Preprint A676/2010
Pricing rules and Arrow Debreu ambiguous valuation

José Heleno Faro | Araujo, Aloisio | Chateauneuf, Alain

**Keywords: **
Pricing rule; frictionless incomplete market; ambiguity; state price; bets; capacity | Lehrer integral; Choquet integral.

This paper considers pricing rules of single-period securities markets with …nitely many states and without arbitrage opportunities. Our main result characterize those pricing rules C that are super-replication prices of a frictionless incomplete asset structure. This characterization relies on the equivalence between the sets of frictionless securities and undominated 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 payo?, then the resulting security is characterized by a higher value priced by C.
We also analyze the special case of pricing rules revealing securities markets admitting a structure of basic assets paying one in some event and nothing otherwise. In this case we show that any security can be priced against a capacity. This risk-neutral capacity, or Arrow-Debreu ambiguous state price, can be viewed as a generalization for incomplete markets of Arrow Debreu price valuation, and the corresponding pricing rule is determined by an integral w.r.t. a subadditive capacity. For instance, a special class of Choquet integral is related to frictionless incomplete markets of Arrow securities and a riskless asset.