Statistical Methods & Applications

, Volume 27, Issue 3, pp 491–513

# Clustering of financial instruments using jump tail dependence coefficient

• Chen Yang
• Wenjun Jiang
• Jiang Wu
• Xin Liu
• Zhichuan Li
Original Paper

## Abstract

In this paper, we propose a new clustering procedure for financial instruments. Unlike the prevalent clustering procedures based on time series analysis, our procedure employs the jump tail dependence coefficient as the dissimilarity measure, assuming that the observed logarithm of the prices/indices of the financial instruments are embedded into multidimensional Lévy processes. The efficiency of our proposed clustering procedure is tested by a simulation study. Finally, with the help of the real data of country indices we illustrate that our clustering procedure could help investors avoid potential huge losses when constructing portfolios.

## Keywords

Clustering analysis Lévy copula Jump tail dependence coefficient Country index

## Notes

### Acknowledgements

The authors are indebted to two anonymous reviewers for comments and suggestions that improved the paper. Jiang Wu is grateful to the support from MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Project No. 10YJC790280).

## References

1. Baragona R (2001) A simulation study on clustering time series with metaheuristic methods. Quaderni di Statistica 3:1–26
2. Billio M, Caporin M (2009) A generalized dynamic conditional correlation model for portfolio risk evaluation. Math Comput Simul 79(8):2566–2578. , http://linkinghub.elsevier.com/retrieve/pii/S0378475408004138
3. Billio M, Caporin M, Gobbo M (2006) Flexible dynamic conditional correlation multivariate GARCH models for asset allocation. Appl Financ Econ Lett 2(2):123–130.
4. Brockwell PJ, Davis RA (eds) (2002) Introduction to time series and forecasting. Springer texts in statistics. Springer, New York. Google Scholar
5. Calinski T, Harabasz J (1974) A dendrite method for cluster analysis. Commun Stat Theory Methods 3(1):1–27.
6. Cont R, Tankov P (2003) Financial modelling with jump processes, vol 2. Financial mathematics series. Chapman and Hall/CRC, Boco Raton.
7. De Luca G, Zuccolotto P (2011) A tail dependence-based dissimilarity measure for financial time series clustering. Adv Data Anal Classif 5(4):323–340.
8. Dobrić J, Schmid F (2005) Nonparametric estimation of the lower tail dependence $$\lambda _L$$ in bivariate copulas. J Appl Stat 32(4):387–407.
9. Durante F, Jaworski P (2010) Spatial contagion between financial markets: a copula-based approach. Appl Stoch Models Bus Ind 26(5):551–564.
10. Durante F, Pappadà R, Torelli N (2014) Clustering of financial time series in risky scenarios. Adv Data Anal Classif 8(4):359–376.
11. Durante F, Pappadà R, Torelli N (2015) Clustering of time series via non-parametric tail dependence estimation. Stat Pap 56(3):701–721.
12. Embrechts M, Arciniegas F, Ozdemir M, Momma M (2001) Scientific data mining with StripMiner/sup TM/. In: SMCia/01. In: Proceedings of the 2001 IEEE mountain workshop on soft computing in industrial applications (Cat. No.01EX504), IEEE, pp 13–16, , http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=936721
13. Engle R (2002) Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional Heteroskedasticity models. J Bus Econ Stat 20(3):339–350
14. Engle R, Sheppard K (2001) Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH. Tech. Rep., National Bureau of Economic Research, Cambridge, MA, , http://www.nber.org/papers/w8554.pdf
15. Grothe O (2013) Jump tail dependence in lévy copula models. Extremes 16(3):303–324
16. Grothe O, Hofert M (2015) Construction and sampling of archimedean and nested archimedean lévy copulas. J Multivar Anal 138:182–198
17. Hubert L, Arabie P (1985) Comparing partitions. J Classif 2(1):193–218
18. Jing BY, Kong XB, Liu Z (2012) Modeling high-frequency financial data by pure jump processes. Ann Stat 40:759–784
19. Kallsen J, Tankov P (2006) Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J Multivar Anal 97(7):1551–1572. , http://linkinghub.elsevier.com/retrieve/pii/S0047259X05002022
20. Kaufman L, Rousseeuw PJ (eds) (1990) Finding groups in data: an introduction to cluster analysis. Wiley Series in Probability and Statistics. Wiley, Hoboken.
21. Kyprianou A (2006) Introductory lectures on fluctuations of Lévy processes with applications. Springer, Berlin
22. Madan DB, Carr PP, Chang EC (1998) The variance gamma process and option pricing. Eur Finance Rev 2(1):79–105
23. Meilă M, Pentney W (2007) Clustering by weighted cuts in directed graphs. Society for Industrial and Applied Mathematics, Philadelphia, pp 135–144, copyright - Copyright Society for Industrial and Applied Mathematics 2007; Last updated - 2012-05-15Google Scholar
24. Nelsen RB (2006) An introduction to Copulas. Springer series in statistics. Springer, New York.
25. Nelsen RB (2007) An introduction to Copulas. Springer, Berlin
26. Poirot J, Tankov P (2007) Monte Carlo option pricing for tempered stable (CGMY) processes. Asia-Pacific Financ Mark 13(4):327–344.
27. Schmidt R, Stadtmüller U (2006) Non-parametric estimation of tail dependence. Scand J Stat 33(2):307–335.
28. Shi J, Malik J (2000) Normalized cuts and image segmentation. IEEE Trans Pattern Anal Mach Intell 22(8):888–905.
29. Sklar A (1959) Fonctions de répartition à $$n$$ dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris 8:229–231
30. Tankov P (2003a) Dependence structure of spectrally positive multidimensional lévy processes. Unpublished manuscriptGoogle Scholar
31. Tankov P (2003b) Financial modelling with jump processes, vol 2. CRC Press, Boca Raton
32. Tankov P (2006) Simulation and option pricing in lévy copula models. Mathematical Modelling of Financial Derivatives, IMA Volumes in Mathematics and Applications, SpringerGoogle Scholar
33. Tankov P (2016) Lévy copulas: review of recent results. In: The fascination of probability, statistics and their applications, Springer, pp 127–151Google Scholar
34. Wagner S, Wagner D (2007) Comparing clusterings: an overview. Universität Karlsruhe, Fakultät für Informatik KarlsruheGoogle Scholar

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

## Authors and Affiliations

• Chen Yang
• 1
• Wenjun Jiang
• 2
• Jiang Wu
• 3
• Xin Liu
• 2
• Zhichuan Li
• 4
1. 1.Economics and Management SchoolWuhan UniversityWuhanPeople’s Republic of China
2. 2.Department of Statistical and Actuarial SciencesUniversity of Western OntarioLondonCanada
3. 3.School of EconomicsCentral University of Finance and EconomicsBeijingPeople’s Republic of China