General Equilibrium Theory and No-Arbitrage

  • Emilio Barucci
  • Claudio Fontana
Chapter
Part of the Springer Finance book series (FINANCE)

Abstract

This chapter deals with general equilibrium theory in a risky environment, where agents interact in a financial market. The chapter starts by presenting the notion of Pareto optimality and its implications in terms of risk sharing. The concept of rational expectations equilibrium is introduced and characterized in the context of a two-period economy. Different financial market structures are considered, with a particular attention to the important case of complete markets. The last part of the chapter is devoted to the fundamental theorem of asset pricing, which relates the absence of arbitrage opportunities to the existence of a strictly positive linear pricing functional. The relation of this important result to the existence of an equilibrium of an economy and its implications for the valuation of financial assets are also discussed.

References

  1. 46.
    Allen, F. and Gale, D. (1994) Financial Innovation and Risk Sharing. MIT Press, Cambridge.Google Scholar
  2. 56.
    Amershi, A. and Stoeckenius, J. (1983) The theory of syndicates and linear sharing rules. Econometrica, 51:1407–1416.CrossRefMATHGoogle Scholar
  3. 76.
    Arrow, K. (1953) The role of securities in the optimal allocation of risk–bearing. Review of Economic Studies, 30:91–96.Google Scholar
  4. 114.
    Balasko, Y. and Cass, D. (1989) The structure of financial equilibrium with exogenous yields: the case of incomplete markets. Econometrica, 57:135–162.MathSciNetCrossRefMATHGoogle Scholar
  5. 269.
    Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica, 30:424–444.CrossRefMATHGoogle Scholar
  6. 287.
    Breeden, D. and Litzenberger, R. (1978) Prices of state-contingent claims implicit in option prices. Journal of Business, 51:621–651.CrossRefGoogle Scholar
  7. 311.
    Brown, D. and Ross, S. (1991) Spanning, valuation and options. Economic Theory, 1:3–12.MathSciNetCrossRefMATHGoogle Scholar
  8. 375.
    Cass, D. (1984) Competitive equilibrium with incomplete financial markets. CARESS working paper, no.84–09, University of Pennsylvania.Google Scholar
  9. 476.
    Connor, G. (1989) Notes on the arbitrage pricing theory. In Theory of Valuation, S. Bhattacharya and G. Constantinides (eds.), Rowman & Littlefield, New Jersey.Google Scholar
  10. 488.
    Constantinides, G. (1982) Intertemporal asset pricing with heterogeneous consumers and without demand aggregation. Journal of Business, 55:253–267.CrossRefGoogle Scholar
  11. 514.
    Dana, R. and Jeanblanc-Picqué, M. (2003) Financial Markets in Continuous Time. Springer, Berlin/Heidelberg.Google Scholar
  12. 535.
    Debreu, G. (1959) Theory of Value. Yale University Press, New Haven.MATHGoogle Scholar
  13. 544.
    Delbaen, F. and Schachermayer, W. (1994) A general version of the fundamental theorem of asset pricing. Mathematische Annalen, 300:463–520.MathSciNetCrossRefMATHGoogle Scholar
  14. 545.
    Delbaen, F. and Schachermayer, W. (1998) The fundamental theorem of asset pricing for unbounded stochastic processes. Mathematische Annalen, 312:215–250.MathSciNetCrossRefMATHGoogle Scholar
  15. 563.
    Detemple, J. and Gottardi, P. (1998) Aggregation, efficiency and mutual fund separation in incomplete markets, Economic Theory, 11:443–455.MathSciNetCrossRefMATHGoogle Scholar
  16. 571.
    Diamond, P. (1967) The role of a stock market in a general equilibrium model with technological uncertainty. The American Economic Review, 57:759–776.Google Scholar
  17. 581.
    Dothan, M. (1990) Prices in Financial Markets. Oxford University Press.Google Scholar
  18. 593.
    Duffie, D. (1996) Dynamic Asset Pricing Theory. Princeton University Press.Google Scholar
  19. 597.
