Advertisement

Finance and Stochastics

, Volume 16, Issue 2, pp 177–206 | Cite as

An example of a stochastic equilibrium with incomplete markets

  • Gordan Žitković
Article

Abstract

We prove existence and uniqueness of stochastic equilibria in a class of incomplete continuous-time financial environments where the market participants are exponential utility maximizers with heterogeneous risk-aversion coefficients and general Markovian random endowments. The incompleteness featured in our setting—the source of which can be thought of as a credit event or a catastrophe—is genuine in the sense that not only the prices, but also the family of replicable claims itself are determined as a part of the equilibrium. Consequently, equilibrium allocations are not necessarily Pareto optimal and the related representative-agent techniques cannot be used. Instead, we follow a novel route based on new stability results for a class of semilinear partial differential equations related to the Hamilton–Jacobi–Bellman equation for the agents’ utility maximization problems. This approach leads to a reformulation of the problem where the Banach fixed-point theorem can be used not only to show existence and uniqueness, but also to provide a simple and efficient numerical procedure for its computation.

Keywords

Equilibrium Exponential utility Incomplete markets Mathematical finance 

Mathematics Subject Classification (2000)2010

91G80 35K59 

JEL Classification

C62 G11 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, R.M., Raimondo, R.C.: Equilibrium in continuous-time financial markets: Endogenously dynamically complete markets. Econometrica 76, 841–907 (2008) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Basak, S., Cuoco, D.: An equilibrium model with restricted stock market participation. Rev. Financ. Stud. 11, 309–341 (1998) CrossRefGoogle Scholar
  3. 3.
    Becherer, D.: Rational hedging and valuation with utility-based preferences. PhD thesis, Technical University of Berlin (2001). http://opus.kobv.de/tuberlin/volltexte/2001/213/pdf/becherer_dirk.pdf
  4. 4.
    Carassus, L., Rásonyi, M.: Convergence of utility indifference prices to the superreplication price. Math. Methods Oper. Res. 64, 145–154 (2006) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Carassus, L., Rásonyi, M.: Optimal strategies and utility-based prices converge when agents’ preferences do. Math. Oper. Res. 32, 102–117 (2007) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Carmona, R. (ed.): Indifference Pricing. Theory and Applications. Princeton Series in Financial Engineering. Princeton University Press, Princeton (2009) MATHGoogle Scholar
  7. 7.
    Dana, R.-A., Pontier, M.: On existence of an Arrow-Radner equilibrium in the case of complete markets. A remark. Math. Oper. Res. 17, 148–163 (1992) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math. Finance 12, 99–123 (2002) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Duffie, D.: Stochastic equilibria: existence, spanning number, and the “no expected financial gain from trade” hypothesis. Econometrica 54, 1161–1183 (1986) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Duffie, D., Huang, C.F.: Implementing Arrow-Debreu equilibria by continuous trading of few long-lived securities. Econometrica 53, 1337–1356 (1985) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Duffie, D., Zame, W.: The consumption-based capital asset pricing model. Econometrica 57, 1279–1297 (1989) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics (New York), vol. 25. Springer, New York (1993) MATHGoogle Scholar
  13. 13.
    Hadamard, J.: Sur les problèmes aux dérivées partielles et leur signification physique. Princet. Univ. Bull. 13, 49–52 (1902) MathSciNetGoogle Scholar
  14. 14.
    Hugonnier, J.: Rational asset pricing bubbles and portfolio constraints. Preprint (2010). http://sfi.epfl.ch/files/content/sites/sfi/files/users/192824/public/papers/Bubbles-Revision1.pdf
  15. 15.
    Jouini, E., Napp, C.: Convergence of utility functions and convergence of optimal strategies. Finance Stoch. 8, 133–144 (2004) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Applications of Mathematics (New York), vol. 39. Springer, New York (1998) MATHGoogle Scholar
  17. 17.
    Karatzas, I., Lakner, P., Lehoczky, J.P., Shreve, S.E.: Equilibrium in a simplified dynamic, stochastic economy with heterogeneous agents. In: Meyer-Wolf, E., Schwartz, A., Zeitouni, O. (eds.) Stochastic Analysis: Liber Amicorum for Moshe Zakai, pp. 245–272. Academic Press, Boston (1991) Google Scholar
  18. 18.
    Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model. Math. Oper. Res. 15, 80–128 (1990) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Equilibrium models with singular asset prices. Math. Finance 1/3, 11–29 (1991) CrossRefGoogle Scholar
  20. 20.
    Kardaras, C., Žitković, G.: Stability of the utility maximization problem with random endowment in incomplete markets. Math. Finance 21, 313–333 (2007) Google Scholar
  21. 21.
    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12. AMS, Providence (1996) MATHGoogle Scholar
  22. 22.
    Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type, vol. 23. AMS, Providence (1967). Translated from the Russian by S. Smith. Translations of Mathematical Monographs Google Scholar
  23. 23.
    Larsen, K.: Continuity of utility-maximization with respect to preferences. Math. Finance 19, 237–250 (2009) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Larsen, K., Žitković, G.: Stability of utility-maximization in incomplete markets. Stoch. Process. Appl. 117, 1642–1662 (2007) MATHCrossRefGoogle Scholar
  25. 25.
    Magill, M., Quinzii, M.: Theory of Incomplete Markets, vol. 1. MIT Press, Cambridge (1996) Google Scholar
  26. 26.
    Owen, M., Žitković, G.: Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19, 129–159 (2009) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Scarf, H.: The approximation of fixed points of a continuous mapping. SIAM J. Appl. Math. 15, 1328–1343 (1967) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Scarf, H.E.: On the computation of equilibrium prices. In: Fellner, W. (ed.) Ten Economic Studies in the Tradition of Irving Fisher, pp. 207–230. Wiley, New York (1967) Google Scholar
  29. 29.
    Wu, Z., Yin, J., Wang, C.: Elliptic & Parabolic Equations. World Scientific, Hackensack (2006) MATHGoogle Scholar
  30. 30.
    Žitković, G.: Financial equilibria in the semimartingale setting: Complete markets and markets with withdrawal constraints. Finance Stoch. 10, 99–119 (2006) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

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

Personalised recommendations