**WELL-POSEDNESS
OF PORTFOLIO CHOICE PROBLEMS WITH RISK MEASURES AND FRICTIONS**

Alejandro Balbás, Beatriz Balbás, Raquel Balbás

**Abstract**

This paper
studies a portfolio choice problem such that the pricing rule may in-corporate transaction
costs and the risk measure is coherent and expectation bounded. We will prove
the necessity of dealing with pricing rules such that there exists
an essentially bounded stochastic discount factor, which must be also bounded
from below by a strictly positive value. Otherwise good deals will be available
to traders, i.e., depending on the selected risk measure, investors can build
portfolios whose (risk, return) will be as close as desired to (-infinity,
infinity) or (0, infinity). This pathologic property still holds for vector
risk measures (i.e., if we minimize a vector valued function whose components
are risk measures). It is worthwhile to point out that essentially bounded
stochastic discount factors are not usual in financial literature. In particular,
the most famous frictionless, complete and arbitrage free pricing models imply
the existence of good deals for every coherent and expectation bounded (scalar
or vector) measure of risk, and the incorporation of transaction costs will not
guarantee the solution of this caveat.

Lecture Notes in Management Science (2011) Vol. 3:
7-12

3rd International Conference on Applied Operational
Research, Proceedings

© Tadbir Operational Research Group Ltd. All rights
reserved.

www.tadbir.ca

ISSN 2008-0050 (Print)

ISSN 1927-0097 (Online)

**ARTICLE OUTLINE**

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**Extended Abstract **

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**References**