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Upper risk bounds in internal factor models with constrained specification sets
1. Department of Quantitative Finance, Albert-Ludwigs University of Freiburg, Platz der Alten Synagoge 1, KG II, 79098 Freiburg i. Br., Germany |
2. Department of Mathematical Stochastics, Albert-Ludwigs University of Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg, Germany |
References:
[1] |
Aas, K., C. Czado, A. Frigessi, and H. Bakken. (2009). Pair-copula constructions of multiple dependence, Insur. Math. Econ. 44, no. 2, 182–198., |
[2] |
Ansari, J. (2019). Ordering risk bounds in partially specified factor models, University of Freiburg, Dissertation., |
[3] |
Ansari, J. and L. Rüschendorf. (2016). Ordering results for risk bounds and cost-efficient payoffs in partially specified risk factor models, Methodol. Comput. Appl. Probab., 1–22., |
[4] |
Ansari, J. and L. Rüschendorf. (2018). Ordering risk bounds in factor models, Depend. Model. 6.1, 259– 287., |
[5] |
Bäuerle, N. and A. Müller. (2006). Stochastic orders and risk measures: consistency and bounds, Insur.Math. Econ. 38, no. 1, 132–148., |
[6] |
Bernard, C. and S. Vanduffel. (2015). A new approach to assessing model risk in high dimensions, J. Bank.Financ. 58, 166–178., |
[7] |
Bernard, C., L. Rüschendorf, and S. Vanduffel. (2017a). Value-at-Risk bounds with variance constraints, J. Risk. Insur. 84, no. 3, 923–959., |
[8] |
Bernard, C., L. Rüschendorf, S. Vanduffel, and R. Wang. (2017b). Risk bounds for factor models, Financ.Stoch. 21, no. 3, 631–659., |
[9] |
Bernard, C., M. Denuit, and S. Vanduffel. (2018). Measuring portfolio risk under partial dependence information, J. Risk Insur. 85, no. 3, 843–863., |
[10] |
Bignozzi, V., G. Puccetti, and L Rüschendorf. (2015). Reducing model risk via positive and negative dependence assumptions, Insur. Math. Econ. 61, 17–26., |
[11] |
Cornilly, D., L. Rüschendorf, and S. Vanduffel. (2018). Upper bounds for strictly concave distortion risk measures on moment spaces, Insur Math Econ. 82, 141–151., |
[12] |
de Schepper, A. and B. Heijnen. (2010). How to estimate the Value at Risk under incomplete information, J. Comput. Appl. Math. 233, no. 9, 2213–2226., |
[13] |
Demarta, S. and A.J. McNeil. (2005). The t copula and related copulas, Int. Stat. Rev. 73, no. 1, 111–129., |
[14] |
Denuit, M., C. Genest, and E. Marceau. (1999). Stochastic bounds on sums of dependent risks, Insur. Math.Econ. 25, no. 1, 85–104., |
[15] |
Embrechts, P. and G. Puccetti. (2006). Bounds for functions of dependent risks, Financ. Stoch. 10, no. 3, 341–352., |
[16] |
Embrechts, P., G. Puccetti, and L. Rüschendorf. (2013). Model uncertainty and VaR aggregation, J. Banking Financ. 37, no. 8, 2750–2764., |
[17] |
Embrechts, P., G. Puccetti, L. Rüschendorf, R. Wang, and A. Beleraj. (2014). An academic response to basel 3.5, Risks 2, no. 1, 25–48., |
[18] |
Embrechts, P., B. Wang, and R. Wang. (2015). Aggregation-robustness and model uncertainty of regulatory risk measures, Financ. Stoch. 19, no. 4, 763–790., |
[19] |
Föllmer, H. and A. Schied. (2010). Convex and coherent risk measures., Encycl. Quant. Financ., 355–363., |
[20] |
Goovaerts, M.J., R. Kaas, and R.J.A. Laeven. (2011). Worst case risk measurement: back to the future?Insur. Math. Econ. 49, no. 3, 380–392., |
[21] |
Hürlimann, W. (2002). Analytical bounds for two Value-at-Risk functionals, ASTIN Bull. 32, no. 2, 235–265., |
[22] |
Hürlimann, W. (2008). Extremal moment methods and stochastic orders, Bol. Asoc. Mat. Venez. 15, no. 2, 153–301., |
[23] |
Kaas, R. and M.J. Goovaerts. (1986). Best bounds for positive distributions with fixed moments, Insur.Math. Econ. 5, 87–95., |
[24] |
McNeil, A.J., R. Frey, and P. Embrechts. (2015). Quantitative Risk Management. Concepts, Techniques and Tools., second edn, Princeton University Press, Princeton., |
[25] |
Müller, A. (1997). Stop-loss order for portfolios of dependent risks, Insur. Math. Econ. 21, no. 3, 219–223., |
[26] |
Müller, A. (2013). Duality theory and transfers for stochastic order relations. In: Stochastic orders in reliability and risk, Springer, New York., |
[27] |
Müller, A. and M. Scarsini. (2001). Stochastic comparison of random vectors with a common copula, Math. Oper. Res. 26, no. 4, 723–740., |
