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

Carole Bernard; Yuntao Liu; Niall MacGillivray; Jinyuan Zhang

Dependence Modeling (2013)

  • Volume: 1, page 37-53
  • ISSN: 2300-2298

Abstract

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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.

How to cite

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Carole Bernard, et al. "Bounds on Capital Requirements For Bivariate Risk with Given Marginals and Partial Information on the Dependence." Dependence Modeling 1 (2013): 37-53. <http://eudml.org/doc/267378>.

@article{CaroleBernard2013,
abstract = {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.},
author = {Carole Bernard, Yuntao Liu, Niall MacGillivray, Jinyuan Zhang},
journal = {Dependence Modeling},
keywords = {Copulas; Fréchet-Hoeffding bounds; Capital requirements; copulas; capital requirements},
language = {eng},
pages = {37-53},
title = {Bounds on Capital Requirements For Bivariate Risk with Given Marginals and Partial Information on the Dependence},
url = {http://eudml.org/doc/267378},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Carole Bernard
AU - Yuntao Liu
AU - Niall MacGillivray
AU - Jinyuan Zhang
TI - Bounds on Capital Requirements For Bivariate Risk with Given Marginals and Partial Information on the Dependence
JO - Dependence Modeling
PY - 2013
VL - 1
SP - 37
EP - 53
AB - 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.
LA - eng
KW - Copulas; Fréchet-Hoeffding bounds; Capital requirements; copulas; capital requirements
UR - http://eudml.org/doc/267378
ER -

References

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  1. [1] Bernard, C., Boyle, P.P., Vanduffel S. (2011). “Explicit Representation of Cost-Efficient Strategies”, Working paper available at SSRN. 
  2. [2] Bernard, C., Chen, J.S., Vanduffel S. (2013). “Optimal Portfolio under Worst-State Scenarios”, Quant. Finance, to appear. Zbl1308.91136
  3. [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. Zbl1259.60022
  4. [4] Bernard, C., Jiang, X., Wang R. (2013). “Risk Aggregation with Dependence Uncertainty”, Working paper. Zbl1291.91090
  5. [5] Bernard, C., Vanduffel S. (2011). “Optimal Investment under Probability Constraints”, AfMath proceedings. 
  6. [6] Boyle, P.P., and W. Tian. 2007, “Portfolio Management with Constraints," Math. Finance, 17(3), 319-343. [WoS] Zbl1186.91191
  7. [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. Zbl1137.62336
  8. [8] Durante, F., Jaworski, P. (2010). “A new characterization of bivariate copulas” Comm. Statist. Theory Methods, 39(16), 2901-2912. Zbl1203.62101
  9. [9] Durante, F., Mesiar, R., Papini, P.-L., Sempi, C. (2007). “2-increasing binary aggregation operators”, Inform. Sci., 177(1), 111-129. [WoS] Zbl1142.68541
  10. [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. [11] Fréchet, M. (1951). “Sur les tableaux de corrélation dont les marges sont données,”Ann. Univ. Lyon Sect.A, Series 3, 14, 53-77. Zbl0045.22905
  12. [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. Zbl0935.62059
  13. [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. [14] Hoeffding, W. (1940). “Masstabinvariante Korrelationstheorie,” Schriften des mathematischen Instituts und des Instituts für angewandte Mathematik der Universität Berlin 5, 179-233. 
  15. [15] Kolesárová, A., Mordelová, J., Muel., E. (2004). “Kernel aggregation operators and their marginals,” Fuzzy Sets Syst., 142(1), 35-50. Zbl1043.03040
  16. [16] Mai, J.-F., Scherer, J., (2012). “Simulating Copulas,” World Scientific, Singapore. 
  17. [17] Meilijson, I., Nadas, A. (1979). “Convex majorization with an application to the length of critical paths,” J. of Appl. Probab., 16, 671-677. Zbl0417.60025
  18. [18] Nelsen, R. (2006). “An introduction to Copulas”, 2nd edition, Springer series in Statistics. Zbl1152.62030
  19. [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] Zbl1008.62605
  20. [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. Zbl1057.62038
  21. [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. Zbl0797.60019
  22. [22] Rüschendorf, L. (1983). “Solution of a Statistical Optimization Problem by Rearrangement Methods”, Biometrika, 30, 55-61. Zbl0535.62008
  23. [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] Zbl1319.62034
  24. [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. [25] Tankov, P., (2011). “Improved Fréchet bounds and model-free pricing of multi-asset options”, J. of Appl. Probab., 48, 389-403. Zbl1219.60016
  26. [26] Tchen, A. H., (1980). “Inequalities for distributions with given margins”, Ann. of Appl. Probab., 8, 814–827. Zbl0459.62010

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