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Watermark options

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  • Rodosthenous, Neofytos
  • Zervos, Mihail

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 non-linear 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

  • Rodosthenous, Neofytos & Zervos, Mihail, 2017. "Watermark options," LSE Research Online Documents on Economics 67859, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:67859
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    File URL: http://eprints.lse.ac.uk/67859/
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    References listed on IDEAS

    as
    1. 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.
    2. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    3. 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.
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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:

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

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