000 8.33E-05 8.36E-03 -7.80E-01 5.47E-02 7.34E+00 1.09E-01 

ret CAT 2000 3.18E-04 8.13E-03 -2.46E-01 5.47E-02 4.49E+00 1.09E-01 

ret GM 2000 -3.06E-04 1.11E-02 5.35E-02 5.47E-02 4.90E+00 1.09E-01 

ret HON 2000 1.17E-04 9.50E-03 2.18E-02 5.47E-02 1.59E+01 1.09E-01 

ret JNJ 2000 1.10E-04 5.31E-03 -1.20E+00 5.47E-02 2.18E+01 1.09E-01 

ret MCD 2000 1.65E-04 7.17E-03 -2.89E-01 5.47E-02 5.39E+00 1.09E-01 

ret WMT 2000 4.62E-05 6.78E-03 2.57E-01 5.47E-02 2.78E+00 1.09E-01 

ret XOM 2000 1.84E-04 6.38E-03 -2.69E-01 5.47E-02 2.53E+00 1.09E-01 

ret HD 2000 -1.54E-04 9.73E-03 -1.77E+00 5.47E-02 2.92E+01 1.09E-01 

ret INTC 2000 -2.25E-04 1.22E-02 -6.62E-01 5.47E-02 8.39E+00 1.09E-01 

ret AIG 2000 -2.52E-04 8.47E-03 -2.79E-01 5.47E-02 6.08E+00 1.09E-01 

Table 5.4: Descriptive statistics of historical return series of Dow Jones Industrial Average 

and 12 assets included during a time interval from 16.08.2000 to 31.07.2008 (sample consisting 

of N = 2000 observations used in chapter (4)) 

172



ret NASDAQ 2000 -1.10E-04 7.37E-03 3.81E-01 5.50E-02 4.73E+00 1.09E-01 

ret AAPL 2000 4.17E-04 1.44E-02 -5.43E+00 5.50E-02 1.19E+02 1.09E-01 

ret INTC 2000 -2.25E-04 1.22E-02 -6.62E-01 5.50E-02 8.39E+00 1.09E-01 

ret MSFT 2000 -3.50E-05 8.70E-03 1.74E-01 5.50E-02 7.79E+00 1.09E-01 

ret CSCO 2000 -2.29E-04 1.29E-02 2.82E-01 5.50E-02 6.67E+00 1.09E-01 

ret CEPH 2000 1.02E-04 1.15E-02 -3.63E-01 5.50E-02 4.50E+00 1.09E-01 

ret PERY 2000 2.76E-04 1.40E-02 -2.20E-01 5.50E-02 1.53E+01 1.09E-01 

ret HSIC 2000 3.87E-04 8.38E-03 2.07E-01 5.50E-02 4.76E+00 1.09E-01 

ret QQQQ 2000 -1.51E-04 8.66E-03 1.92E-01 5.50E-02 5.05E+00 1.09E-01 

ret ONNN 2000 -1.64E-04 2.16E-02 2.02E-01 5.50E-02 6.32E+00 1.09E-01 

Table 5.5: Descriptive statistics of historical return series of NASDAQ Composite and 9 assets 

included during a time interval from 15.08.2000 to 31.07.2008 (sample consisting of N = 2000 

observations used in chapter (4)) 



