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Financial economics without probabilistic prior assumptions

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  • Frank Riedel

Abstract

The treatment of uncertainty in general equilibrium theory in the style of Arrow and Debreu does not require a prior probability on the state space. Finance models nevertheless treat payoffs as random variables, implicitly or explicitly using a known probability distribution. In the light of Knightian uncertainty, we might challenge such an assumption on the probabilistic sophistication of our market model. The present paper shows that one can still develop a sound model of arbitrage pricing under complete Knightian uncertainty as long as certain continuity conditions are met. The pricing functional given by an arbitrage-free market can be identified with a full support martingale measure (instead of equivalent martingale measure). We relate the no-arbitrage theory to economic equilibrium by establishing a variant of the Harrison–Kreps theorem on viability and no arbitrage. Finally, we consider (super) hedging of contingent claims and embed it in a classical infinite-dimensional linear programming problem. Copyright Springer-Verlag Italia 2015

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  • Frank Riedel, 2015. "Financial economics without probabilistic prior assumptions," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 38(1), pages 75-91, April.
    • Handle: RePEc:spr:decfin:v:38:y:2015:i:1:p:75-91
    DOI: 10.1007/s10203-014-0159-0
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    References listed on IDEAS

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    Cited by:

    1. Gianluca Cassese, 2021. "Complete and competitive financial markets in a complex world," Finance and Stochastics, Springer, vol. 25(4), pages 659-688, October.
    2. Ariel Neufeld & Julian Sester, 2023. "Neural networks can detect model-free static arbitrage strategies," Papers 2306.16422, arXiv.org, revised Aug 2024.
    3. Hölzermann, Julian, 2020. "Pricing Interest Rate Derivatives under Volatility Uncertainty," Center for Mathematical Economics Working Papers 633, Center for Mathematical Economics, Bielefeld University.
    4. Matteo Burzoni & Marco Frittelli & Marco Maggis, 2016. "Universal arbitrage aggregator in discrete-time markets under uncertainty," Finance and Stochastics, Springer, vol. 20(1), pages 1-50, January.
    5. Matteo Burzoni & Marco Frittelli & Marco Maggis, 2016. "Universal arbitrage aggregator in discrete-time markets under uncertainty," Finance and Stochastics, Springer, vol. 20(1), pages 1-50, January.
    6. Ariel Neufeld & Antonis Papapantoleon & Qikun Xiang, 2023. "Model-Free Bounds for Multi-Asset Options Using Option-Implied Information and Their Exact Computation," Management Science, INFORMS, vol. 69(4), pages 2051-2068, April.
    7. Zhaoxu Hou & Jan Obłój, 2018. "Robust pricing–hedging dualities in continuous time," Finance and Stochastics, Springer, vol. 22(3), pages 511-567, July.
    8. Matteo Burzoni & Marco Maggis, 2019. "Arbitrage-free modeling under Knightian Uncertainty," Papers 1909.04602, arXiv.org, revised Apr 2020.
    9. Lorenzo Bastianello & Alain Chateauneuf & Bernard Cornet, 2022. "Put-Call Parities, absence of arbitrage opportunities and non-linear pricing rules," Papers 2203.16292, arXiv.org.
    10. Gianluca Cassese, 2017. "Asset pricing in an imperfect world," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(3), pages 539-570, October.
    11. Jan Obłój & Johannes Wiesel, 2021. "A unified framework for robust modelling of financial markets in discrete time," Finance and Stochastics, Springer, vol. 25(3), pages 427-468, July.
    12. Matteo Burzoni & Marco Frittelli & Zhaoxu Hou & Marco Maggis & Jan Obłój, 2019. "Pointwise Arbitrage Pricing Theory in Discrete Time," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 1034-1057, August.
    13. Christian Bender & Sebastian Ferrando & Alfredo Gonzalez, 2021. "Model-Free Finance and Non-Lattice Integration," Papers 2105.10623, arXiv.org.
    14. Alessandro Doldi & Marco Frittelli, 2021. "Real-Valued Systemic Risk Measures," Mathematics, MDPI, vol. 9(9), pages 1-24, April.
    15. Matteo Burzoni & Frank Riedel & H. Mete Soner, 2021. "Viability and Arbitrage Under Knightian Uncertainty," Econometrica, Econometric Society, vol. 89(3), pages 1207-1234, May.
    16. Matteo Burzoni & Mario Sikic, 2018. "Robust martingale selection problem and its connections to the no-arbitrage theory," Papers 1801.03574, arXiv.org, revised Nov 2018.
    17. Ariel Neufeld & Antonis Papapantoleon & Qikun Xiang, 2020. "Model-free bounds for multi-asset options using option-implied information and their exact computation," Papers 2006.14288, arXiv.org, revised Jan 2022.
    18. Tolulope Fadina & Thorsten Schmidt, 2019. "Default Ambiguity," Risks, MDPI, vol. 7(2), pages 1-17, June.
    19. Alessandro Doldi & Marco Frittelli & Emanuela Rosazza Gianin, 2024. "On entropy martingale optimal transport theory," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 47(1), pages 1-42, June.
    20. Matteo Burzoni & Marco Frittelli & Zhaoxu Hou & Marco Maggis & Jan Ob{l}'oj, 2016. "Pointwise Arbitrage Pricing Theory in Discrete Time," Papers 1612.07618, arXiv.org, revised Feb 2018.
    21. Julian Holzermann, 2020. "Pricing Interest Rate Derivatives under Volatility Uncertainty," Papers 2003.04606, arXiv.org, revised Nov 2021.
    22. Huy N. Chau, 2020. "On robust fundamental theorems of asset pricing in discrete time," Papers 2007.02553, arXiv.org, revised Apr 2024.
    23. Bruno Bouchard & Marcel Nutz, 2016. "Consistent price systems under model uncertainty," Finance and Stochastics, Springer, vol. 20(1), pages 83-98, January.
    24. Bruno Bouchard & Marcel Nutz, 2016. "Consistent price systems under model uncertainty," Finance and Stochastics, Springer, vol. 20(1), pages 83-98, January.
    25. Tolulope Fadina & Thorsten Schmidt, 2018. "Ambiguity in defaultable term structure models," Papers 1801.10498, arXiv.org, revised Apr 2018.

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    More about this item

    Keywords

    Probability-free finance; Fundamental theorem of asset pricing; Full support martingale measure; Superhedging; Infinite-dimensional linear programming; G12 ; D53;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets

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