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Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences

30 Pages Posted: 7 Feb 2011 Last revised: 14 Mar 2013

Patrick Cheridito

ETH Zurich; Swiss Finance Institute

Ulrich Horst

Humboldt University of Berlin

Michael Kupper

Humboldt University of Berlin - Department of Mathematics

Traian A. Pirvu

McMaster University; University of British Columbia (UBC)

Date Written: July 18, 2012

Abstract

We propose a general discrete-time framework for deriving equilibrium prices of financial securities. It allows for heterogeneous agents, unspanned random endowments and convex trading constraints. We give a dual characterization of equilibria and provide general results on their existence and uniqueness. In the special case where all agents have preferences of the same type, and in equilibrium, all random endowments are replicable by trading in the fi nancial market, we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by nitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic di fference equations, which in the continuous-time limit becomes a multidimensional backward stochastic di fferential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it.

Keywords: Competitive equilibrium, incomplete markets, heterogenous agents, trading constraints, one-fund theorem

JEL Classification: D52, D53, G12

Suggested Citation

Cheridito, Patrick and Horst, Ulrich and Kupper, Michael and Pirvu, Traian A., Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences (July 18, 2012). Available at SSRN: https://ssrn.com/abstract=1755610 or http://dx.doi.org/10.2139/ssrn.1755610

Patrick Cheridito (Contact Author)

ETH Zurich ( email )

Department of Mathematics
8092 Zurich
Switzerland

Swiss Finance Institute ( email )

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

Ulrich Horst

Humboldt University of Berlin ( email )

Unter den Linden 6
Berlin, Berlin 10099
Germany

Michael Kupper

Humboldt University of Berlin - Department of Mathematics ( email )

Unter den Linden
Berlin, D-10099
Germany

Traian Adrian Pirvu

McMaster University ( email )

1280 Main Street West
Hamilton, Ontario L8S 4M4
Canada

University of British Columbia (UBC) ( email )

2329 West Mall
Vancouver, British Columbia BC V6T 1Z4
Canada

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