Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 31, 2013

Bounds on Capital Requirements For Bivariate Risk with Given Marginals and Partial Information on the Dependence

  • Carole Bernard EMAIL logo , Yuntao Liu , Niall MacGillivray and Jinyuan Zhang
From the journal Dependence Modeling


Nelsen et al. [20] find bounds for bivariate distribution functions when there are constraints on the values of its quartiles. Tankov [25] generalizes this work by giving explicit expressions for the best upper and lower bounds for a bivariate copula when its values on a compact subset of [0; 1]2 are known. He shows that they are quasi-copulas and not necessarily copulas. Tankov [25] and Bernard et al. [3] both give sufficient conditions for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that both bounds are simultaneously copulas. Furthermore, we develop a novel application to quantitative risk management by computing bounds on a bivariate risk measure. This can be useful in optimal portfolio selection, in reinsurance, in pricing bivariate derivatives or in determining capital requirements when only partial information on dependence is available.


[1] Bernard, C., Boyle, P.P., Vanduffel S. (2011). “Explicit Representation of Cost-Efficient Strategies”, Working paper available at SSRN. 10.2139/ssrn.1561272Search in Google Scholar

[2] Bernard, C., Chen, J.S., Vanduffel S. (2013). “Optimal Portfolio under Worst-State Scenarios”, Quant. Finance, to appear. Search in Google Scholar

[3] Bernard, C., Jiang, X., Vanduffel S. (2012). Note on“ Improved Fréchet bounds and model-free pricing of multi-asset options” by Tankov (2011)”, J. of Appl. Probab., 49(3), 866-875. 10.1239/jap/1346955339Search in Google Scholar

[4] Bernard, C., Jiang, X., Wang R. (2013). “Risk Aggregation with Dependence Uncertainty”, Working paper. 10.1016/j.insmatheco.2013.11.005Search in Google Scholar

[5] Bernard, C., Vanduffel S. (2011). “Optimal Investment under Probability Constraints”, AfMath proceedings. Search in Google Scholar

[6] Boyle, P.P., and W. Tian. 2007, “Portfolio Management with Constraints," Math. Finance, 17(3), 319-343. Search in Google Scholar

[7] Carley, H., Taylor, M.D. (2002). “A new proof of Sklar’s Theorem” in C.M. Cuadras, J. Fortiana and J.A. Rodriguez- Lallena, editors, Distributions with Given Marginals and Statistical Modelling, 29-34, Kluwer Acad. Publ., Dodrecht. 10.1007/978-94-017-0061-0_4Search in Google Scholar

[8] Durante, F., Jaworski, P. (2010). “A new characterization of bivariate copulas” Comm. Statist. Theory Methods, 39(16), 2901-2912. 10.1080/03610920903151459Search in Google Scholar

[9] Durante, F., Mesiar, R., Papini, P.-L., Sempi, C. (2007). “2-increasing binary aggregation operators”, Inform. Sci., 177(1), 111-129. Search in Google Scholar

[10] Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). “Model uncertainty and VaR aggregation”. J. of Banking and Finance, 37(8), 2750-2764. 10.1016/j.jbankfin.2013.03.014Search in Google Scholar

[11] Fréchet, M. (1951). “Sur les tableaux de corrélation dont les marges sont données,”Ann. Univ. Lyon Sect.A, Series 3, 14, 53-77. Search in Google Scholar

[12] Genest, C., Quesada-Molina, J.J., Rodri´guez, J.A., Sempi, C. (1999). “A characterization of quasi-copulas”, J. of Multivariate Anal., 69(2), 193-205. 10.1006/jmva.1998.1809Search in Google Scholar

[13] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E. (2009). “Aggregation functions,” Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York, (No. 127). Search in Google Scholar

[14] Hoeffding, W. (1940). “Masstabinvariante Korrelationstheorie,” Schriften des mathematischen Instituts und des Instituts für angewandte Mathematik der Universität Berlin 5, 179-233. Search in Google Scholar

[15] Kolesárová, A., Mordelová, J., Muel., E. (2004). “Kernel aggregation operators and their marginals,” Fuzzy Sets Syst., 142(1), 35-50. Search in Google Scholar

[16] Mai, J.-F., Scherer, J., (2012). “Simulating Copulas,” World Scientific, Singapore. 10.1142/p842Search in Google Scholar

[17] Meilijson, I., Nadas, A. (1979). “Convex majorization with an application to the length of critical paths,” J. of Appl. Probab., 16, 671-677. 10.2307/3213097Search in Google Scholar

[18] Nelsen, R. (2006). “An introduction to Copulas”, 2nd edition, Springer series in Statistics. Search in Google Scholar

[19] Nelsen, R., Quesada-Molina, J., Rodriguez-Lallena, J. and Úbeda-Flores, M. (2001). “Bounds on Bivariate Distribution Functions with Given Margins and Measures of Associations”, Comm. Statist. Theory Methods. 30(6), 1155-1162. Search in Google Scholar

[20] Nelsen, R., Quesada-Molina, J., Rodriguez Lallena, J. and Ubeda-Flores, M. (2004). “Best Possible Bounds on Sets of Bivariate Distribution Functions”, J. of Multivariate Anal., 90, 348-358. 10.1016/j.jmva.2003.09.002Search in Google Scholar

[21] Rachev, S.T. and Rüschendorf, L. (1994). “Solution of some transportation problems with relaxed or additional constraints”, SIAM J. Control Optim., 32, 673-689. 10.1137/S0363012991221365Search in Google Scholar

[22] Rüschendorf, L. (1983). “Solution of a Statistical Optimization Problem by Rearrangement Methods”, Biometrika, 30, 55-61. Search in Google Scholar

[23] Sadooghi-Alvandi, S. M., Shishebor, Z., Mardani-Fard, H.A. (2013). “Sharp bounds on a class of copulas with known values at several points" Communications Statist. Theory Methods, 42(12), 2215-2228. 10.1080/03610926.2011.607529Search in Google Scholar

[24] Stoeber, J. and Czado, C. (2012). “Detecting regime switches in the dependence structure of high dimensional financial data”, forthcoming in Comput. Statist. Data Anal.. Search in Google Scholar

[25] Tankov, P., (2011). “Improved Fréchet bounds and model-free pricing of multi-asset options”, J. of Appl. Probab., 48, 389-403. 10.1239/jap/1308662634Search in Google Scholar

[26] Tchen, A. H., (1980). “Inequalities for distributions with given margins”, Ann. of Appl. Probab., 8, 814–827.Search in Google Scholar

Received: 2013-05-05
Accepted: 2013-10-08
Published Online: 2013-12-31
Published in Print: 2013-01-01

©2013 Versita Sp. z o.o.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 1.3.2024 from
Scroll to top button