    Duffie, D. and Rahi, R. (1995) Financial market innovation and security design: an introduction. Journal of Economic Theory, 65:1–42.CrossRefMATHGoogle Scholar
  20. 598.
    Duffie, D. and Shafer, W. (1985) equilibrium in incomplete markets I. A basic model of generic existence. Journal of Mathematical Economics, 14:285–300.MathSciNetCrossRefMATHGoogle Scholar
  21. 611.
    Dybvig, P. and Ross, S. (1992) Arbitrage. In The New Palgrave Dictionary of Money and Finance, J. Eatwell, M. Milgate and P. Newman (eds.), Palgrave Macmillan, London.Google Scholar
  22. 683.
    Fernholz, R. (2002) Stochastic Portfolio Theory. Springer, New York.CrossRefMATHGoogle Scholar
  23. 721.
    Föllmer, H. and Schied, A. (2011) Stochastic Finance - An Introduction in Discrete Time. 3rd edition. Walter de Gruyter, Berlin.CrossRefMATHGoogle Scholar
  24. 724.
    Fontana, C. (2015) Weak and strong no-arbitrage conditions for continuous financial markets. International Journal of Theoretical and Applied Finance, 18:1550005.MathSciNetCrossRefMATHGoogle Scholar
  25. 746.
    Fuydenberg, D. and Tirole, J. (1991) Game Theory. MIT Press, Cambridge.Google Scholar
  26. 763.
    Geanakoplos, J. (1990) An introduction to general equilibrium with incomplete markets. Journal of Mathematical Economics, 19:1–38.MathSciNetCrossRefMATHGoogle Scholar
  27. 764.
    Geanakoplos, J. and Mas-Colell, A. (1989) Real indeterminancy with financial assets. Journal of Economic Theory, 47:22–38.MathSciNetCrossRefMATHGoogle Scholar
  28. 765.
    Geanakoplos, J. and Polemarchakis, H. (1986) Existence, regularity and constrained suboptimality of competitive allocations when the asset market is incomplete. In Essays in Honor of K. Arrow, W. Heller, R. Starr and D. Starrett (eds.), Cambridge University Press, 65–95.Google Scholar
  29. 800.
    Gollier, C. (2001) The Economics of Risk and Time. MIT Press, Cambridge.MATHGoogle Scholar
  30. 822.
    Green, J. and Jarrow, R. (1987) Spanning and completeness in markets with contingent claims. Journal of Economic Theory, 41:202–210.MathSciNetCrossRefMATHGoogle Scholar
  31. 874.
    Hahn, F. (1973) On the Notion of Equilibrium in Economics. Cambridge University Press.Google Scholar
  32. 904.
    Harrison, M. and Kreps, D. (1979) Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 28:183–91.MathSciNetMATHGoogle Scholar
  33. 905.
    Harrison, M. and Pliska, S. (1981) Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, 11:215–260.MathSciNetCrossRefMATHGoogle Scholar
  34. 906.
    Hart, O. (1975) On the optimality of equilibrium when the market structure is incomplete. Journal of Economic Theory, 11:418–443.MathSciNetCrossRefMATHGoogle Scholar
  35. 937.
    Hens, T. and Pilgrim, B. (2003) General Equilibrium Foundations of Finance: Structure of Incomplete Markets Models, Kluwer Academic Publishers.Google Scholar
  36. 971.
    Huang, C. and Litzenberger, R. (1988) Foundations for Financial Economics. North-Holland, Amsterdam.MATHGoogle Scholar
  37. 1018.
    Jarrow, R. (1986) The relationship between arbitrage and first order stochastic dominance. Journal of Finance, 41:159–172.MathSciNetCrossRefGoogle Scholar
  38. 1130.
    Krasa, S. (1987) Existence of competitive equilibrium for option markets. Journal of Economic Theory, 47:413–431.MathSciNetCrossRefMATHGoogle Scholar
  39. 1131.
    Krasa, S. and Werner, J. (1991) Equilibrium with options: existence and indeterminacy. Journal of Economic Theory, 54:305–320.MathSciNetCrossRefMATHGoogle Scholar
  40. 1182.
    Lengwiler, Y. (2004) Microfoundations of Financial Economics. Princeton University Press.Google Scholar
  41. 1191.
    LeRoy, S. and Werner, J. (2001) Principles of Financial Economics, Cambridge University Press.Google Scholar
  42. 1286.
    Magill, M. and Quinzii, M. (1998) Theory of Incomplete Markets. MIT Press, Cambridge.MATHGoogle Scholar
