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MULTIVARIATE COMPOSITE COPULAS

Published online by Cambridge University Press:  03 November 2021

Jiehua Xie ,
Jun Fang ,
Jingping Yang  and
Lan Bu
Jiehua Xie*
Affiliation:
School of Business Administration Nanchang Institute of TechnologyJiangxi 330099, P. R. China. E-Mail: [email protected]
Jun Fang
Affiliation:
Department of Financial Mathematics Peking UniversityBeijing 100871, P. R. China. E-Mail: [email protected]
Jingping Yang*
Affiliation:
LMEQF, Department of Financial Mathematics Peking UniversityBeijing 100871, P. R. China. E-Mail: [email protected]
Lan Bu
Affiliation:
Department of Financial Mathematics Peking UniversityBeijing 100871, P. R. China. E-Mail: [email protected]
*
E-Mail: [email protected]
E-Mail: [email protected]
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Abstract

In this paper, we present a method for generating a copula by composing two arbitrary n-dimensional copulas via a vector of bivariate functions, where the resulting copula is named as the multivariate composite copula. A necessary and sufficient condition on the vector guaranteeing the composite function to be a copula is given, and a general approach to construct the vector satisfying this necessary and sufficient condition via bivariate copulas is provided. The multivariate composite copula proposes a new framework for the construction of flexible multivariate copula from existing ones, and it also includes some known classes of copulas. It is shown that the multivariate composite copula has a clear probability structure, and it satisfies the characteristic of uniform convergence as well as the reproduction property for its component copulas. Some properties of multivariate composite copulas are discussed. Finally, numerical illustrations and an empirical example on financial data are provided to show the advantages of the multivariate composite copula, especially in capturing the tail dependence.

Keywords

Copula constructionmultivariate composite copulauniform convergencereproduction propertyC02
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 52 , Issue 1 , January 2022 , pp. 145 - 184
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The International Actuarial Association

