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Watermark options

Author

Listed:
  • Neofytos Rodosthenous

    (Queen Mary University of London)

  • Mihail Zervos

    (London School of Economics)

Abstract

We consider a new family of derivatives whose payoffs become strictly positive when the price of their underlying asset falls relative to its historical maximum. We derive the solution to the discretionary stopping problems arising in the context of pricing their perpetual American versions by means of an explicit construction of their value functions. In particular, we fully characterise the free-boundary functions that provide the optimal stopping times of these genuinely two-dimensional problems as the unique solutions to highly nonlinear first order ODEs that have the characteristics of a separatrix. The asymptotic growth of these free-boundary functions can take qualitatively different forms depending on parameter values, which is an interesting new feature.

Suggested Citation

  • Neofytos Rodosthenous & Mihail Zervos, 2017. "Watermark options," Finance and Stochastics, Springer, vol. 21(1), pages 157-186, January.
    • Handle: RePEc:spr:finsto:v:21:y:2017:i:1:d:10.1007_s00780-016-0319-x
    DOI: 10.1007/s00780-016-0319-x
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    References listed on IDEAS

    as
    1. Guo, Xin & Zervos, Mihail, 2010. "[pi] options," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1033-1059, July.
    2. Curdin Ott, 2014. "Bottleneck options," Finance and Stochastics, Springer, vol. 18(4), pages 845-872, October.
    3. A. M. G. Cox & David Hobson & Jan Ob{l}'oj, 2007. "Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping," Papers math/0702173, arXiv.org, revised Nov 2008.
    4. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    5. Min Dai & Yue Kuen Kwok, 2006. "Characterization Of Optimal Stopping Regions Of American Asian And Lookback Options," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 63-82, January.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Optimal stopping; Running maximum process; Variational inequality; Two-dimensional free-boundary problem; Separatrix;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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