[28] |
Müller, A. and M. Scarsini. (2006). Stochastic order relations and lattices of probability measures, SIAM J. Optim. 16, no. 4, 1024–1043., |
[29] |
Müller, A. and D. Stoyan. (2002). Comparison Methods for Stochastic Models and Risks, Wiley, Chichester., |
[30] |
Nelsen, R.B. (2006). An introduction to copulas, 2nd ed, Springer, New York., |
[31] |
Nelsen, R.B., J.J. Quesada-Molina, J.A. Rodríguez-Lallena, and M. Úbeda-Flores. (2001). Bounds on bivariate distribution functions with given margins and measures of association, Commun. Stat.Theory Methods 30, no. 6, 1155–1162., |
[32] |
Puccetti, G. and L. Rüschendorf. (2012a). Bounds for joint portfolios of dependent risks, Stat. Risk. Model.Appl. Financ. Insur. 29, no. 2, 107–132., |
[33] |
Puccetti, G. and L. Rüschendorf. (2012b). Computation of sharp bounds on the distribution of a function of dependent risks, J. Comput. Appl. Math 236, no. 7, 1833–1840., |
[34] |
Puccetti, G. and L. Rüschendorf. (2013). Sharp bounds for sums of dependent risks, J. Appl. Probab. 50, no. 1, 42–53., |
[35] |
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, no. 3, 245–266., |
[36] |
Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process, J. Stat. Plann. Inference 139, no. 11, 3921–3927., |
[37] |
Rüschendorf, L. (2013). Mathematical Risk Analysis, Springer, New York., |
[38] |
Rüschendorf, L. (2017a). Improved Hoeffding–Fréchet bounds and applications to VaR estimates. In:Copulas and Dependence Models with Applications. Contributions in Honor of Roger B. Nelsen(M. Úbeda Flores, E. de Amo Artero, F. Durante, and J. Fernández Sánchez, eds.), Springer, Cham. https://doi.org/10.1007/978-3-319-64221-5 12., |
[39] |
Rüschendorf, L. (2017b). Risk bounds and partial dependence information. In: From Statistics to Mathematical Finance, Springer, Festschrift in honour of Winfried Stute, Cham., |
[40] |
Rüschendorf, L. and J. Witting. (2017). VaR bounds in models with partial dependence information on subgroups, Depend Model 5, 59–74., |
[41] |
Shaked, M. and J.G. Shantikumar. (2007). Stochastic Orders, Springer, New York., |
[42] |
Tian, R. (2008). Moment problems with applications to Value-at-Risk and portfolio management, Georgia State University, Dissertation., |
show all references
References:
[1] |
Aas, K., C. Czado, A. Frigessi, and H. Bakken. (2009). Pair-copula constructions of multiple dependence, Insur. Math. Econ. 44, no. 2, 182–198., |
[2] |
Ansari, J. (2019). Ordering risk bounds in partially specified factor models, University of Freiburg, Dissertation., |
[3] |
Ansari, J. and L. Rüschendorf. (2016). Ordering results for risk bounds and cost-efficient payoffs in partially specified risk factor models, Methodol. Comput. Appl. Probab., 1–22., |
[4] |
Ansari, J. and L. Rüschendorf. (2018). Ordering risk bounds in factor models, Depend. Model. 6.1, 259– 287., |
[5] |
Bäuerle, N. and A. Müller. (2006). Stochastic orders and risk measures: consistency and bounds, Insur.Math. Econ. 38, no. 1, 132–148., |
[6] |
Bernard, C. and S. Vanduffel. (2015). A new approach to assessing model risk in high dimensions, J. Bank.Financ. 58, 166–178., |
[7] |
Bernard, C., L. Rüschendorf, and S. Vanduffel. (2017a). Value-at-Risk bounds with variance constraints, J. Risk. Insur. 84, no. 3, 923–959., |
[8] |
Bernard, C., L. Rüschendorf, S. Vanduffel, and R. Wang. (2017b). Risk bounds for factor models, Financ.Stoch. 21, no. 3, 631–659., |
[9] |
Bernard, C., M. Denuit, and S. Vanduffel. (2018). Measuring portfolio risk under partial dependence information, J. Risk Insur. 85, no. 3, 843–863., |
[10] |
Bignozzi, V., G. Puccetti, and L Rüschendorf. (2015). Reducing model risk via positive and negative dependence assumptions, Insur. Math. Econ. 61, 17–26., |
[11] |
Cornilly, D., L. Rüschendorf, and S. Vanduffel. (2018). Upper bounds for strictly concave distortion risk measures on moment spaces, Insur Math Econ. 82, 141–151., |
[12] |
de Schepper, A. and B. Heijnen. (2010). How to estimate the Value at Risk under incomplete information, J. Comput. Appl. Math. 233, no. 9, 2213–2226., |
[13] |
Demarta, S. and A.J. McNeil. (2005). The t copula and related copulas, Int. Stat. Rev. 73, no. 1, 111–129., |
