Annals of Finance

, 6:107 | Cite as

Partial equilibria with convex capital requirements: existence, uniqueness and stability

  • Michail Anthropelos
  • Gordan Žitković
Research Article


We consider several risk-averse financial agents who negotiate the price of a bundle of contingent claims in an incomplete semimartingale model of a financial market. Assuming that the agents’ risk preferences are modeled by convex capital requirements, we define and analyze their demand functions and propose a notion of a partial equilibrium price. In addition to sufficient conditions for the existence and uniqueness, we also show that the equilibrium prices are stable with respect to misspecifications of agents’ risk preferences.


Acceptance sets Convex capital requirements Incomplete markets Mutually agreeable claims Partial equilibrium allocation Partial equilibrium price Stability of equilibria 

Mathematics Subject Classification (2000)

Primary 91B70 Secondary 91B30 60G35 

JEL Classification

G11 G12 C62 


  1. Anthropelos, M., Žitković, G.: On agents’ agreement and partial-equilibrium pricing in incomplete markets (2008, to appear in Mathematical Finance). arxiv:0803.2198Google Scholar
  2. Artzner P., Delbaen F., Eber J.M., Heath D.: Coherent measures of risk. Math Financ 9(3), 203–228 (1999)CrossRefGoogle Scholar
  3. Barrieu P., El Karoui N.: Inf-convolution of risk measures and optimal risk tranfer. Financ Stoch 9, 269–298 (2005)CrossRefGoogle Scholar
  4. Barrieu, P., Karoui, N.E.: Optimal derivatives design under dynamic risk measures. In: Mathematics of finance, Contemp. Math., vol. 351, pp. 13–25. Providence: Amer. Math. Soc. (2004)Google Scholar
  5. Barrieu P., Scandolo G.: General pareto optimal allocation and applications to multi-period risks. Astin Bull 1(38), 105–136 (2008)CrossRefGoogle Scholar
  6. Biagini S., Frittelli M.: A unified framework for utility maximization problems: an Orlicz space approach. Ann Appl Probab 18(3), 929–966 (2008)CrossRefGoogle Scholar
  7. Borch K.: Equilibrium in reinsurance market. Econometrica 30(3), 424–444 (1962)CrossRefGoogle Scholar
  8. Borwein, J.M., Lewis, A.S.: Convex analysis and nonlinear optimization. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 3 (Theory and examples). New York: Springer (2000)Google Scholar
  9. Bühlmann H.: An economic premium principle. Astin Bull 11, 52–60 (1980)Google Scholar
  10. Bühlmann H.: The general economic premium principle. Astin Bull 15, 13–21 (1984)Google Scholar
  11. Bühlmann H., Jewell W.S.: Optimal risk exchanges. Astin Bull 10, 243–262 (1979)Google Scholar
  12. Burgert C., Rüschendorf L.: Allocation of risks and equilibrium in markets with finitely many traders. Insur Math Econ 42, 177–188 (2008)CrossRefGoogle Scholar
  13. Carasus, L., Rásonyi, M.: Optimal strategies and utility-based prices converge when agents’ preferences do (2005, preprint)Google Scholar
  14. Carmona, R. (eds): Indifference Pricing: Theory and Applications. Princeton University Press, Princeton (2008)Google Scholar
  15. Carr P., Geman H., Madan D.: Pricing and hedging in incomplete markets. J Financial Econ 62(1), 131–167 (2001)CrossRefGoogle Scholar
  16. Cvitanić J., Schachermayer W., Wang H.: Utility maximization in incomplete markets with random endowment. Finance Stoch 5, 237–259 (2001)CrossRefGoogle Scholar
  17. Delbaen, F.: Coherent risk measures on general probability spaces. In: Advances in Finance and Stochastics, pp. 1–37. Berlin: Springer (2002)Google Scholar
  18. Delbaen F., Grandits P., Rheinländer T., Samperi D., Schweizer M., Stricker C.: Exponential hedging and entropic penalties. Math. Financ 12(2), 99–123 (2002)CrossRefGoogle Scholar
  19. Delbaen F., Schachermayer W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463–520 (1994)CrossRefGoogle Scholar
