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2013 | 1 | 37-53
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Bounds on Capital Requirements For Bivariate Risk with Given Marginals and Partial Information on the Dependence

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Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
1
Strony
37-53
Opis fizyczny
Daty
otrzymano
2013-05-05
zaakceptowano
2013-10-08
online
2013-10-21
Twórcy
autor
Bibliografia
  • [1] Bernard, C., Boyle, P.P., Vanduffel S. (2011). “Explicit Representation of Cost-Efficient Strategies”, Working paperavailable at SSRN.
  • [2] Bernard, C., Chen, J.S., Vanduffel S. (2013). “Optimal Portfolio under Worst-State Scenarios”, Quant. Finance, toappear.
  • [3] Bernard, C., Jiang, X., Vanduffel S. (2012). Note on“ Improved Fréchet bounds and model-free pricing of multi-assetoptions” by Tankov (2011)”, J. of Appl. Probab., 49(3), 866-875.
  • [4] Bernard, C., Jiang, X., Wang R. (2013). “Risk Aggregation with Dependence Uncertainty”, Working paper.
  • [5] Bernard, C., Vanduffel S. (2011). “Optimal Investment under Probability Constraints”, AfMath proceedings.
  • [6] Boyle, P.P., and W. Tian. 2007, “Portfolio Management with Constraints," Math. Finance, 17(3), 319-343.[WoS]
  • [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.
  • [8] Durante, F., Jaworski, P. (2010). “A new characterization of bivariate copulas” Comm. Statist. Theory Methods,39(16), 2901-2912.
  • [9] Durante, F., Mesiar, R., Papini, P.-L., Sempi, C. (2007). “2-increasing binary aggregation operators”, Inform. Sci.,177(1), 111-129.[WoS]
  • [10] Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). “Model uncertainty and VaR aggregation”. J. of Banking andFinance, 37(8), 2750-2764.[WoS]
  • [11] Fréchet, M. (1951). “Sur les tableaux de corrélation dont les marges sont données,”Ann. Univ. Lyon Sect.A, Series3, 14, 53-77.
  • [12] Genest, C., Quesada-Molina, J.J., Rodri´guez, J.A., Sempi, C. (1999). “A characterization of quasi-copulas”, J. ofMultivariate Anal., 69(2), 193-205.
  • [13] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E. (2009). “Aggregation functions,” Encyclopedia of Mathematics andits Applications. Cambridge University Press, New York, (No. 127).
  • [14] Hoeffding, W. (1940). “Masstabinvariante Korrelationstheorie,” Schriften des mathematischen Instituts und desInstituts für angewandte Mathematik der Universität Berlin 5, 179-233.
  • [15] Kolesárová, A., Mordelová, J., Muel., E. (2004). “Kernel aggregation operators and their marginals,” Fuzzy SetsSyst., 142(1), 35-50.
  • [16] Mai, J.-F., Scherer, J., (2012). “Simulating Copulas,” World Scientific, Singapore.
  • [17] Meilijson, I., Nadas, A. (1979). “Convex majorization with an application to the length of critical paths,” J. of Appl.Probab., 16, 671-677.
  • [18] Nelsen, R. (2006). “An introduction to Copulas”, 2nd edition, Springer series in Statistics.
  • [19] Nelsen, R., Quesada-Molina, J., Rodriguez-Lallena, J. and Úbeda-Flores, M. (2001). “Bounds on Bivariate DistributionFunctions with Given Margins and Measures of Associations”, Comm. Statist. Theory Methods. 30(6),1155-1162.[WoS]
  • [20] Nelsen, R., Quesada-Molina, J., Rodriguez Lallena, J. and Ubeda-Flores, M. (2004). “Best Possible Bounds on Setsof Bivariate Distribution Functions”, J. of Multivariate Anal., 90, 348-358.
  • [21] Rachev, S.T. and Rüschendorf, L. (1994). “Solution of some transportation problems with relaxed or additionalconstraints”, SIAM J. Control Optim., 32, 673-689.
  • [22] Rüschendorf, L. (1983). “Solution of a Statistical Optimization Problem by Rearrangement Methods”, Biometrika,30, 55-61.
  • [23] Sadooghi-Alvandi, S. M., Shishebor, Z., Mardani-Fard, H.A. (2013). “Sharp bounds on a class of copulas with knownvalues at several points" Communications Statist. Theory Methods, 42(12), 2215-2228.[WoS]
  • [24] Stoeber, J. and Czado, C. (2012). “Detecting regime switches in the dependence structure of high dimensionalfinancial data”, forthcoming in Comput. Statist. Data Anal..
  • [25] Tankov, P., (2011). “Improved Fréchet bounds and model-free pricing of multi-asset options”, J. of Appl. Probab., 48,389-403.
  • [26] Tchen, A. H., (1980). “Inequalities for distributions with given margins”, Ann. of Appl. Probab., 8, 814–827.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_demo-2013-0002
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