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A variational inequality arising from optimal exercise perpetual executive stock options

  • XIN LAI (a1), XINFU CHEN (a2), CONG QIN (a3) and WANGHUI YU (a3) (a4)
Abstract

We investigate a degenerate parabolic variational inequality arising from optimal continuous exercise perpetual executive stock options. It is also shown in Qin et al. (Continuous-Exercise Model for American Call Options with Hedging Constraints, working paper, available at SSRN: http://dx.doi.org/10.2139/ssrn.2757541) that to make this problem non-trivial the stock's growth rate must be no smaller than the discount rate. Well-posedness of the problem is established in Lai et al. (2015, Mathematical analysis of a variational inequality modeling perpetual executive stock options, Euro. J. Appl. Math., 26 (2015), 193–213), Qin et al. (2015, Regularity free boundary arising from optimal continuous exercise perpetual executive stock options, Interfaces and Free Boundaries, 17 (2015), 69–92), Song & Yu (2011, A parabolic variational inequality related to the perpetual American executive stock options, Nonlinear Analysis, 74 (2011), 6583-6600) for the case when the underlying stock's expected return rate is smaller than the discount rate. In this paper, we consider the remaining case: the discount rate is bigger than the growth rate but no bigger than the return rate. The existence of a unique classical solution as well as a continuous and strictly decreasing free boundary is proved.

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Lai was supported by Scientific Research Foundations of Civil Aviation University of China(No. 2015QD06X). Chen thanks the support from NSF grant DMS-1008905. Qin acknowledges support from CSC.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] J. N. Carpenter (1998) The exercise and valuation of executive stock options. J. Financ. Econ. 48 (2), 127158.

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[10] X. Lai , X. Chen , M. Wang , C. Qin & W. Yu (2015) Mathematical analysis of a variational inequality modeling perpetual executive stock options. Euro. J. Appl. Math. 26 (2), 193213.

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[16] L. Song & W. Yu (2011) A parabolic variational inequality related to the perpetual American executive stock options. Nonlinear Anal. 74 (17), 65836600.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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