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Pricing early exercise contracts in incomplete markets

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  • A. Oberman
  • T. Zariphopoulou

Abstract

We present a utility-based methodology for the valuation of early exercise contracts in incomplete markets. Incompleteness stems from nontraded assets on which the contracts are written. This methodology takes into account the individual’s attitude towards risk and yields nonlinear pricing rules. The early exercise indifference prices solve a quasilinear variational inequality with an obstacle term. They are also shown to satisfy an optimal stopping problem with criterion given by their European indifference price counterpart. A class of numerical schemes are developed for the variational inequalities and a general approach for solving numerically nonlinear equations arising in incomplete markets is discussed. Copyright Springer-Verlag Berlin/Heidelberg 2003

Suggested Citation

  • A. Oberman & T. Zariphopoulou, 2003. "Pricing early exercise contracts in incomplete markets," Computational Management Science, Springer, vol. 1(1), pages 75-107, December.
    • Handle: RePEc:spr:comgts:v:1:y:2003:i:1:p:75-107
    DOI: 10.1007/s10287-003-0005-2
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    Citations

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    Cited by:

    1. Tim Leung & Ronnie Sircar, 2009. "Accounting For Risk Aversion, Vesting, Job Termination Risk And Multiple Exercises In Valuation Of Employee Stock Options," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 99-128, January.
    2. Jan Hendrik Witte & Christoph Reisinger, 2011. "Penalty Methods for the Solution of Discrete HJB Equations -- Continuous Control and Obstacle Problems," Papers 1105.5954, arXiv.org, revised Dec 2011.
    3. Konstandatos, Otto, 2020. "Fair-value analytical valuation of reset executive stock options consistent with IFRS9 requirements," Annals of Actuarial Science, Cambridge University Press, vol. 14(1), pages 188-218, March.
    4. Dandan Song & Zhaojun Yang, 2014. "Utility-Based Pricing, Timing and Hedging of an American Call Option Under an Incomplete Market with Partial Information," Computational Economics, Springer;Society for Computational Economics, vol. 44(1), pages 1-26, June.
    5. Mustafa Pınar, 2011. "Gain–loss based convex risk limits in discrete-time trading," Computational Management Science, Springer, vol. 8(3), pages 299-321, August.
    6. Erhan Bayraktar & Arash Fahim, 2011. "A Stochastic Approximation for Fully Nonlinear Free Boundary Parabolic Problems," Papers 1109.5752, arXiv.org, revised Nov 2013.
    7. Mahan Tahvildari, 2021. "Forward indifference valuation and hedging of basis risk under partial information," Papers 2101.00251, arXiv.org.
    8. M. R. Grasselli, 2005. "Nonlinearity, correlation and the valuation of employee stock options," Papers math/0511234, arXiv.org.
    9. Giorgia Callegaro & Luciano Campi & Valeria Giusto & Tiziano Vargiolu, 2017. "Utility indifference pricing and hedging for structured contracts in energy markets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(2), pages 265-303, April.
    10. Vicky Henderson & Gechun Liang, 2011. "A Multidimensional Exponential Utility Indifference Pricing Model with Applications to Counterparty Risk," Papers 1111.3856, arXiv.org, revised Sep 2015.
    11. Henderson, Vicky & Hobson, David, 2007. "Horizon-unbiased utility functions," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1621-1641, November.
    12. Yunhong Li & Zuo Quan Xu & Xun Yu Zhou, 2023. "Robust utility maximization with intractable claims," Papers 2304.06938, arXiv.org, revised Jul 2023.
    13. Xiaoshan Chen & Qingshuo Song & Fahuai Yi & George Yin, 2011. "Indifference Pricing of American Option Underlying Illiquid Stock under Exponential Forward Performance," Papers 1201.0075, arXiv.org.
    14. Yunhong Li & Zuo Quan Xu & Xun Yu Zhou, 2023. "Robust utility maximisation with intractable claims," Finance and Stochastics, Springer, vol. 27(4), pages 985-1015, October.

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