Publication: Good deals in markets with friction
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2013-06
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Taylor & Francis
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Abstract
This paper studies an optimization problem involving pay-offs of (perhaps dynamic) investment
strategies. The pay-off is the decision variable, the expected pay-off is maximized and its risk is
minimized. The pricing rule may incorporate transaction costs and the risk measure is continuous,
coherent and expectation bounded.We will prove the necessity of dealing with pricing rules such that
there exists an essentially bounded stochastic discount factor that must also be 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 choose pay-offs whose (risk, return) will be as close as desired
to (−1,1) or (−1,1). This pathological property still holds for vector risk measures (i.e. if we
minimize a vector-valued function whose components are risk measures). It is worth pointing out that,
essentially, bounded stochastic discount factors are not usual in the financial literature. In particular,
the most famous frictionless, complete and arbitrage-free pricing models imply the existence of good
deals for every continuous, coherent and expectation bounded (scalar or vector) measure of risk, and
the incorporation of transaction costs will not guarantee the solution of this caveat
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Keywords
Risk measures, Transaction costs, Portfolio optimization, Arbitrage relationship
Bibliographic citation
Quantitative Finance, vol. 13, no. 6, pp. 827-836