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Pricing derivatives of American and game type in incomplete markets

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  • Jan Kallsen
  • Christoph Kühn

Abstract

In this paper the neutral valuation approach is applied to American and game options in incomplete markets. Neutral prices occur if investors are utility maximizers and if derivative supply and demand are balanced. Game contingent claims are derivative contracts that can be terminated by both counterparties at any time before expiration. They generalize American options where this right is limited to the buyer of the claim. It turns out that as in the complete case, the price process of American and game contingent claims corresponds to a Snell envelope or to the value of a Dynkin game, respectively. On the technical level, an important role is played by $\sigma$ -sub- and $\sigma$ -supermartingales. We characterize these processes in terms of semimartingale characteristics. Copyright Springer-Verlag Berlin/Heidelberg 2004

Suggested Citation

  • Jan Kallsen & Christoph Kühn, 2004. "Pricing derivatives of American and game type in incomplete markets," Finance and Stochastics, Springer, vol. 8(2), pages 261-284, May.
    • Handle: RePEc:spr:finsto:v:8:y:2004:i:2:p:261-284
    DOI: 10.1007/s00780-003-0110-7
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    Citations

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    Cited by:

    1. Yan Dolinsky, 2020. "On Shortfall Risk Minimization for Game Options," Papers 2002.01528, arXiv.org.
    2. Hamadène, S. & Wang, H., 2009. "BSDEs with two RCLL reflecting obstacles driven by Brownian motion and Poisson measure and a related mixed zero-sum game," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2881-2912, September.
    3. Gapeev, Pavel V. & Li, Libo, 2022. "Perpetual American standard and lookback options with event risk and asymmetric information," LSE Research Online Documents on Economics 114940, London School of Economics and Political Science, LSE Library.
    4. Miryana Grigorova & Marie-Claire Quenez & Yuan Peng, 2023. "Non-linear non-zero-sum Dynkin games with Bermudan strategies," Papers 2311.01086, arXiv.org.
    5. Huang, Haishi, 2010. "Convertible Bonds: Default Risk and Uncertain Volatility," Bonn Econ Discussion Papers 09/2010, University of Bonn, Bonn Graduate School of Economics (BGSE).
    6. Hsuan-Ku Liu, 2013. "The pricing formula for cancellable European options," Papers 1304.5962, arXiv.org, revised Sep 2014.
    7. Ivan Guo & Marek Rutkowski, 2017. "Arbitrage-free pricing of multi-person game claims in discrete time," Finance and Stochastics, Springer, vol. 21(1), pages 111-155, January.
    8. Gapeev Pavel V. & Kühn Christoph, 2005. "Perpetual convertible bonds in jump-diffusion models," Statistics & Risk Modeling, De Gruyter, vol. 23(1/2005), pages 15-31, January.
    9. Kallsen Jan & Kühn Christoph, 2006. "On utility-based derivative pricing with and without intermediate trades," Statistics & Risk Modeling, De Gruyter, vol. 24(4/2006), pages 1-20, October.
    10. Gapeev, Pavel V., 2008. "The integral option in a model with jumps," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2623-2631, November.
    11. Said Hamadene & Jianfeng Zhang, 2008. "The Continuous Time Nonzero-sum Dynkin Game Problem and Application in Game Options," Papers 0810.5698, arXiv.org.
    12. Yuri Kifer, 2012. "Dynkin Games and Israeli Options," Papers 1209.1791, arXiv.org.
    13. Jan Kallsen & Johannes Muhle-Karbe, 2009. "Utility maximization in models with conditionally independent increments," Papers 0911.3608, arXiv.org.
    14. Zaevski, Tsvetelin S., 2020. "Discounted perpetual game put options," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    15. Baurdoux, Erik J. & Kyprianou, Andreas E., 2004. "Further calculations for Israeli options," LSE Research Online Documents on Economics 23916, London School of Economics and Political Science, LSE Library.
    16. Tumellano Sebehela, 2017. "Game Options," Annals of Financial Economics (AFE), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 1-17, September.
    17. Rubbenstroth, Bodo, 2019. "Game Options under Knightian Uncertainty in Discrete Time," Center for Mathematical Economics Working Papers 619, Center for Mathematical Economics, Bielefeld University.
    18. Mark Owen & Gordan Zitkovic, 2007. "Optimal Investment with an Unbounded Random Endowment and Utility-Based Pricing," Papers 0706.0478, arXiv.org, revised Sep 2007.
    19. Tianyang Nie & Edward Kim & Marek Rutkowski, 2018. "Arbitrage-Free Pricing of Game Options in Nonlinear Markets," Papers 1807.05448, arXiv.org.
    20. Huang, Haishi, 2010. "Convertible Bonds: Risks and Optimal Strategies," Bonn Econ Discussion Papers 07/2010, University of Bonn, Bonn Graduate School of Economics (BGSE).
    21. Johannes Gerer & Gregor Dorfleitner, 2016. "A Note On Utility Indifference Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(06), pages 1-17, September.
    22. Erhan Bayraktar & Song Yao, 2015. "On the Robust Dynkin Game," Papers 1506.09184, arXiv.org, revised Sep 2016.
    23. Mark P. Owen & Gordan Žitković, 2009. "Optimal Investment With An Unbounded Random Endowment And Utility‐Based Pricing," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 129-159, January.
    24. Klebert Kentia & Christoph Kuhn, 2017. "Nash equilibria for game contingent claims with utility-based hedging," Papers 1707.09351, arXiv.org, revised Sep 2018.
    25. Ivan Guo & Marek Rutkowski, 2014. "Arbitrage Pricing of Multi-person Game Contingent Claims," Papers 1405.2718, arXiv.org.

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