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References

Albrecher, H., Constantinescu, C. and Loisel, S. (2011) Explicit ruin formulas for models with dependence among risks. Insurance: Mathematics and Economics, 48(2), 265–270.Google Scholar
Alvoni, E., Papini, P.L. and Spizzichino, F. (2009) On a class of transformations of copulas and quasi-copulas. Fuzzy Sets and Systems, 160(3), 334–343.CrossRefGoogle Scholar
Baker, R. (2008) An order-statistics-based method for constructing multivariate distributions with fixed marginals. Journal of Multivariate Analysis, 99(10), 2312–2327.CrossRefGoogle Scholar
Brigo, D., Capponi, A. and Pallavicini, A. (2014) Arbitrage-free bilateral counterparty risk valuation under collateralization and application to credit default swaps. Mathematical Finance, 24(1), 125–146.CrossRefGoogle Scholar
Chen, X. and Fan, Y. (2006) Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. Journal of Econometrics, 135(1), 125–154.CrossRefGoogle Scholar
Cherubini, U., Gobbi, F., Mulinacci, S. and Romagnoli, S. (2012) Dynamic Copula Methods in Finance. Chichester: John Wiley and Sons.Google Scholar
Coval, J., Jurek, J. and Stafford, E. (2009) The economics of structured finance. Journal of Economic Perspectives, 23, 3–25.CrossRefGoogle Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002a) The concept of comonotonicity in actuarial science and finance: Application. Insurance: Mathematics and Economics, 31(2), 133–161.Google Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002b) The concept of comonotonicity in actuarial science and finance: Theory. Insurance: Mathematics and Economics, 31(1), 3–33.Google Scholar
Donnelly, C. and Embrechts, P. (2010). The devil is in the tails: actuarial mathematics and the subprime mortgage crisis. ASTIN Bulletin, 40(1), 1–33.CrossRefGoogle Scholar
Dou, X., Kuriki, S., Lin, G.D. and Richards, D. (2016) EM algorithms for estimating the Bernstein copula. Computational Statistics and Data Analysis, 93, 228–245.CrossRefGoogle Scholar
Durante, F., Foschi, R. and Sarkoci, P. (2010) Distorted copulas: Constructions and tail dependence. Communications in Statistics: Theory and Methods, 39, 2288–2301.CrossRefGoogle Scholar
Durante, F. and Sempi, C. (2016) Principles of Copula Theory. Boca Raton, FL: Chapman & Hall/CRC.Google Scholar
Durrett, R. (2010) Probability: Theory and Examples. New York: Cambridge University Press.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer.CrossRefGoogle Scholar
Fang, J., Jiang, F., Liu, Y. and Yang, J. (2020) Copula-based Markov process. Insurance: Mathematics and Economics, 91, 166–187.Google Scholar
Frees, E.W. and Valdez, E.A. (1998) Understanding relationships using copulas. North American Actuarial Journal, 2(1), 1–25.CrossRefGoogle Scholar
Genest, C. and Rivest, L. (2001) On the multivariate probability integral transformation. Statistics and Probability Letters, 53, 391–399.CrossRefGoogle Scholar
Janssen, P., Swanepoel, J. and Veraverbeke, N. (2012) Large sample behavior of the Bernstein copula estimator. Journal of Statistical Planning and Inference, 142(5), 1189–1197.CrossRefGoogle Scholar
Joe, H. (2014) Dependence Modeling with Copulas. London: CRC Press.CrossRefGoogle Scholar
Jondeau, E. and Rockinger, M. (2006) The Copula-GARCH model of conditional dependencies: An international stock market application. Journal of International Money and Finance, 25(5), 827–853.CrossRefGoogle Scholar
Klement, E.P., Mesiar, R. and Pap, E. (2005) Transformations of copulas. Kybernetika, 41(4), 425–434.Google Scholar
Lakhal, L., Rivest, L.-P. and Abdous, B. (2008) Estimating survival and association in a semicompeting risks model. Biometrics, 64(1), 180–188.CrossRefGoogle Scholar
Li, D. (2000) On default correlation: A copula function approach. Journal of Fixed Income, 9, 43–54.CrossRefGoogle Scholar
Liebscher, E. (2008) Construction of asymmetric multivariate copulas. Journal of Multivariate Analysis, 99(10), 2234–2250.CrossRefGoogle Scholar
Lin, F., Peng, L., Xie, J. and Yang, J. (2018) Stochastic distortion and its transformed copula. Insurance Mathematics and Economics, 79, 148–166.CrossRefGoogle Scholar
Mazo, G., Girard, S. and Forbes, F. (2015) A class of multivariate copulas based on products of bivariate copulas. Journal of Multivariate Analysis, 140, 363–376.CrossRefGoogle Scholar
McNeil, A., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques and Tools . Revised edition. Princeton: Princeton University Press.Google Scholar
McNeil, A.J. and Nešlehová, J. (2009) Multivariate archimedean copulas, d-monotone functions and $\ell_1$ -norm symmetric distributions. The Annals of Statistics, 3059–3097.Google Scholar
Morillas, P.M. (2005) A method to obtain new copulas from a given one. Metrika, 61(2), 169–184.CrossRefGoogle Scholar
Nelsen, R.B. (2006) An Introduction to Copulas, 2nd Edition. New York: Springer Science & Business Media.Google Scholar
Sancetta, A. (2007) Online forecast combinations of distributions: Worst case bounds. Journal of Econometrics, 141(2), 621–651.CrossRefGoogle Scholar
Sancetta, A. and Satchell, S. (2004) The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory, 20(3), 535–562.CrossRefGoogle Scholar
Scheffer, M. and Weiÿ, G.N. (2017) Smooth nonparametric Bernstein vine copulas. Quantitative Finance, 17(1), 139–156.CrossRefGoogle Scholar
Schloegl, L. and O’Kane, D. (2005) A note on the large homogeneous portfolio approximation with the Student-t copula. Finance and Stochastics, 9(4), 577–584.CrossRefGoogle Scholar
Schönbucher, P.J. (2003) Credit Derivatives Pricing Models: Models, Pricing and Implementation. New York: John Wiley and Sons.Google Scholar
Sklar, M. (1959) Fonctions de repartition an dimensions et leurs marges. Publications de l’Institut de statistique de l’Universite Paris, 8, 229–231.Google Scholar
Valdez, E.A. and Xiao, Y. (2011) On the distortion of a copula and its margins. Scandinavian Actuarial Journal, 2011(4), 292–317.CrossRefGoogle Scholar
Xie, J., Lin, F. and Yang, J. (2017) On a generalization of Archimedean copula family. Statistics and Probability Letters, 125, 121–129.CrossRefGoogle Scholar
Xie, J., Yang, J. and Zhu, W. (2019) A family of transformed copulas with a singular component. Fuzzy Sets and Systems, 354, 20–47.CrossRefGoogle Scholar
Yang, J., Chen, Z., Wang, F. and Wang, R. (2015) Composite Bernstein copulas. ASTIN Bulletin, 45(2), 445–475.CrossRefGoogle Scholar
Zhao, Z. and Zhang, Z. (2018) Semiparametric dynamic max-copula model for multivariate time series. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(2), 409–432.CrossRefGoogle Scholar