Abstract

Monetary risk measures are studied here in terms of acceptable outcomes, normalized non-negative prices, and resulting shortfalls. At center stage stand convex analysis, saddle functions and associated max-min formulae. The latter comply with common sense and established theory.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Convex risk measures
  • normalized prices
  • max-min shortfall
  • asset pricing
  • ambiguity aversion

Mathematics Subject Classification

46N10, 47H07, 91B30, 91G80

References

  1. C. D. Aliprantis, K. C. Border: Infinite Dimensional Analysis, Springer, Berlin (1994).https://doi.org/10.1007/978-3-662-03004-2Google Scholar
  2. P. Artzner, F. Delbaen, J.-M. Eber, D. Heath: Coherent measures of risk, Math. Finance 9 (1999) 203–228.https://doi.org/10.1111/1467-9965.00068Google Scholar
  3. J.-P. Aubin, I. Ekeland: Applied Nonlinear Analysis, J. Wiley & Sons, New York (1984).Google Scholar
  4. A. Ben-Tal, M. Teboulle: Expected utility, penalty functions, and duality in stochastic nonlinear programming, Manage. Sci. 32(11) (1986) 1145–1466.https://doi.org/10.1287/mnsc.32.11.1445Google Scholar
  5. A. Ben-Tal, A. Ben-Israel: A recourse certainty equivalent for decisions under uncertainty, Ann. Oper. Res. 30 (1991) 3–44.https://doi.org/10.1007/BF02204807Google Scholar
  6. K. H. Borch: Equilibrium in a reinsurance market, Econometrica 30 (1962) 424–444.https://doi.org/10.2307/1909887Google Scholar
  7. J. M. Borwein, A. S. Lewis: Convex Analysis and Nonlinear Optimization, Springer, Berlin (2000).https://doi.org/10.1007/978-1-4757-9859-3Google Scholar
  8. C. Burgert, L. Rüschendorf: Consistent risk measures for portfolio vectors, Insur. Math. Econ. 38 (2006) 289–297.https://doi.org/10.1016/j.insmatheco.2005.08.008Google Scholar
  9. S. Cerreia-Vioglio, F. Maccheroni, M. Marinacci, L. Montrucchio: Complete monotone quasiconcave duality, Math. Oper. Res. 36 (2011) 321–339.https://doi.org/10.1287/moor.1110.0483Google Scholar
  10. S. Cerreia-Vioglio, F. Macchehroni, M. Marinacci, L. Montrucchio: Risk measures: rationality and diversification, Math. Finance 21 (2011) 743–774.https://doi.org/10.1111/j.1467-9965.2010.00450.xGoogle Scholar
  11. S. Cerreia-Vioglio, F. Maccheroni, M. Marinacci, L. Montrucchio: Uncertainty averse preferences, J. Econ. Theory 146 (2011) 1275–1330.https://doi.org/10.1016/j.jet.2011.05.006Google Scholar
  12. J. H. Cochrane: Asset Pricing, Princeton University Press, Princeton (2005).Google Scholar
  13. T. M. Cover, J. A. Thomas: Elements of Information Theory, John Wiley & Sons, New York (1991).Google Scholar
  14. F. Delbaen: Coherent risk measures on general probability spaces, in: Advances in Finance and Stochastics, K. Sandmann, P. Schönbucher (eds.), Springer, Berlin (2002) 1–38.https://doi.org/10.1007/978-3-662-04790-3_1Google Scholar
  15. D. Denneberg: Non-Additive Measure and Integral, Kluwer, Dordrecht (1994).https://doi.org/10.1007/978-94-017-2434-0Google Scholar
  16. S. Drapeau, M. Kupper: Risk preferences and their robust representation, Math. Oper. Res. 38(1) (2013) 28–62.https://doi.org/10.1287/moor.1120.0560Google Scholar
  17. I. Ekeland, W. Schachermayer: Law invariant risk measures on L∞ (R), Stat. Risk. Model. 28(3) (2011) 195–225.https://doi.org/10.1524/stnd.2011.1099Google Scholar
  18. D. Ellsberg: Risk, ambiguity, and the Savage axioms, Q. J. Econ. 75 (1961) 643–669.https://doi.org/10.2307/1884324Google Scholar
  19. S. D. Flåm: Option pricing by mathematical programming, Optimization 57(1) (2008) 165–182.https://doi.org/10.1080/02331930701779054Google Scholar
  20. S. D. Flåm: Exchanges and measures of risks, Math. Financ. Econ. 5 (2011) 249–267.https://doi.org/10.1007/s11579-012-0062-9Google Scholar
  21. S. D. Flåm: On sharing of risk and resources, in: Optimization Theory and Related Topics (Haifa, 2010), S. Reich et al. (ed.), Contemp. Math. 568, Amer. Math. Soc., Providence (2012) 53–68.https://doi.org/10.1090/conm/568/11276Google Scholar
  22. H. Föllmer, A. Schied: Stochastic Finance, 2nd. Ed., de Gruyter, Berlin (2004).https://doi.org/10.1515/9783110212075Google Scholar
