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A variational inequality arising from optimal exercise perpetual executive stock options†

Published online by Cambridge University Press:  23 February 2017

XIN LAI ,
XINFU CHEN ,
CONG QIN  and
WANGHUI YU
XIN LAI
Affiliation:
College of Science, Civil Aviation University of China, Tianjin 300300, P.R. China email: [email protected]
XINFU CHEN
Affiliation:
Department of Mathematics, University of Pittsburgh, PA 15260, USA email: [email protected]
CONG QIN
Affiliation:
Center for Financial Engineering, Soochow University, Suzhou 215006, P.R. China
WANGHUI YU
Affiliation:
Center for Financial Engineering, Soochow University, Suzhou 215006, P.R. China School of Mathematic Science, Soochow University, Suzhou 215006, P.R. China emails: [email protected], [email protected]
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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.

Keywords

Variational inequalityfree boundaryoptimal exerciseperpetual executive stock option
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 1 , February 2018 , pp. 55 - 77
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

†

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.

References

[1] Carpenter, J. N. (1998) The exercise and valuation of executive stock options. J. Financ. Econ. 48 (2), 127–158.Google Scholar
[2] Chen, X. & Dai, M. (2013) Characterization of optimal strategy for multi-asset investment and consumption with transaction costs. Siam J. Math. Fina. 4 (1), 857–883.Google Scholar
[3] Chen, X., Hu, B. & Liang, J. Finite difference scheme convergence rate for approximated free boundaries of American options, preprint.Google Scholar
[4] Dixit, A. & Pindyck, R. (1994) Investment Under Uncertainty, Princeton University Press, Princeton.CrossRefGoogle Scholar
[5] Friedman, A. (1982) Variational Principles and Free Boundary Problems, John Wiley & Sons, New York.Google Scholar
[6] Grasselli, M. & Henderson, V. (2009) Risk aversion and block exercise of executive stock options. J. Econ. Dyn. Control 33 (1), 109–127.Google Scholar
[7] Hall, B. J. & Murphy, K. J. (2002) Stock option for undiversified executives. J. Account. Econ. 33 (1), 3–42.Google Scholar
[8] Ingersoll, J. (2006) The subjective and objective evaluation of incentive stock options. J. Bus. 79 (2), 453–487.Google Scholar
[9] Jain, A. & Subramanian, A. (2004) The intertemporal exercise and valuation of employee options. Account. Rev. 79 (3), 705–743.Google Scholar
[10] Lai, X., Chen, X., Wang, M., Qin, C. & Yu, W. (2015) Mathematical analysis of a variational inequality modeling perpetual executive stock options. Euro. J. Appl. Math. 26 (2), 193–213.CrossRefGoogle Scholar
[11] Lambert, R., Larcker, D. & Verrecchia, R. (1991) Portfolio considerations in valuing executive compensation. J. Account. Res. 29 (1), 129–149.Google Scholar
[12] Leung, T. & Sircar, R. (2009) Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options. Math. Finance 19 (1), 99–128.Google Scholar
[13] Qin, C., Chen, X., Lai, X. & Yu, W. (2015) Regularity free boundary arising from optimal continuous exercise perpetual executive stock options. Interfaces Free Boundaries 17 (1), 69–92.Google Scholar
[14] Qin, C., Chen, X., Lai, X. & Yu, W. A Continuous-Exercise Model for American Call Options with Hedging Constraints, working paper, available at SSRN: http://dx.doi.org/10.2139/ssrn.2757541 Google Scholar
[15] Rogers, L. C. G. & Scheinkman, J. (2007) Optimal exercise of executive stock options. Finance Stoch 11 (3), 357–372.Google Scholar
[16] Song, L. & Yu, W. (2011) A parabolic variational inequality related to the perpetual American executive stock options. Nonlinear Anal. 74 (17), 6583–6600.Google Scholar