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Equilibrium pricing in incomplete markets under translation invariant preferences

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  • Cheridito, Patrick
  • Horst, Ulrich
  • Kupper, Michael
  • Pirvu, Traian A.

Abstract

We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by trading in the financial 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 finitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multi-dimensional backward stochastic differential 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. As an example we simulate option prices in the presence of stochastic volatility, demand pressure and short-selling constraints.

Suggested Citation

  • Cheridito, Patrick & Horst, Ulrich & Kupper, Michael & Pirvu, Traian A., 2011. "Equilibrium pricing in incomplete markets under translation invariant preferences," SFB 649 Discussion Papers 2011-083, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
  • Handle: RePEc:zbw:sfb649:sfb649dp2011-083
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    3. Traian A. Pirvu & Huayue Zhang, 2012. "A Multi Period Equilibrium Pricing Model," Papers 1205.6193, arXiv.org.
    4. Wayne King Ming Chan, 2015. "RAROC-Based Contingent Claim Valuation," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 21, July-Dece.
    5. Myšičková, Alena & Song, Song & Majer, Piotr & Mohr, Peter N. C. & Heekeren, Hauke R. & Härdle, Wolfgang Karl, 2011. "Risk patterns and correlated brain activities: Multidimensional statistical analysis of fMRI data with application to risk patterns," SFB 649 Discussion Papers 2011-085, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
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    9. Ulrich Horst & Michael Kupper & Andrea Macrina & Christoph Mainberger, 2013. "Continuous equilibrium in affine and information-based capital asset pricing models," Annals of Finance, Springer, vol. 9(4), pages 725-755, November.

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    Keywords

    competitive equilibrium; incomplete markets; heterogenous agents; trading constraints; backward stochastic difference equations;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets
    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets

    Statistics

    Access and download statistics

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