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Financial economics without probabilistic prior assumptions

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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.

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Notes

  1. See the related work by Jörg Vorbrink Vorbrink (2010) and Beissner (2012) on the foundations of Finance under uncertain volatility.

  2. In discrete time, the fundamental theorem was first proved by Harrison and Kreps (1979), Kreps (1981) and Yan (1980). It was improved by Dalang et al. (1990). The definitive treatment in continuous time was obtained by Delbaen and Schachermayer (1994, 1998).

  3. Note added in proof: This paper was first widely circulated in 2011 (arXiv:1107.1078). In the meantime, the topic attracted more interest and several other papers on the fundamental theorem under Knightian uncertainty were written afterward, including Acciaio et al. (to appear) and Bouchard and Nutz (to appear). Marcel Nutz and I discussed the topic in detail when he visited Bielefeld in 2011 and I visited Columbia in 2013.

  4. Including the author of this note and Walter Willinger.

  5. A set \(A \subset \mathbb N\) has nonzero density if \(\limsup \frac{|A\cap \{1,\ldots ,n\}|}{n}>0\).

  6. The support of a probability measure \(Q\) is the smallest closed set \(F\subset \Omega \) with \(Q(F)=1.\)

  7. One might ask, of course, why one cannot work with just measurable payoffs. The answer is as follows: There would be no hope to develop a reasonable counterpart of the fundamental theorem; the mathematical reason is that there are no strictly positive functionals on the space of all bounded measurable functions. More economically speaking, when you have a continuum of states, it is impossible to assign a positive price to all Arrow securities that pay off \(1\) unit of the consumption in exactly one state, and nothing in all other states.

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  1. Center for Mathematical Economics, Bielefeld University, Bielefeld, Germany

    Frank Riedel

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  1. Frank Riedel
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Correspondence to Frank Riedel.

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First Version: July 6 2011 (arXiv:1107.1078v1). This paper is dedicated to the students of the class “Advanced Topics in Finance” at Bielefeld University in Summer 2011. Comments by an excellent referee of this journal are gratefully acknowledged. I also wish to thank Hans Föllmer, Bernard Cornet, Jean-Marc Bonnisseau, Filipe da Martins-Rocha, Francesco Russo, and seminar participants in Bielefeld, Dauphine, and Sorbonne as well of the NBER Workshop on General Equilibrium Theory in Iowa 2011 for comments and discussions. All errors are mine. Financial Support through the German Research Foundation, International Graduate College “Stochastics and Real World Models,” Research Training Group EBIM, “Economic Behavior and Interaction Models,” and Grant Ri-1128-4-1 is gratefully acknowledged. The paper was revised while I was visiting Princeton University. I thank the department for Operations Research and Financial Engineering, and especially Patrick Cheridito, for the hospitality and excellent working conditions.

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Riedel, F. Financial economics without probabilistic prior assumptions. Decisions Econ Finan 38, 75–91 (2015). https://doi.org/10.1007/s10203-014-0159-0

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