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Exchanges and measures of risks

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Abstract

When investors or reinsurers measure economic risk in monetary terms, they operate as though utility were transferable. A main purpose of this paper is to show that transferability largely facilitates attainment, analysis, computation and modelling of equilibrium in exchange economies. To wit, under reasonable and weak assumptions, it is shown that duality delivers an equilibrium price, and that simple bilateral barters may ensure market clearing. If, however, underlying beliefs about future states are strictly incompatible, equilibrium cannot generally exist.

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References

  1. Artzner P., Delbaen F., Eber J.-M., Heath D.: Coherent measures of risk. Math. Financ. 9, 203–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barrieu P., El Karoui N.: Inf-convolution of risk measures and optimal risk transfer. Financ. Stoch. 9, 269–298 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benveniste A., Métivier M., Priouret P.: Adaptive algorithms and stochastic approximations. Springer-Verlag, Berlin (1990)

    MATH  Google Scholar 

  4. Borch K.H.: Equilibrium in a reinsurance market. Econometrica. 30, 424–444 (1962)

    Article  MATH  Google Scholar 

  5. Borwein J.M., Lewis A.S.: Convex analysis and nonlinear optimization. Springer, Berlin (2000)

    MATH  Google Scholar 

  6. Carr P., Geman H., Madan D.B.: Pricing and hedging in incomplete markets. J. Financ. Econ. 62, 131–167 (2001)

    Article  Google Scholar 

  7. Cheridito P., Li T.: Dual characterization of properties of risk measures on Orlicz hearts. Math. Financ. Econ. 2, 29–55 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cochrane J.H.: Asset pricing. Princeton University Press, Princeton (2005)

    Google Scholar 

  9. Delbaen F.: Coherent risk measures on general probability spaces. In: Sandmann, K., Schönbucher, P. Advances in finance and stochastics, essays in honour of Dieter Sondermann, pp. 1–38. Springer, Berlin (2002)

    Google Scholar 

  10. Ekeland I., Temam R.: Analyse convexe et problèmes variationnels. Dunod, Paris (1974)

    MATH  Google Scholar 

  11. Feldmann A.M.: Bilateral trading processes, pairwise optimality, and Pareto optimality. Rev. Econ. Stud. 4, 463–473 (1973)

    Google Scholar 

  12. Filipović D., Svindland G.: Optimal capital and risk allocation for law- and cash-invariant convex functions. Financ. Stoch. 12, 423–439 (2008)

    Article  MATH  Google Scholar 

  13. Filipović D., Kupper M.: Equilibrium prices for monetary utility functions. Int. J. Theor. Appl. Financ. 11, 325–343 (2008)

    Article  MATH  Google Scholar 

  14. Föllmer H., Schied A.: Stochastic finance. 2nd edn. Gruyter, Berlin (2004)

    Book  MATH  Google Scholar 

  15. Fritelli M., Rosazza Gianin E.: Putting order in risk measure. J. Bank. Financ. 26, 1473–1486 (2002)

    Article  Google Scholar 

  16. Fritelli M., Scandolo G.: Risk measures and capital requirements for processes. Math. Financ. 16(4), 589–612 (2006)

    Article  Google Scholar 

  17. Gale D.: Strategic foundations of general equilibrium. Cambridge University Press, Melbourne (2000)

    Book  MATH  Google Scholar 

  18. Gintis H.: The dynamics of general equilibrium. Econ. J. 117, 1280–1309 (2007)

    Article  Google Scholar 

  19. Greco G.H., Flores-Franulič A., Román-Flores H.: Fenchel equalities and bilinear minmax equalities. Math. Scand. 98, 217–228 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Heath D., Ku H.: Pareto equilibria with coherent measures of risk. Math. Financ. 14(2), 163–172 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jaschke S., Küchler U.: Coherent risk measures and good-deal bounds. Financ. Stoch. 5(2), 181–200 (2001)

    Article  MATH  Google Scholar 

  22. Jouini E., Schachermayer W., Touzi N.: Optimal risk sharing for law invariant monetary utility functions. Math. Financ. 18(2), 269–292 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kakutani S.: Ein Beweis des Satzes von M. Eidelheit über konvexe Mengen. Proc. Imp. Acad. Tokyo 13, 93–94 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kirman A.P.: Whom or what does the representative individual represent. J. Econ. Perspect. 6(2), 117–132 (1992)

    Google Scholar 

  25. Madden P.J.: Efficient sequences of non-monetary exchange. Rev. Econ. Stud. 42(4), 581–596 (1975)

    Article  MATH  Google Scholar 

  26. Magill M., Quinzii M.: Theory of incomplete markets. MIT Press, Cambridge (1996)

    Google Scholar 

  27. Mas-Colell A.: The theory of general equilibrium analysis—a differentiable approach. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  28. Mordukhovich B.S., Nam N.M., Yen N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116, 369–396 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Müller A., Stoyan D.: Comparison methods for stochastic models and risks. Wiley, New York (2002)

    MATH  Google Scholar 

  30. Obstfeld M., Rogoff K.: Foundations of international macroeconomics. MIT Press, Cambridge (1996)

    Google Scholar 

  31. Osborne M.J., Rubinstein A.: Bargaining and markets. Academic Press, New York (1990)

    MATH  Google Scholar 

  32. Pflug G.: Subdifferential representations of risk measures. Math. Program. B 108, 339–354 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rockafellar R.T., Wets J.-B: Variational analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  34. Roth A.E., Sotomayor M.O.: Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, New York (1990)

    MATH  Google Scholar 

  35. Rubinstein A., Wolinsky A.: Decentralized trading, strategic behavior and the Walrasian outcome. Rev. Econ. Stud. 57, 63–78 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ruszczynski A.: Nonlinear Optimization. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  37. Saari D.G.: Iterative price mechanisms. Econometrica 53, 1117–1131 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  38. Schied A.: Optimal investments for risk- and ambiguity-averse preferences: a duality approach. Financ. Stoch. 11, 107–129 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Smith V.L.: Microeconomic systems as an experimental science. Am. Econ. Rev. 72(5), 923–955 (1982)

    Google Scholar 

  40. Starr R.M.: The structure of exchange in barter and monetary economies. Q. J. Econ. 86(2), 290–302 (1972)

    Article  MATH  Google Scholar 

  41. Starr R.M.: Decentralized non-monetary trade. Econometrica 44(5), 1087–1089 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tesfatsion L., Judd K.L.: Handbook of computational economics II: agent-based computational economics. North-Holland, Amsterdam (2006)

    MATH  Google Scholar 

  43. Valentine F.A.: Convex sets. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  44. Varian H.R.: Microeconomic analysis. Norton, New York (1992)

    Google Scholar 

  45. Zălinescu, C.: Convex analysis in general vector spaces. World Scientific (2002)

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  1. Department of Economics, University of Bergen, 5007, Bergen, Norway

    Sjur Didrik Flåm

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Flåm, S.D. Exchanges and measures of risks. Math Finan Econ 5, 249–267 (2011). https://doi.org/10.1007/s11579-012-0062-9

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