×

Coherent multiperiod risk adjusted values and Bellman’s principle. (English) Zbl 1132.91484

Summary: Starting with a time-0 coherent risk measure defined for “value processes”, we also define risk measurement processes. Two other constructions of measurement processes are given in terms of sets of test probabilities. These latter constructions are identical and are related to the former construction when the sets fulfill a stability condition also met in multiperiod treatment of ambiguity as in decision-making. We finally deduce risk measurements for the final value of locked-in positions and repeat a warning concerning Tail-Value-at-Risk.

MSC:

91B30 Risk theory, insurance (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Artzner, P. (2002a). Conditional Value at Risk: Is it Good in the Multiperiod Case? IIR Conference on Volatility and Risk, London, Feb. 18–19.
[2] Artzner, P. (2002b). Multiperiod Risk Measurement: Where are we? Quantitative Finance Seminar, Fields Institute, U. Toronto, Nov. 25, http://www.fields.utoronto.ca/audio/02-03/finance_seminar/artzner/.
[3] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1997). ”Thinking Coherently,” Risk, 10, 68–71.
[4] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999a). ”Coherent Risk Measures.” Mathematical Finance, 9, 203–228. · Zbl 0980.91042 · doi:10.1111/1467-9965.00068
[5] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999b). Risk Management and Capital Allocation with Coherent Measures of Risk. http://symposium.wiwi.uni-karlsruhe.de/8thpapers/artzner.ps. · Zbl 0980.91042
[6] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (2001). Coherent Measures of Multiperiod Risk. Presentation at the Workshop New Ideas in Risk Management, Carnegie Mellon University, Aug. 25–26.
[7] Artzner, P., F. Delbaen, and P. Koch-Medina. (2005). ”Risk Measures and Efficient Use of Capital,” Working Paper ETH, Zürich. · Zbl 1203.91110
[8] Bennet, O. (2001). ”Reinventing RAROC.” Risk, 14, 112–113.
[9] Cheridito, P., F. Delbaen, and M. Kupper. (2002). Convex Measures of Risk for Càdlàg Processes. Working Paper ETH, Zürich. · Zbl 1114.91047
[10] Cvitanić, J. and I. Karatzas. (1999). ”On Dynamic Measures of Risk.” Finance and Stochastics, 3, 451–482. · Zbl 0982.91030 · doi:10.1007/s007800050071
[11] Chow, Y.S., H. Robbins, and D. Siegmund. (1972). Great Expectations: The Theory of Optimal Stopping. Boston: Houghton Mifflin, reprinted by Dover, New York (1992). · Zbl 0233.60044
[12] Delbaen, F. (2000). Coherent Risk Measures on General Probability Spaces. Advances in Finance and Stochastics, Essays in Honour of Dieter Sondermann. New York: Springer.
[13] Delbaen, F. (2001). ”The Structure of m-Stable Sets and in Particular of the Set of Risk Neutral Measures”. Working Paper ETH, Zürich. · Zbl 1121.60043
[14] Delbaen, F. (2002). Coherent Risk Measures. Lectures given at the Cattedra Galileiana, March 2000. Pisa: Scuola Normale Superiore.
[15] Dothan, M. (1990). Prices in Financial Markets. New York: Oxford University Press. · Zbl 0744.90010
[16] Embrechts, P. (1995). A Survival Kit to Quantile Estimation. Zürich: UBS Quant Workshop.
[17] Epstein, L. and M. Schneider. (2003). ”Recursive Multiple-Priors.” Journal of Economic Theory, 113, 1–31, earlier versions June 2001, April 2002. · Zbl 1107.91360
[18] Föllmer, H. and A. Schied. (2002). ”Convex Measures of Risk and Trading Constraints.” Finance and Stochastics, 6, 429–447. · Zbl 1041.91039 · doi:10.1007/s007800200072
[19] Föllmer, H. and A. Schied. (2004). Stochastic Finance, 2nd ed. Berlin: de Gruyter. · Zbl 1126.91028
[20] Gilboa, I. and D. Schmeidler. (1989). ”Maxmin Expected Utility with Non-Unique Prior.” Journal of Mathematical Economics, 18, 141–153. · Zbl 0675.90012 · doi:10.1016/0304-4068(89)90018-9
[21] Heath, D. (1998) Coherent Measures of Risk. Documents from the 5th Annual Conference on Risk Management, International Center for Business Information, Geneva, December 8th.
[22] Jaschke, S. and U. Küchler. (2001). ”Coherent Risk Measures and Good-Deal Bounds.” Finance and Stochastics, 5, 181–200. · Zbl 0993.91023 · doi:10.1007/PL00013530
[23] Nakano, Y. (2003). ”Minimizing Coherent Risk Measures of Shortfall in Discrete-time Models with Cone Constraints.” Applied Mathematical Finance, 10, 163–181. · Zbl 1090.91054 · doi:10.1080/1350486032000102924
[24] Neveu, J. (1972). Martingales à temps discret. Paris: Masson, English transl. Discrete-Parameter Martingales (1975), North-Holland. · Zbl 0235.60010
[25] Riedel, F. (2004). ”Dynamic Coherent Risk Measures.” Stochastic Processes and Applications, 112, 185–200, earlier version November 2002. · Zbl 1114.91055
[26] Roorda, B., J. Engwerda, and H. Schumacher. (2002). Coherent Acceptability Measures in Multiperiod Models. Working Paper U. of Twente and Tilburg U., June, later version July 2003. · Zbl 1107.91059
[27] Wang, T. (1996). A Characterization of Dynamic Risk Measures. Working Paper Faculty of Commerce and Business Administration, U.B.C.
[28] Wang, T. (1999). A Class of Dynamic Risk Measures. Working Paper Faculty of Commerce and Business Administration, U.B.C, later version 2002.
[29] Wang, T. (2003). ”Conditional Preferences and Updating,” Journal of Economic Theory, 108, 286–321, earlier version March 2002. · Zbl 1040.91033
[30] Wilkie, D., H. Waters, and S. Yang. (2003). ”Reserving, Pricing and Hedging for Policies with Guaranteed Annuity Options.” Paper presented to the Faculty of Actuaries, Edinburgh, 20 January 2003. British Actuarial Journal, 9, 263–425.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.