  43. 1287.
    Magill, M. and Shafer, W. (1985) Characterization of generically complete real asset structures. Journal of Mathematical Economics, 19:167–194.CrossRefMATHGoogle Scholar
  44. 1288.
    Magill, M. and Shafer, W. (1991) Incomplete markets. In Handbook of Mathematical Economics (Vol.4), W. Hildenbrand and H. Sonnenschein (eds.), Elsevier Science Publisher.Google Scholar
  45. 1310.
    Mas-Colell, A., Whinston, M. and Green, J. (1995) Microeconomic Theory. Oxford University Press.Google Scholar
  46. 1331.
    Merton, R. (1974) On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 29:449–470.Google Scholar
  47. 1344.
    Milne, F. (1979) Consumer preferences, linear demand functions and aggregation in competitive asset markets. Review of Economic Studies, 50:407–417.MathSciNetCrossRefMATHGoogle Scholar
  48. 1346.
    Milne, F. (1995) Finance Theory and Asset Pricing. Clarendon Press, Oxford.Google Scholar
  49. 1348.
    Modigliani, F. and Miller, M. (1958) The cost of capital, corporate finance, and the theory of investment. The American Economic Review, 48:261–297.Google Scholar
  50. 1349.
    Modigliani, F. and Miller, M. (1963) Corporate income taxes and the cost of capital: a correction. The American Economic Review, 53:433–443.Google Scholar
  51. 1363.
    Munk, C. (2013) Financial Asset Pricing Theory. Oxford University Press.Google Scholar
  52. 1365.
    Nachman, D. (1989) Spanning and completeness with options. Review of Financial Studies, 1:311–328.CrossRefGoogle Scholar
  53. 1420.
    Platen, E. and Heath, D. (2006) A Benchmark Approach to Quantitative Finance. Springer, Berlin/Heidelberg.CrossRefMATHGoogle Scholar
  54. 1424.
    Polemarchakis, H. and Ku, B. (1990) Options and equilibrium. Journal of Mathematical Economics, 19:107–112.MathSciNetCrossRefMATHGoogle Scholar
  55. 1437.
    Radner, R. (1972) Existence of equilibrium of plans, prices, and price expectations in a sequence of markets. Econometrica, 40:289–303.MathSciNetCrossRefMATHGoogle Scholar
  56. 1439.
    Radner, R. (1982) Equilibrium under uncertainty. In Handbook of Mathematical Economics (Vol.I), K. Arrow and M. Intriligator (eds.), Elsevier Science Publisher.Google Scholar
  57. 1464.
    Ross, S. (1976) The arbitrage pricing theory of capital asset pricing. Journal of Economic Theory, 13:341–360.MathSciNetCrossRefGoogle Scholar
  58. 1465.
    Ross, S. (1976) Options and efficiency. Quarterly Journal of Economics, 90:75–89.CrossRefMATHGoogle Scholar
  59. 1485.
    Rubinstein, M. (1974) An aggregation theorem for securities markets. Journal of Financial Economics, 1:225–244.CrossRefGoogle Scholar
  60. 1501.
    Schachermayer, W. (2010) The fundamental theorem of asset pricing. In Encyclopedia of Quantitative Finance, Vol. 2, R. Cont (ed.), Wiley, Chichester, 792–801.Google Scholar
  61. 1568.
    Stiglitz, J. (1982) The inefficiency of the stock market equilibrium. Review of Economic Studies, 49:241–261.MathSciNetCrossRefMATHGoogle Scholar
  62. 1599.
    Townsend, R. (1994) Risk and insurance in village India. Econometrica, 62:539–591.CrossRefGoogle Scholar
  63. 1653.
    Werner, J. (1985) Equilibrium in economies with incomplete financial markets. Journal of Economic Theory, 36:110–119.MathSciNetCrossRefMATHGoogle Scholar
  64. 1660.
    Willard, G. and Dybvig, P. (1999) Empty promises and arbitrage. Review of Financial Studies, 12:807–834.CrossRefGoogle Scholar
  65. 1661.
    Wilson, R. (1968) The theory of syndacates. Econometrica, 36:119–132.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd. 2017

Authors and Affiliations

  • Emilio Barucci
    • 1
  • Claudio Fontana
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris Diderot (Paris 7)ParisFrance

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