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

• Autorzy: Carole Bernard , Yuntao Liu , Niall MacGillivray , Jinyuan Zhang
• Języki publikacji: EN

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.

Słowa kluczowe

EN

Copulas; Fréchet-Hoeffding bounds; Capital requirements

Kategorie tematyczne

• 62H99: None of the above, but in this section
• 62E99: None of the above, but in this section
• 62P05: Applications to actuarial sciences and financial mathematics

• Wydawca: De Gruyter Open
• Czasopismo: Dependence Modeling
• Rocznik: 2013
• Tom: 1
• Strony: 37-53

Daty

otrzymano

2013-05-05

zaakceptowano

2013-10-08

online

2013-10-21

Twórcy

autor

Carole Bernard

• Department of Statistics and Actuarial Science
  at the University of Waterloo

autor

Yuntao Liu

• University of California

autor

Niall MacGillivray

• University of Waterloo

autor

Jinyuan Zhang

• University of British Columbia

Bibliografia

• [1] Bernard, C., Boyle, P.P., Vanduffel S. (2011). “Explicit Representation of Cost-Efficient Strategies”, Working paper available at SSRN.
• [2] Bernard, C., Chen, J.S., Vanduffel S. (2013). “Optimal Portfolio under Worst-State Scenarios”, Quant. Finance, to appear.
• [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.
• [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 and Finance, 37(8), 2750-2764. [WoS]
• [11] Fréchet, M. (1951). “Sur les tableaux de corrélation dont les marges sont données,”Ann. Univ. Lyon Sect.A, Series 3, 14, 53-77.
• [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.
• [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).
• [14] Hoeffding, W. (1940). “Masstabinvariante Korrelationstheorie,” Schriften des mathematischen Instituts und des Instituts für angewandte Mathematik der Universität Berlin 5, 179-233.
• [15] Kolesárová, A., Mordelová, J., Muel., E. (2004). “Kernel aggregation operators and their marginals,” Fuzzy Sets Syst., 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 Distribution Functions 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 Sets of 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 additional constraints”, 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 known values 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 dimensional financial 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.

Identyfikatory

DOI

10.2478/demo-2013-0002