Consistent price systems under model uncertainty
type: Item
instance of: scholarly article
title: Consistent price systems under model uncertainty
zbMATH Open document ID: 1369.91200
author: Bruno Bouchard
author: Marcel Nutz
published in: Finance and Stochastics
publication date: 2016-03-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1408.5510
review text: The authors give a version of the fundamental theorem of asset pricing for discrete-time models with transaction costs and uncertainty. The investment opportunities in the model are described by a set-valued stochastic process. For each trading day there is given a solvency cone \(K_t\) -- a random set of positions that can be turned into nonnegative ones by immediate exchange (accounting for transaction costs). The \(K_t\) is a convex cone generated by a finite number of vectors. The set \(K^\ast_t\backslash\{0\}\) is interpreted as the set of possible frictionless prices for assets. For each moment, there is given a nonempty convex set of probability measures \(\mathcal{P}_t\) that represents the model uncertainty. The authors define two properties of the model. The first one, \(\text{NA}_2(\mathcal{P})\), is an augmentation of the no-arbitrage condition. It states that \(\zeta\in K_{t+1}\) implies \(\zeta\in K_t\) (\(\mathcal{P}\)-quasi surely). The second one, PCE (price system is extendable), states that for each measurable random variable \(Y\) taking values in \(\operatorname{int} K^\ast_t\) there exists a probability measure \(Q\) and a stochastic process \((Z_s)_{s=t,\dots,T}\) starting as \(Y\), such that \(Z_s\in K^\ast_s\) and \(Z_s\) is a \(Q\)-martingale. The main result of the article states that \(\text{NA}_2(\mathcal{P})\) and PCE are equivalent.
reviewed by: Paweł Kliber
Mathematics Subject Classification ID: 91G99
Mathematics Subject Classification ID: 91B25
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 6560417
zbMATH Keywords: fundamental theorem of asset pricing
zbMATH Keywords: transaction costs
zbMATH Keywords: model uncertainty
zbMATH Keywords: solvency cone
zbMATH Keywords: set-valued stochastic processes
zbMATH Keywords: random sets
MaRDI profile type: Publication
OpenAlex ID: W1597670103
arXiv ID: 1408.5510
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DOI: 10.1007/S00780-015-0286-7
description: scientific article; zbMATH DE number 6560417