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Multi-currency reserving for coherent risk measures

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  • Saul Jacka
  • Seb Armstrong
  • Abdel Berkaoui

Abstract

We examine the problem of dynamic reserving for risk in multiple currencies under a general coherent risk measure. The reserver requires to hedge risk in a time-consistent manner by trading in baskets of currencies. We show that reserving portfolios in multiple currencies $\mathbf{V}$ are time-consistent when (and only when) a generalisation of Delbaen's m-stability condition \cite{D06}, termed optional $\V$-m-stability, holds. We prove a version of the Fundamental Theorem of Asset Pricing in this context. We show that this problem is equivalent to dynamic trading across baskets of currencies (rather than just pairwise trades) in a market with proportional transaction costs and with a frictionless final period.

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  • Saul Jacka & Seb Armstrong & Abdel Berkaoui, 2017. "Multi-currency reserving for coherent risk measures," Papers 1712.01319, arXiv.org, revised Dec 2017.
    • Handle: RePEc:arx:papers:1712.01319
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    References listed on IDEAS

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    1. Tomasz R. Bielecki & Igor Cialenco & Marcin Pitera, 2016. "A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective," Papers 1603.09030, arXiv.org, revised Jan 2017.
    2. Saul Jacka & Seb Armstrong & Abdelkarem Berkaoui, 2017. "On representing and hedging claims for coherent risk measures," Papers 1703.03638, arXiv.org, revised Feb 2018.
    3. Tomasz R. Bielecki & Igor Cialenco & Rodrigo Rodriguez, 2015. "No-Arbitrage Pricing For Dividend-Paying Securities In Discrete-Time Markets With Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 25(4), pages 673-701, October.
    4. Jakša Cvitanić & Ioannis Karatzas, 1996. "Hedging And Portfolio Optimization Under Transaction Costs: A Martingale Approach12," Mathematical Finance, Wiley Blackwell, vol. 6(2), pages 133-165, April.
    5. Tomasz R. Bielecki & Igor Cialenco & Marcin Pitera, 2014. "A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time," Papers 1409.7028, arXiv.org, revised Sep 2017.
    6. 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.
    7. Detlefsen, Kai & Scandolo, Giacomo, 2005. "Conditional and dynamic convex risk measures," SFB 649 Discussion Papers 2005-006, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    8. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
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    10. Saul Jacka & Abdelkarem Berkaoui & Jon Warren, 2008. "No arbitrage and closure results for trading cones with transaction costs," Finance and Stochastics, Springer, vol. 12(4), pages 583-600, October.
    11. Saul Jacka & Abdelkarem Berkaoui & Jon Warren, 2006. "No-arbitrage and closure results for trading cones with transaction costs," Papers math/0602178, arXiv.org, revised Apr 2008.
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