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Mathematics and Financial Economics

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01.06.2013

Efficient portfolios in financial markets with proportional transaction costs

verfasst von: Luciano Campi, Elyès Jouini, Vincent Porte

Erschienen in: Mathematics and Financial Economics | Ausgabe 3/2013

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Abstract

In this article, we characterize efficient portfolios, i.e. portfolios which are optimal for at least one rational agent, in a very general multi-currency financial market model with proportional transaction costs. In our setting, transaction costs may be random, time-dependent, have jumps and the preferences of the agents are modeled by multivariate expected utility functions. We provide a complete characterization of efficient portfolios, generalizing earlier results of Dybvig (Rev Financ Stud 1:67–88, 1988) and Jouini and Kallal (J Econ Theory 66: 178–197, 1995). We basically show that a portfolio is efficient if and only if it is cyclically anticomonotonic with respect to at least one consistent price system that prices it. Finally, we introduce the notion of utility price of a given contingent claim as the minimal amount of a given initial portfolio allowing any agent to reach the claim by trading, and give a dual representation of it as the largest proportion of the market price necessary for all agents to reach the same expected utility level.

Vorheriger Artikel Indifference pricing for CRRA utilities

Nächster Artikel Optimal consumption and investment strategies with partial and private information in a multi-asset setting

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i.e., such a space supports a uniform random variable \(U\) on \((0,1)\).

Since \(Y_{0}(\omega )\in \partial U(X_{0}(\omega ))\) for all \(\omega \in A\), we have \(\partial U(X_{0}(\omega ))\ne \emptyset \). Thus, the fact that \( {\text{ dom}}\,(\partial U)\subset {\text{ dom}}\,(U)\) implies that \(X_{0}(\omega )\in {\text{ dom}}\,(\partial U)\subset {\text{ dom}}\,(U)\), so that \(U(X_{0}(\omega ))>-\infty \).

Indeed, assume that \(\pi (X_{0},x_{0})<1\) for some \(X\in \mathcal B ^{U}(X_{0})\). Since \(X_{0}\) is efficient for some \(x_{0}\) and \(U\in \mathcal U \), we have that \(X_{0}\) is a maximizer for an agent having utility function \(U\) and an initial portfolio \(x_{0}\). Moreover, by definition of \( \pi (X_{0},x_{0})\), the initial portfolio \(\pi (X_{0},x_{0})x_{0}<x_{0}\) leads to \(X\) as well. In other terms, the initial wealth \(x_{0}\) may lead to the terminal portfolio \(X_{0}+(1-\pi (X_{0},x_{0}))x_{0}>X\). Since \(U\) is strictly increasing, this contradicts the fact that \(X_{0}\) is a maximizer. Thus, when \(X_{0}\) is efficient its utility price.

A construction of such a random vector \(X^{\prime }\) goes as follows: take a random variable \(U\) with uniform distribution on \((0,1)\) and set \(X^{\prime }=\tilde{X}_{0}\) on \(\Omega _{0}\) and, for \(i=1,\ldots ,n,\,X_{\mid \Omega _{i}}^{\prime }=F_{i+1}^{-1}(U_{i})\) where \(U_{i}\) is the restriction of \(U\) on \(\Omega _{i}\) and \(F_{i+1}\) is the c.d.f. of the restriction of \(\tilde{X} _{0}\) on \(\Omega _{i+1}\). It is easy to verify that \(X^{\prime }\) satisfies the properties listed above. Notice that to perform such a construction we need the assumption that \(X_{0}\) has a continuous c.d.f..

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Titel: Efficient portfolios in financial markets with proportional transaction costs
verfasst von: Luciano Campi
Elyès Jouini
Vincent Porte
Publikationsdatum: 01.06.2013
Verlag: Springer-Verlag
Erschienen in: Mathematics and Financial Economics / Ausgabe 3/2013
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-013-0099-4

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