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Mathematics and Financial Economics

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01-06-2015

A note on utility-based pricing in models with transaction costs

Authors: Mark H. A. Davis, Daisuke Yoshikawa

Published in: Mathematics and Financial Economics | Issue 3/2015

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Abstract

This paper considers the utility-based and indifference pricing in a market with transaction costs. The utility maximization problem, including contingent claims in the market with transaction costs, has been widely researched. In this paper, closely following the results of Bouchard (Financ Stoch 6:495–516, 2002), we consider the market equilibrium of contingent claims. This is done by specifying the utility function as exponential utility and, thus, determining equilibrium in the market with transaction costs. Unlike Davis and Yoshikawa (Math Finan Econ, 2015), we use the strong assumption to deduce the equilibrium at which trade does not occur (zero trade equilibrium). It implicitly shows that transaction costs may generate a non-zero trade equilibrium under a weaker assumption.

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By introducing ${\mathcal {X}}_U(x)$, Bouchard proves the existence of a solution under looser conditions; that is, he addresses the problem $\sup _{X^{L} \in {\mathcal {X}}_U(x)} {\mathbb {E}}U[l(X^{x,L} - B q)]$. The definition of ${\mathcal {X}}_U(x)$ is given as follows: for $X \in L^0\big ({\mathcal {F}}_T,{\mathbb {R}}^d\big )$, which is a member of ${\mathcal {X}}_U(x)$, there exists a sequence $\big (X_k\big )_k \in {\mathcal {X}}(x)$ such that

$$\begin{aligned} X_k \rightarrow X \ P-a.s. \text{ and } {\mathbb {E}}\bigg [U\big (l(X_k - B)\big )\bigg ] \rightarrow {\mathbb {E}}\bigg [U\big (l(X - B)\big )\bigg ],\ \text{ as } k \rightarrow \infty . \end{aligned}$$

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Title: A note on utility-based pricing in models with transaction costs
Authors: Mark H. A. Davis
Daisuke Yoshikawa
Publication date: 01-06-2015
Publisher: Springer Berlin Heidelberg
Published in: Mathematics and Financial Economics / Issue 3/2015
Print ISSN: 1862-9679
Electronic ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-015-0143-7