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[pi] options

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  • Guo, Xin
  • Zervos, Mihail

Abstract

We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual American-type call options, which include the perpetual American, Russian and lookback-American call options as special cases. We solve this genuinely two-dimensional optimal stopping problem by means of an explicit construction of its value function. In particular, we fully characterise the free-boundary that provides the optimal strategy, and which involves the analysis of a highly nonlinear ordinary differential equation (ODE). In accordance with other optimal stopping problems involving a running maximum process that have been studied in the literature, it turns out that the associated variational inequality has an uncountable set of solutions that satisfy the so-called principle of smooth fit.

Suggested Citation

  • Guo, Xin & Zervos, Mihail, 2010. "[pi] options," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1033-1059, July.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:7:p:1033-1059
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    References listed on IDEAS

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    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.
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    Cited by:

    1. Neofytos Rodosthenous & Mihail Zervos, 2017. "Watermark options," Finance and Stochastics, Springer, vol. 21(1), pages 157-186, January.
    2. Gapeev, Pavel V. & Rodosthenous, Neofytos, 2016. "Perpetual American options in diffusion-type models with running maxima and drawdowns," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2038-2061.

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