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A continuous selection for optimal portfolios under convex risk measures does not always exist

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  • Michel Baes
  • Cosimo Munari

Abstract

One of the crucial problems in mathematical finance is to mitigate the risk of a financial position by setting up hedging positions of eligible financial securities. This leads to focusing on set-valued maps associating to any financial position the set of those eligible payoffs that reduce the risk of the position to a target acceptable level at the lowest possible cost. Among other properties of such maps, the ability to ensure lower semicontinuity and continuous selections is key from an operational perspective. It is known that lower semicontinuity generally fails in an infinite-dimensional setting. In this note we show that neither lower semicontinuity nor, more surprisingly, the existence of continuous selections can be a priori guaranteed even in a finite-dimensional setting. In particular, this failure is possible under arbitrage-free markets and convex risk measures.

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  • Michel Baes & Cosimo Munari, 2017. "A continuous selection for optimal portfolios under convex risk measures does not always exist," Papers 1711.00370, arXiv.org.
    • Handle: RePEc:arx:papers:1711.00370
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    References listed on IDEAS

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    1. J. P. Crouzeix, 1980. "Conditions for Convexity of Quasiconvex Functions," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 120-125, February.
    2. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    3. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2014. "Beyond cash-additive risk measures: when changing the numéraire fails," Finance and Stochastics, Springer, vol. 18(1), pages 145-173, January.
    4. Artzner, Philippe & Delbaen, Freddy & Koch-Medina, Pablo, 2009. "Risk Measures and Efficient use of Capital 1," ASTIN Bulletin, Cambridge University Press, vol. 39(1), pages 101-116, May.
    5. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2012. "Beyond cash-additive risk measures: when changing the num\'{e}raire fails," Papers 1206.0478, arXiv.org, revised Feb 2014.
    6. Michel Baes & Pablo Koch-Medina & Cosimo Munari, 2017. "Existence, uniqueness and stability of optimal portfolios of eligible assets," Papers 1702.01936, arXiv.org, revised Dec 2017.
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