The risk-neutral non-additive probability with market frictions

Alain Chateauneuf and Bernard Cornet
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Bernard Cornet: KU - University of Kansas [Lawrence], CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École nationale des ponts et chaussées - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement

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Abstract: The fundamental theory of asset pricing has been developed under the two main assumptions that markets are frictionless and have no arbitrage opportunities. In this case the market enforces that replicable assets are valued by a linear function of their payoffs, or as the discounted expectation with respect to the so-called risk-neutral probability. Important evidence of the presence of frictions in financial markets has led to study market pricing rules in such a framework. Recently, Cerreia-Vioglio et al. (J Econ Theory 157:730–762, 2015) have extended the Fundamental Theorem of Finance by showing that, with markets frictions, requiring the put–call parity to hold, together with the mild assumption of translation invariance, is equivalent to the market pricing rule being represented as a discounted Choquet expectation with respect to a risk-neutral nonadditive probability. This paper continues this study by characterizing important properties of the (unique) risk-neutral nonadditive probability vf associated with a Choquet pricing rule f, when it is not assumed to be subadditive. First, we show that the observed violation of the call–put parity, a condition considered by Chateauneuf et al. (Math Financ 6:323–330, 1996) similar to the put–call parity in Cerreia-Vioglio et al. (2015), is consistent with the existence of bid-ask spreads. Second, the balancedness of vf—or equivalently the non-vacuity of its core—is characterized by an arbitrage-free condition that eliminates all the arbitrage opportunities that can be obtained by splitting payoffs in parts; moreover the (nonempty) core of vf consists of additive probabilities below vf whose associated (standard) expectations are all below the Choquet pricing rule f. Third, by strengthening again the previous arbitrage-free condition, we show the existence of a strictly positive risk-neutral probability below vf, which allows to recover the standard formulation of the Fundamental Theorem of Finance for frictionless markets.

Keywords: Market frictions; Risk-neutral nonadditive probability; Absence of arbitrage opportunities; Choquet pricing; Put–call parity (search for similar items in EconPapers)
Date: 2022
Note: View the original document on HAL open archive server: https://hal.science/hal-03722945v1
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Published in Economic Theory Bulletin, 2022, 10 (1), pp.13-25. ⟨10.1007/s40505-022-00216-4⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:cesptp:hal-03722945

DOI: 10.1007/s40505-022-00216-4

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