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Economic Theory

, Volume 49, Issue 1, pp 1–35 | Cite as

Pricing rules and Arrow–Debreu ambiguous valuation

  • Aloisio Araujo
  • Alain Chateauneuf
  • José Heleno Faro
Research Article

Abstract

This paper considers pricing rules of single-period securities markets with finitely many states. Our main result characterizes those pricing rules C that are super-replication prices of a frictionless and arbitrage-free incomplete asset structure with a bond. This characterization relies on the equivalence between the sets of frictionless securities and securities priced by C. The former captures securities without bid-ask spreads, while the second captures the class of securities where, if some of its delivers is replaced by a higher payoff, then the resulting security is characterized by a higher value priced by C. We also analyze the special case of pricing rules associated with securities markets admitting a structure of basic assets paying one in some event and nothing otherwise. In this case, we show that the pricing rule can be characterized in terms of capacities. This Arrow–Debreu ambiguous state price can be viewed as a generalization for incomplete markets of Arrow–Debreu state price valuation. Also, some interesting cases are given by pricing rules determined by an integral w.r.t. a risk-neutral capacity. For instance, incomplete markets of Arrow securities and a bond are revealed by a Choquet integral w.r.t. a special risk-neutral capacity.

Keywords

Pricing rule Frictionless incomplete market Ambiguity State price Capacity, Lehrer integral Choquet integral 

JEL Classification

D52 D53 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Aloisio Araujo
    • 1
    • 2
  • Alain Chateauneuf
    • 3
  • José Heleno Faro
    • 4
  1. 1.IMPARio de JaneiroBrazil
  2. 2.EPGE/Getulio Vargas FoundationRio de JaneiroBrazil
  3. 3.Paris School of Economics, CES, U. de Paris IParis Cedex 13France
  4. 4.Insper Institute of Education and ResearchSão PauloBrazil

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