  23. H. Föllmer, A. Schied: Convex measures of risk and trading constraints, Finance Stoch. 6(4) (2002) 429–447.https://doi.org/10.1007/s007800200072Google Scholar
  24. M. Frittelli, R. E. Gianin: Putting order in risk measure, J. Banking Finance 26 (2002) 1473–1486.https://doi.org/10.1016/S0378-4266(02)00270-4Google Scholar
  25. I. Gilboa, D. Schmeidler: Max-min expected utility with a non-unique prior, J. Math. Econ. 18 (1989) 141–153.https://doi.org/10.1016/0304-4068(89)90018-9Google Scholar
  26. G. H. Greco: Minmax convex pairs, J. Convex Analysis 13(1) (2006) 101–111.https://doi.org/10.68381/jca13007Google Scholar
  27. G. H. Greco, A. Flores-Franulič, H. Román-Flores: Fenchel equalities and bilinear min-max equalities, Math. Scand. 98 (2006) 217–228.https://doi.org/10.7146/math.scand.a-14992Google Scholar
  28. L. Hansen, T. Sargent: Robust control and model uncertainty, Amer. Econ. Rev. 91 (2001) 60–66.https://doi.org/10.1257/aer.91.2.60Google Scholar
  29. D. Heath, H. Ku: Pareto equilibria with coherent measures of risk, Math. Finance 14(2) (2004) 163–172.https://doi.org/10.1111/j.0960-1627.2004.00187.xGoogle Scholar
  30. A. J. King: Duality and martingales: a stochastic programming perspective on contingent claims, Math. Program., Ser. B. 93(3) (2002) 543–562.https://doi.org/10.1007/s101070100257Google Scholar
  31. S. Kou, X. Peng, C. C. Heyde: External risk measures and Basel accords, Math. Oper. Res. 38(3) (2013) 393–417.https://doi.org/10.1287/moor.1120.0577Google Scholar
  32. V. Krätschmer: Robust representation of convex risk measures by probability measures, Finance Stoch. 9 (2005) 597–608.https://doi.org/10.1007/s00780-005-0160-0Google Scholar
  33. P. A. Krokhmal: Higher moment coherent risk measures, in: Quantitative Fund Management, M. A. H. Dempster et al. (ed.), Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Boca Raton (2009) 271–298.https://doi.org/10.1201/9781420081923.ch13Google Scholar
  34. F. LeRoy, J. Werner: Principles of Financial Economics, Cambridge University Press, Cambridge (2001).https://doi.org/10.1017/CBO9780511753787Google Scholar
  35. F. Maccheroni, M. Marinacci, A. Rustichini: Ambiguity aversion, robustness and the variational representation of preferences, Econometrica 74(6) (2006) 1447–1498.https://doi.org/10.1111/j.1468-0262.2006.00716.xGoogle Scholar
  36. M. Magill, M. Quinzii: Theory of Incomplete Markets, MIT Press, Cambridge (1996).Google Scholar
  37. A. Mas-Colell, M. D. Whinston, J. R. Green: Microeconomic Theory, Oxford University Press, New York (1995).Google Scholar
  38. J. Orihuela, M. Ruiz Galán: Lebesgue property for convex risk measures on Orlicz spaces, Math. Financ. Econ. 6 (2012) 15–35.https://doi.org/10.1007/s11579-012-0058-5Google Scholar
  39. K. Owari: On the Lebesgue property of monotone convex functions, Math. Financ. Econ. 8 (2014) 159–167.https://doi.org/10.1007/s11579-013-0111-zGoogle Scholar
  40. G. Pflug: Subdifferential representations of risk measures, Math. Program., Ser. B 108 (2006) 339–354.https://doi.org/10.1007/s10107-006-0714-8Google Scholar
  41. S. R. Pliska: Introduction to Mathematical Finance, Blackwell, Oxford (1997).Google Scholar
  42. R. T. Rockafellar, S. Uryasev: Conditional value-at-risk for general loss distributions, J. Banking Finance 26(7) (2002) 1443–1471.https://doi.org/10.1016/S0378-4266(02)00271-6Google Scholar
  43. P. A. Samuelson: General proof that diversification pays, J. Financial Quant. Anal. 2 (1967) 1–13.https://doi.org/10.2307/2329779Google Scholar
  44. A. Schied: Optimal investments for risk- and ambiguity-averse preferences: a duality approach, Finance Stoch. 11 (2007) 107–129.https://doi.org/10.1007/s00780-006-0024-2Google Scholar
  45. F. A. Valentine: Convex Sets, McGraw-Hill, New York (1964).Google Scholar
  46. R. Wilson: The theory of syndicates, Econometrica 36(1) (1968) 119–132.https://doi.org/10.2307/1909607Google Scholar
  47. M. E. Yaari: The dual theory of choice under risk, Econometrica 55 (1987) 95–115.https://doi.org/10.2307/1911158Google Scholar

Suggested citation

S. D. Flåm. “Risk Measures, Convexity, and Max-Min Shortfalls.” Journal of Convex Analysis 22 (2015), No. 3, 603–626. https://doi.org/10.68381/jca22030

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.