ret AORD 1398 1.69E-04 3.77E-03 -5.59E-01 6.54E-02 7.26E+00 1.31E-01 

ret CBA 1398 -4.00E-04 2.16E-02 -3.39E+01 6.54E-02 1.23E+03 1.31E-01 

ret BBG 1398 1.62E-04 7.50E-03 5.27E-02 6.54E-02 4.72E+00 1.31E-01 

ret AAC 1398 3.68E-04 9.67E-03 2.74E-01 6.54E-02 4.39E+00 1.31E-01 

ret IIF 1398 -4.79E-05 6.18E-03 -2.72E-01 6.54E-02 8.10E+00 1.31E-01 

ret MOF 1398 -8.33E-05 6.99E-03 -9.28E-01 6.54E-02 1.09E+01 1.31E-01 

ret NCM 1398 -8.33E-05 6.99E-03 -9.28E-01 6.54E-02 1.09E+01 1.31E-01 

ret ABP 1398 9.02E-05 6.13E-03 -3.07E-02 6.54E-02 5.47E+00 1.31E-01 

ret AGK 1398 8.17E-05 7.05E-03 -8.13E+00 6.54E-02 1.69E+02 1.31E-01 

ret BLD 1398 7.02E-05 6.97E-03 1.36E-01 6.54E-02 1.79E+00 1.31E-01 

ret NDO 1398 1.10E-03 5.53E-02 2.16E-01 6.54E-02 2.16E+01 1.31E-01 

ret CEY 1398 4.57E-04 1.07E-02 -1.14E+00 6.54E-02 1.34E+01 1.31E-01 

ret CSR 1398 -3.21E-04 1.84E-02 -2.58E+01 6.54E-02 8.49E+02 1.31E-01 

ret FCL 1398 -2.35E-05 9.85E-03 -1.69E+00 6.54E-02 3.58E+01 1.31E-01 

ret AXA 1398 2.43E-04 7.77E-03 7.45E-01 6.54E-02 8.15E+00 1.31E-01 

Table 5.6: Descriptive statistics of historical return series of AORD (ASX) and 14 assets 

included during a time interval from February 2003 to end of August 2008 (sample consisting 

of N = 1398 observations used in chapter (4)) 

173



A 2000 -1.14E-04 2.30E-03 5.24E-01 5.50E-02 3.43E+00 1.09E-01 

Eur 2000 -1.05E-04 1.93E-03 1.00E-03 5.50E-02 2.63E+00 1.09E-01 

GBP 2000 -5.30E-05 1.74E-03 5.30E-02 5.50E-02 2.14E+00 1.09E-01 

BrR 2000 -1.80E-04 3.88E-03 -1.15E+00 5.50E-02 2.36E+01 1.09E-01 

CHFR 2000 -8.80E-05 2.18E-03 -1.24E-01 5.50E-02 2.58E+00 1.09E-01 

CHY 2000 -3.60E-05 3.05E-04 -1.25E+01 5.50E-02 3.39E+02 1.09E-01 

indoRu 2000 0.00E+00 2.41E-03 1.03E+00 5.50E-02 3.20E+01 1.09E-01 

indRu 2000 -4.10E-05 9.75E-04 -8.20E-02 5.50E-02 7.63E+00 1.09E-01 

JPY 2000 -4.40E-05 1.93E-03 -1.97E-01 5.50E-02 2.64E+00 1.09E-01 

Table 5.7: Descriptive statistics of historical currency exchange rate series during a time 

interval from 19.10.2002 to 10.04.2008 (sample consisting of N = 2000 observations used in 

chapter (4)) 



ret synth1 2507 6.86E-05 1.45E-02 -1.59E+00 4.89E-02 3.45E+01 9.77E-02 

ret synth2 2507 2.93E-04 1.83E-02 -5.45E-01 4.89E-02 1.20E+01 9.77E-02 

ret synth3 2507 2.30E-04 1.30E-02 1.70E-01 4.89E-02 1.12E+01 9.77E-02 

ret synth4 2507 -7.74E-05 1.28E-02 1.17E-01 4.89E-02 5.03E+00 9.77E-02 

ret synth5 2507 6.61E-05 1.72E-02 -3.41E-02 4.89E-02 3.26E+00 9.77E-02 

ret synth6 2507 1.35E-04 1.30E-02 1.74E-01 4.89E-02 7.80E+00 9.77E-02 

Table 5.8: Descriptive statistics of 6 exemplary synthetic samples (sample consisting of N = 

2507 observations used in chapter (4)) 

174

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