[14] |
Denuit, M., C. Genest, and E. Marceau. (1999). Stochastic bounds on sums of dependent risks, Insur. Math.Econ. 25, no. 1, 85–104., |
[15] |
Embrechts, P. and G. Puccetti. (2006). Bounds for functions of dependent risks, Financ. Stoch. 10, no. 3, 341–352., |
[16] |
Embrechts, P., G. Puccetti, and L. Rüschendorf. (2013). Model uncertainty and VaR aggregation, J. Banking Financ. 37, no. 8, 2750–2764., |
[17] |
Embrechts, P., G. Puccetti, L. Rüschendorf, R. Wang, and A. Beleraj. (2014). An academic response to basel 3.5, Risks 2, no. 1, 25–48., |
[18] |
Embrechts, P., B. Wang, and R. Wang. (2015). Aggregation-robustness and model uncertainty of regulatory risk measures, Financ. Stoch. 19, no. 4, 763–790., |
[19] |
Föllmer, H. and A. Schied. (2010). Convex and coherent risk measures., Encycl. Quant. Financ., 355–363., |
[20] |
Goovaerts, M.J., R. Kaas, and R.J.A. Laeven. (2011). Worst case risk measurement: back to the future?Insur. Math. Econ. 49, no. 3, 380–392., |
[21] |
Hürlimann, W. (2002). Analytical bounds for two Value-at-Risk functionals, ASTIN Bull. 32, no. 2, 235–265., |
[22] |
Hürlimann, W. (2008). Extremal moment methods and stochastic orders, Bol. Asoc. Mat. Venez. 15, no. 2, 153–301., |
[23] |
Kaas, R. and M.J. Goovaerts. (1986). Best bounds for positive distributions with fixed moments, Insur.Math. Econ. 5, 87–95., |
[24] |
McNeil, A.J., R. Frey, and P. Embrechts. (2015). Quantitative Risk Management. Concepts, Techniques and Tools., second edn, Princeton University Press, Princeton., |
[25] |
Müller, A. (1997). Stop-loss order for portfolios of dependent risks, Insur. Math. Econ. 21, no. 3, 219–223., |
[26] |
Müller, A. (2013). Duality theory and transfers for stochastic order relations. In: Stochastic orders in reliability and risk, Springer, New York., |
[27] |
Müller, A. and M. Scarsini. (2001). Stochastic comparison of random vectors with a common copula, Math. Oper. Res. 26, no. 4, 723–740., |
[28] |
Müller, A. and M. Scarsini. (2006). Stochastic order relations and lattices of probability measures, SIAM J. Optim. 16, no. 4, 1024–1043., |
[29] |
Müller, A. and D. Stoyan. (2002). Comparison Methods for Stochastic Models and Risks, Wiley, Chichester., |
[30] |
Nelsen, R.B. (2006). An introduction to copulas, 2nd ed, Springer, New York., |
[31] |
Nelsen, R.B., J.J. Quesada-Molina, J.A. Rodríguez-Lallena, and M. Úbeda-Flores. (2001). Bounds on bivariate distribution functions with given margins and measures of association, Commun. Stat.Theory Methods 30, no. 6, 1155–1162., |
[32] |
Puccetti, G. and L. Rüschendorf. (2012a). Bounds for joint portfolios of dependent risks, Stat. Risk. Model.Appl. Financ. Insur. 29, no. 2, 107–132., |
[33] |
Puccetti, G. and L. Rüschendorf. (2012b). Computation of sharp bounds on the distribution of a function of dependent risks, J. Comput. Appl. Math 236, no. 7, 1833–1840., |
[34] |
Puccetti, G. and L. Rüschendorf. (2013). Sharp bounds for sums of dependent risks, J. Appl. Probab. 50, no. 1, 42–53., |
[35] |
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, no. 3, 245–266., |
[36] |
Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process, J. Stat. Plann. Inference 139, no. 11, 3921–3927., |
[37] |
Rüschendorf, L. (2013). Mathematical Risk Analysis, Springer, New York., |
[38] |
Rüschendorf, L. (2017a). Improved Hoeffding–Fréchet bounds and applications to VaR estimates. In:Copulas and Dependence Models with Applications. Contributions in Honor of Roger B. Nelsen(M. Úbeda Flores, E. de Amo Artero, F. Durante, and J. Fernández Sánchez, eds.), Springer, Cham. https://doi.org/10.1007/978-3-319-64221-5 12., |
[39] |
Rüschendorf, L. (2017b). Risk bounds and partial dependence information. In: From Statistics to Mathematical Finance, Springer, Festschrift in honour of Winfried Stute, Cham., |
[40] |
Rüschendorf, L. and J. Witting. (2017). VaR bounds in models with partial dependence information on subgroups, Depend Model 5, 59–74., |
[41] |
Shaked, M. and J.G. Shantikumar. (2007). Stochastic Orders, Springer, New York., |
[42] |
Tian, R. (2008). Moment problems with applications to Value-at-Risk and portfolio management, Georgia State University, Dissertation., |
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