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Concave Distortion Semigroups

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  • Alexander Cherny
  • Damir Filipovi'c

Abstract

The problem behind this paper is the proper measurement of the degree of quality/acceptability/distance to arbitrage of trades. We are narrowing the class of coherent acceptability indices introduced by Cherny and Madan (2007) by imposing an additional mathematical property. For this, we introduce the notion of a concave distortion semigroup as a family $(\Psi_t)_{t\ge0}$ of concave increasing functions $[0,1]\to[0,1]$ satisfying the semigroup property $$ \Psi_s\circ\Psi_t=\Psi_{s+t},\quad s,t\ge0. $$ The goal of the paper is the investigation of these semigroups with regard to the following aspects: representation of distortion semigroups; properties of distortion semigroups desirable from the economical or mathematical perspective; determining which concave distortions belong to some distortion semigroup.

Suggested Citation

  • Alexander Cherny & Damir Filipovi'c, 2011. "Concave Distortion Semigroups," Papers 1104.0508, arXiv.org.
    • Handle: RePEc:arx:papers:1104.0508
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    References listed on IDEAS

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    1. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
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    3. Robert B. Israel & Jeffrey S. Rosenthal & Jason Z. Wei, 2001. "Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 245-265, April.
    4. 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.
    5. A. Cherny, 2006. "Weighted V@R and its Properties," Finance and Stochastics, Springer, vol. 10(3), pages 367-393, September.
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    Cited by:

    1. Michael R. Tehranchi, 2020. "A Black–Scholes inequality: applications and generalisations," Finance and Stochastics, Springer, vol. 24(1), pages 1-38, January.

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