Multi-portfolio time consistency for set-valued convex and coherent risk measures

详细信息    查看全文

• 作者：Zachary Feinstein (1)
  Birgit Rudloff (2)
  
• 关键词：Dynamic risk measures ; Transaction costs ; Set ; valued risk measures ; Time consistency ; Multi ; portfolio time consistency ; Stability ; 91B30 ; 46A20 ; 46N10 ; 26E25 ; G32 ; C61 ; G15 ; G28
• 刊名：Finance and Stochastics
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：19
• 期：1
• 页码：67-107
• 全文大小：1,196 KB
• 参考文献：1. Acciaio, B., Penner, I.: Dynamic risk measures. In: Di Nunno, G., 脴ksendal, B. (eds.) Advanced Mathematical Methods for Finance, pp.聽1鈥?4. Springer, Berlin (2011) CrossRef
  2. Ararat, C., Hamel, A.H., Rudloff, B.: Set-valued shortfall and divergence risk measures. Preprint (2014). http://arxiv.org/pdf/1405.4905.pdf
  3. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Thinking coherently. Risk 10, 68鈥?1 (1997)
  4. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203鈥?28 (1999) CrossRef
  5. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., Ku, H.: Coherent multiperiod risk adjusted values and Bellman鈥檚 principle. Ann. Oper. Res. 152, 5鈥?2 (2007) CrossRef
  6. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Systems & Control. Birkh盲user, Basel (1990)
  7. Ben Tahar, I., L茅pinette, E.: Vector-valued risk measure processes. Int. J. Theor. Appl. Finance 17, 1450011 (2014) CrossRef
  8. Bion-Nadal, J.: Conditional risk measures and robust representation of convex risk measures. Ecole Polytechnique, CMAP. Preprint no. 557 (2004). http://www.cmap.polytechnique.fr/preprint/repository/557.pdf
  9. Bion-Nadal, J.: Dynamic risk measures: time consistency and risk measures from BMO martingales. Finance Stoch. 12, 219鈥?44 (2008) CrossRef
  10. Bion-Nadal, J.: Time consistent dynamic risk processes. In: Stochastic Processes and Their Applications, vol.聽119, pp.聽633鈥?54 (2009)
  11. Cheridito, P., Delbaen, F., Kupper, M.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57鈥?06 (2006) CrossRef
  12. Cheridito, P., Kupper, M.: Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Finance 14, 137鈥?62 (2011) CrossRef
  13. Cheridito, P., Stadje, M.: Time-inconsistency of VaR and time-consistent alternatives. Finance Res. Lett. 6, 40鈥?6 (2009) CrossRef
  14. Delbaen, F.: The structure of / m-stable sets and in particular of the set of risk neutral measures. In: 脡mery, M., Yor, M. (eds.) Memoriam Paul-Andr茅 Meyer. Lecture Notes in Mathematics, vol.聽1874, pp.聽215鈥?58. Springer, Berlin (2006) CrossRef
  15. Delbaen, F., Peng, S., Rosazza Gianin, E.: Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14, 449鈥?72 (2010) CrossRef
  16. Detlefsen, K., Scandolo, G.: Conditional and dynamic convex risk measures. Finance Stoch. 9, 539鈥?61 (2005) CrossRef
  17. Feinstein, Z., Rudloff, B.: Time consistency of dynamic risk measures in markets with transaction costs. Quant. Finance 13, 1473鈥?489 (2013) CrossRef
  18. Feinstein, Z., Rudloff, B.: A comparison of techniques for dynamic multivariate risk measures. In: Hamel, A., Heyde, F., L枚hne, A., Rudloff, B., Schrage, C. (eds.) Set Optimization and Applications in Finance鈥擳he State of the Art. Springer Proceedings in Mathematics & Statistics Springer, Berlin (2014). http://arxiv.org/pdf/1305.2151.pdf
  19. F枚llmer, H., Penner, I.: Convex risk measures and the dynamics of their penalty functions. Stat. Decis. 24, 61鈥?6 (2006)
  20. F枚llmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6, 429鈥?47 (2002) CrossRef
  21. F枚llmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. De Gruyter Studies in Mathematics. de Gruyter, Berlin (2011) CrossRef
  22. Frittelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Bank. Finance 26, 1473鈥?486 (2002) CrossRef
