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.
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We thank David Cass (in memoriam), Ehud Lehrer, Erio Castagnoli, Fabio Maccheroni, Fulvio Ortu, Jean-Marc Tallon, Monique Florenzano, Rodrigo Novinski, and specially an anonymous referee for very useful suggestions and comments. We are also very grateful to Bernard Cornert by detected an error in an earlier version of this paper. Chateauneuf thanks IMPA for the generous financial support from the Franco-Brazilian Scientific Cooperation. Faro gratefully acknowledges the financial support from CNPq-Brazil, the financial support from Franco-Brazilian Scientific Cooperation and CERMSEM at the University of Paris I for their hospitality.
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Araujo, A., Chateauneuf, A. & Faro, J.H. Pricing rules and Arrow–Debreu ambiguous valuation. Econ Theory 49, 1–35 (2012). https://doi.org/10.1007/s00199-011-0660-4
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DOI: https://doi.org/10.1007/s00199-011-0660-4
Keywords
- Pricing rule
- Frictionless incomplete market
- Ambiguity
- State price
- Capacity, Lehrer integral
- Choquet integral