Approximation of bivariate copulas by patched bivariate Fréchet copulas

Yanting Zheng; Jingping Yang; Jianhua Z. Huang

doi:10.1016/j.insmatheco.2010.11.002

Approximation of bivariate copulas by patched bivariate Fréchet copulas

Yanting Zheng, Jingping Yang, Jianhua Z. Huang

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

Bivariate Fréchet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums. © 2010 Elsevier B.V.

Original language	English (US)
Pages (from-to)	246-256
Number of pages	11
Journal	Insurance: Mathematics and Economics
Volume	48
Issue number	2
DOIs	https://doi.org/10.1016/j.insmatheco.2010.11.002
State	Published - Mar 2011
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: We thank the reviewer for his helpful comments. Yang's research was partly supported by the National Basic Research Program (973 Program) of China (2007CB814905) and the National Natural Science Foundation of China (Grants No. 10871008). Yang also thanks National Science Foundation (DMS-0630950) of the US for supporting his visit to Texas A&M University through the Virtual Center for Collaboration between Statisticians in the US and China, where some initial ideas of the project was developed. Huang's research was partly supported by the National Cancer Institute (CA57030) and the National Science Foundation (DMS-0907170) of the US, and by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1016/j.insmatheco.2010.11.002

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title = "Approximation of bivariate copulas by patched bivariate Fr{\'e}chet copulas",

abstract = "Bivariate Fr{\'e}chet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums. {\textcopyright} 2010 Elsevier B.V.",

author = "Yanting Zheng and Jingping Yang and Huang, {Jianhua Z.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: We thank the reviewer for his helpful comments. Yang's research was partly supported by the National Basic Research Program (973 Program) of China (2007CB814905) and the National Natural Science Foundation of China (Grants No. 10871008). Yang also thanks National Science Foundation (DMS-0630950) of the US for supporting his visit to Texas A&M University through the Virtual Center for Collaboration between Statisticians in the US and China, where some initial ideas of the project was developed. Huang's research was partly supported by the National Cancer Institute (CA57030) and the National Science Foundation (DMS-0907170) of the US, and by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - Bivariate Fréchet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums. © 2010 Elsevier B.V.

AB - Bivariate Fréchet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums. © 2010 Elsevier B.V.

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Approximation of bivariate copulas by patched bivariate Fréchet copulas

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