  23. Frittelli, M., Rosazza Gianin, E.: Dynamic convex risk measures. In: Szeg枚, G.P. (ed.) New Risk Measures for the 21st Century, pp.聽227鈥?48. Wiley, New York (2004)
  24. G枚pfert, A., Riahi, H., Tammer, C., Z膬linescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics. Springer, Berlin (2003)
  25. Hamel, A.H.: A duality theory for set-valued functions I: Fenchel conjugation theory. Set-Valued Var. Anal. 17, 153鈥?82 (2009) CrossRef
  26. Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1, 66鈥?5 (2010) CrossRef
  27. Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5, 1鈥?8 (2011) CrossRef
  28. Hamel, A.H., Rudloff, B.: Continuity and finite-valuedness of set-valued risk measures. In: Tammer, C., Heyde, F. (eds.) Festschrift in Celebration of Prof. Dr. Wilfried Greckschs鈥?60th Birthday, pp.聽46鈥?4. Shaker, Aachen (2008)
  29. Hamel, A.H., Rudloff, B., Yankova, M.: Set-valued average value at risk and its computation. Math. Financ. Econ. 7, 229鈥?46 (2013) CrossRef
  30. Heyde, F., Schrage, C.: Continuity concepts for set-valued functions and a fundamental duality formula for set-valued optimization. J. Math. Anal. Appl. 397, 772鈥?84 (2013) CrossRef
  31. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis: Volume I: Theory. Mathematics and Its Applications. Springer, Berlin (1997) CrossRef
  32. Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stoch. 8, 531鈥?52 (2004)
  33. Kabanov, Y.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 3, 237鈥?48 (1999) CrossRef
  34. Kabanov, Y.M., Safarian, M.: Markets with Transaction Costs: Mathematical Theory. Springer, Berlin (2009)
  35. Kl枚ppel, S., Schweizer, M.: Dynamic indifference valuation via convex risk measures. Math. Finance 17, 599鈥?27 (2007) CrossRef
  36. L枚hne, A.: Vector Optimization with Infimum and Supremum. Vector Optimization. Springer, Berlin (2011) CrossRef
  37. L枚hne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios in markets with transaction costs. Int. J. Theor. Appl. Finance 17, 1450012 (2014) CrossRef
  38. Pennanen, T., Penner, I.: Hedging of claims with physical delivery under convex transaction costs. SIAM J. Financ. Math. 1, 158鈥?78 (2010) CrossRef
  39. Riedel, F.: Dynamic coherent risk measures. In: Stochastic Processes and Their Applications, vol.聽112, pp.聽185鈥?00 (2004)
  40. Ruszczynski, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res. 31, 544鈥?61 (2006) CrossRef
  41. Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14, 19鈥?8 (2004) CrossRef
  42. Yan, J.-A.: On the commutability of essential infimum and conditional expectation operations. Chin. Sci. Bull. 30, 1013鈥?018 (1985)
  
• 作者单位：Zachary Feinstein (1)
  Birgit Rudloff (2)
  
  1. Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO, 63130, USA
  2. Department of Operations Research and Financial Engineering, Bendheim Center for Finance, Princeton University, Princeton, NJ, 08544, USA
  
• ISSN：1432-1122

文摘

Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on \(L^{p}({\varOmega,\mathcal{F},\mathbb{P}; \mathbb{R}^{d}})\) with image space in the power set of \(L^{p}({\varOmega,\mathcal{F}_{t},\mathbb{P}; \mathbb{R}^{d}})\) . In the convex case, multi-portfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the coherent case, multi-portfolio time consistency is equivalent to a generalized version of stability of the dual variables. As examples, the set-valued entropic risk measure with constant risk aversion coefficient is shown to satisfy the cocycle condition for its minimal penalty functions; the set of superhedging portfolios is shown to have in markets with proportional transaction costs the stability property and to satisfy in markets with convex transaction costs the composed cocycle condition; and a multi-portfolio time-consistent version of the set-valued average value at risk, the composed AV@R, is given, and its dual representation deduced.