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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access June 6, 2018

Risk bounds with additional information on functionals of the risk vector

  • L. Rüschendorf EMAIL logo
From the journal Dependence Modeling

Abstract

We consider the problem of determining risk bounds for the Value at Risk for risk vectors X where besides the marginal distributions also information on the distribution or on the expectation of some functionals Tj(X), 1 ≤ j ≤ m, is available. In particular this formulation includes the case where information on subgroup sums or maxima or on the correlations or covariances is available. Based on the method of dual bounds we obtain improved risk bounds compared to the marginal case. In general the explicit calculation of the dual bounds poses a challenge. We discuss various forms of relaxation of these bounds which are accessible and in some cases even lead to sharp bounds.

References

[1] Acciaio, B., M. Beiglböck, F. Penkner, and W. Schachermayer (2016). A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance 26(2), 233-251.10.1111/mafi.12060Search in Google Scholar

[2] Beiglböck, M., P. Henry-Labordère, and F. Penkner (2013). Model-independent bounds for option prices - a mass transport approach. Finance Stoch. 17(3), 477-501.10.1007/s00780-013-0205-8Search in Google Scholar

[3] Bernard, C., L. Rüschendorf, and S. Vanduffel (2017). Value-at-risk bounds with variance constraints. J. Risk. Insur. 84(3), 923-959.10.1111/jori.12108Search in Google Scholar

[4] Bernard, C., L. Rüschendorf, S. Vanduffel, and R.Wang (2017). Risk bounds for factor models. Finance Stoch. 21(3), 631-659.10.1007/s00780-017-0328-4Search in Google Scholar

[5] Bernard, C. and S. Vanduffel (2015). A new approach to assessing model risk in high dimensions. J. Bank. Financ. 58, 166-178.10.1016/j.jbankfin.2015.03.007Search in Google Scholar

[6] Bignozzi, V., G. Puccetti, and L. Rüschendorf (2015). Reducing model risk via positive and negative dependence assumptions. Insurance Math. Econom. 61, 17-26.10.1016/j.insmatheco.2014.11.004Search in Google Scholar

[7] Denuit, M., J. Genest, and É.Marceau (1999). Stochastic bounds on sums of dependent risks. InsuranceMath. Econom. 25(1), 85-104.10.1016/S0167-6687(99)00027-XSearch in Google Scholar

[8] Embrechts, P., A. Höing, and A. Juri (2003). Using copulae to bound the Value-at-Risk for functions of dependent risks. Finance Stoch. 7(2), 145-167.10.1007/s007800200085Search in Google Scholar

[9] Embrechts, P. and G. Puccetti (2006a). Bounds for functions of dependent risks. Finance Stoch. 10(3), 341-352.10.1007/s00780-006-0005-5Search in Google Scholar

[10] Embrechts, P. and G. Puccetti (2006b). Bounds for functions of multivariate risks. J. Multivariate Anal. 97(2), 526-547.10.1016/j.jmva.2005.04.001Search in Google Scholar

[11] Embrechts, P., G. Puccetti, and L. Rüschendorf (2013). Model uncertainty and VaR aggregation. J. Bank. Financ. 37(8), 2750-2764.10.1016/j.jbankfin.2013.03.014Search in Google Scholar

[12] Li, L., H. Shao, R. Wang, and J. Yang (2018). Worst-case Range Value-at-Risk with partial information. SIAM J. Finan. Math. 9(1), 190-218.10.1137/17M1126138Search in Google Scholar

[13] Lux, T. and A. Papapantoleon (2016). Model-free bounds on Value-at-Risk using partial dependence information. Available at https://arxiv.org/abs/1610.09734.Search in Google Scholar

[14] Lux, T. and A. Papapantoleon (2017). Improved Fréchet-Hoeffding bounds on d-copulas and applications in model-free finance. Ann. Appl. Probab. 27(6), 3633-3671.10.1214/17-AAP1292Search in Google Scholar

[15] Lux, T. and L. Rüschendorf (2017). Value-at-Risk bounds with two-sided dependence information. Available at https://ssrn.com/abstract=3086256.Search in Google Scholar

[16] Popescu, I. (2005). A semideffnite programming approach to optimal-moment bounds for convex classes of distributions. Math. Oper. Res. 30(3), 632-657.10.1287/moor.1040.0137Search in Google Scholar

[17] Puccetti, G. and L. Rüschendorf (2012). Bounds for joint portfolios of dependent risks. Stat. Risk Model. 29(2), 107-132.10.1524/strm.2012.1117Search in Google Scholar

[18] Puccetti, G. and L. Rüschendorf (2013). Sharp bounds for sums of dependent risks. J. Appl. Probab. 50(1), 42-53.10.1239/jap/1363784423Search in Google Scholar

[19] Puccetti, G., L. Rüschendorf, and D. Manko (2016). VaR bounds for joint portfolios with dependence constraints. Depend. Model. 4(1), 368-381.Search in Google Scholar

[20] Puccetti, G., L. Rüschendorf, D. Small, and S. Vanduffel (2017). Reduction of Value-at-Risk bounds via independence and variance information. Scand. Actuar. J. 2017(3), 245-266.Search in Google Scholar

[21] Rockafellar, R. T. and S. Uryasev (2000). Optimization of conditional value-at-risk. J. Risk 2(3), 21-41.10.21314/JOR.2000.038Search in Google Scholar

[22] Rüschendorf, L. (1980). Inequalities for the expectation of ff-monotone functions. Z. Wahrscheinlichkeitstheorie Verw. Geb. 54(3), 341-349.10.1007/BF00534351Search in Google Scholar

[23] Rüschendorf, L. (2005). Stochastic ordering of risks, influence of dependence, and a.s. constructions. In N. Balakrishnan, I. G. Bairamov, and O. L. Gebizlioglu (Eds.), Advances on Models, Characterization and Applications, pp. 19-56. Chapman & Hall/CRC, Boca Raton FL.10.1201/9781420028690.ch2Search in Google Scholar

[24] Rüschendorf, L. (2013).Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer, Heidelberg. 10.1007/978-3-642-33590-7Search in Google Scholar

[25] Rüschendorf, L. (2017a). Improved Hoeffding-Fréchet bounds and applications to VaR estimates. In M. Úbeda Flores, E. de Amo Artero, F. Durante, and J. Fernández Sánchez (Eds.), Copulas and Dependence Models with Applications, pp. 181-202. Springer, Cham.10.1007/978-3-319-64221-5_12Search in Google Scholar

[26] Rüschendorf, L. (2017b). Risk bounds and partial dependence information. In D. Ferger, W. González Manteiga, T. Schmidt, and J.-L.Wang (Eds.), FromStatistics toMathematical Finance. Festschrift in Honour ofWinfried Stute, pp. 345-366. Springer, Cham.10.1007/978-3-319-50986-0_17Search in Google Scholar

[27] Rüschendorf, L. and J. Witting (2017). VaR bounds in models with partial dependence information on subgroups. Depend. Model. 5(1), 59-74.10.1515/demo-2017-0004Search in Google Scholar

[28] Wang, B. and R. Wang (2016). Joint mixability. Math. Oper. Res. 41(3), 808-826.10.1287/moor.2015.0755Search in Google Scholar

[29] Williamson, R. C. and T. Downs (1990). Probabilistic arithmetic. I. Numerical methods for calculationg convolutions and dependency bounds. Internat. J. Approx. Reason. 4(2), 89-158.Search in Google Scholar

Received: 2017-11-29
Accepted: 2018-04-09
Published Online: 2018-06-06

© 2018 L. Rüschendorf, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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