  20. Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Berlin: Springer (1993)Google Scholar
  21. Ekeland, I., Témam, R.: Convex analysis and variational problems. Classics in Applied Mathematics, vol. 28. english edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999). Translated from the FrenchGoogle Scholar
  22. Filipović D., Kupper M.: Equilibrium prices for monetary utility functions. Int J Theor Appl Financ 11(3), 325–343 (2008)CrossRefGoogle Scholar
  23. Föllmer H., Schied A.: Convex measures of risk and trading constraints. Financ Stoch 6(4), 429–447 (2002)CrossRefGoogle Scholar
  24. Föllmer, H., Schied, A.: Robust preferences and convex measures of risk. In: Advances in Finance and Stochastics, pp. 39–56. Berlin: Springer (2002)Google Scholar
  25. Föllmer, H., Schied, A.: Stochastic Finance. de Gruyter Studies in Mathematics, vol. 27, extended edn. Berlin: Walter de Gruyter & Co. (2004). An introduction in discrete timeGoogle Scholar
  26. Frittelli M., Scandolo G.: Risk measures and capital requirements for processes. Math Financ 16(4), 589–612 (2006)Google Scholar
  27. Gerber H.: Pareto optimal risk exchanges and related decision problems. Astin Bull 10, 22–33 (1978)Google Scholar
  28. Hadamard J.: Sur les Problèmes Aux dérivées Partielles et leur Signification Physique, pp. 49–52. Princeton University Bulletin, Princeton (1902)Google Scholar
  29. Heath D., Ku E.: Pareto equilibria with coherent measures of risk. Math Financ 14(2), 163–172 (2004)CrossRefGoogle Scholar
  30. Hugonnier J., Kramkov D.: Optimal investment with random endowments in incomplete markets. Ann Appl Probab 14(2), 845–864 (2004)CrossRefGoogle Scholar
  31. Jouini E., Napp C.: Convergence of utility functions and convergence of optimal strategies. Financ Stoch 8(1), 133–144 (2004)CrossRefGoogle Scholar
  32. Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. In: Advances in Mathematical Economics. Adv. Math. Econ., vol. 9, pp. 49–71. Tokyo: Springer (2006)Google Scholar
  33. Kardaras, C., Žitković, G.: Stability of the utility maximization problem with random endowment in incomplete markets (2007, to appear in Mathematical Finance). arxiv:0706.0482Google Scholar
  34. Klöppel S., Schweizer M.: Dynamic utility indifference valuation via convex risk measures. Math Financ 17, 599–627 (2007)CrossRefGoogle Scholar
  35. Larsen K., Žitković G.: Stability of utility-maximization in incomplete markets. Stoch Process Appl 117(11), 1642–1662 (2007)CrossRefGoogle Scholar
  36. Lucchetti R.: Convexity and Well-Posed Problems. Springer, New York (2006)Google Scholar
  37. Mania M., Schweizer M.: Dynamic exponential utility indifference valuation. Ann Appl Probab 15(3), 2113–2143 (2005)CrossRefGoogle Scholar
  38. Owen, M., Žitković, G.: Optimal investment with an unbounded random endowment and utility-based pricing (2006, to appear in Mathematical Finance). arxiv:0706.0478Google Scholar
  39. Pal, S.: On capital requirements and optimal strategies to achieve acceptability. Ph.D. thesis, Columbia University (2006)Google Scholar
  40. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)Google Scholar
  41. Rockafellar, R.T., Wets, R.J.B.: Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Berlin: Springer (1998)Google Scholar
  42. Schachermayer, W.: Portfolio optimization in incomplete financial markets. Cattedra Galileiana. [Galileo Chair]. Scuola Normale Superiore, Classe di Scienze, Pisa (2004)Google Scholar
  43. Wyler E.: Pareto-optimal risk exchanges and a system of differential equations: a duality theorem. Astin Bull 20, 23–32 (1990)CrossRefGoogle Scholar
  44. Xu M.: Risk measure pricing and hedging in incomplete markets. Ann Financ 2, 51–71 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at AustinAustinUSA

Personalised recommendations