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The Minimal Entropy Martingale Measure in a Market of Traded Financial and Actuarial Risks

Author

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  • Jan Dhaene

    (KU Leuven, Leuven, Belgium; University of the Freestate, Bloemfontein, South Africa; and Duisenberg school of finance, Amsterdam, the Netherlands)

  • Ben Stassen

    (KU Leuven, Leuven, Belgium)

  • Pierre Devolder

    (UC de Louvain, Louvain-La-Neuve, Belgium)

  • Michel Vellekoop

    (University of Amsterdam, Amsterdam, the Netherlands)

Abstract

In arbitrage-free but incomplete markets, the equivalent martingale measure Q for pricing traded assets is not uniquely determined. A possible approach when it comes to choosing a particular pricing measure is to consider the one that is ‘closest’to the physical probability measure P, where closeness is measured in terms of relative entropy. In this paper, we determine the minimal entropy martingale measure in a market where securities are traded with payoffs depending on two types of risks, which we will call financial and actuarial risks, respectively. In case only purely financial and purely actuarial securities are traded, we prove that financial and actuarial risks are independent under the physical measure if and only if these risks are independent under the entropy measure. Moreover, in such a market the entropy measure of the combined financial-actuarial world is the product measure of the entropy measures of the financial and the actuarial subworlds, respectively.

Suggested Citation

  • Jan Dhaene & Ben Stassen & Pierre Devolder & Michel Vellekoop, 2014. "The Minimal Entropy Martingale Measure in a Market of Traded Financial and Actuarial Risks," Tinbergen Institute Discussion Papers 14-104/IV/78, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20140104
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    References listed on IDEAS

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    1. Dhaene, Jan & Kukush, Alexander & Luciano, Elisa & Schoutens, Wim & Stassen, Ben, 2013. "On the (in-)dependence between financial and actuarial risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 522-531.
    2. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52, January.
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    2. Yuan Hu & Abootaleb Shirvani & W. Brent Lindquist & Frank J. Fabozzi & Svetlozar T. Rachev, 2020. "Option Pricing Incorporating Factor Dynamics in Complete Markets," JRFM, MDPI, vol. 13(12), pages 1-33, December.
    3. Yuan Hu & Abootaleb Shirvani & W. Brent Lindquist & Frank J. Fabozzi & Svetlozar T. Rachev, 2020. "Option Pricing Incorporating Factor Dynamics in Complete Markets," Papers 2011.08343, arXiv.org.
    4. Arık, Ayşe & Uğur, Ömür & Kleinow, Torsten, 2023. "The impact of simultaneous shocks to financial markets and mortality on pension buy-out prices," ASTIN Bulletin, Cambridge University Press, vol. 53(2), pages 392-417, May.
    5. Dhaene, Jan & Stassen, Ben & Barigou, Karim & Linders, Daniël & Chen, Ze, 2017. "Fair valuation of insurance liabilities: Merging actuarial judgement and market-consistency," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 14-27.
    6. José L. Vilar-Zanón & Olivia Peraita-Ezcurra, 2019. "A linear goal programming method to recover risk neutral probabilities from options prices by maximum entropy," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 259-276, June.

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