Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T07:22:04.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical analysis of a variational inequality modelling perpetual executive stock options

Published online by Cambridge University Press:  07 January 2015

XIN LAI ,
XINFU CHEN ,
MINGXIN WANG ,
CONG QIN  and
WANGHUI YU
XIN LAI
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People's Republic of China email: [email protected]
XINFU CHEN
Affiliation:
Department of Mathematics, University of Pittsburgh, PA 15260, USA email: [email protected]
MINGXIN WANG
Affiliation:
Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, People's Republic of China email: [email protected]
CONG QIN
Affiliation:
Center for Financial Engineering, Soochow University, Suzhou 215006, People's Republic of China email: [email protected], [email protected]
WANGHUI YU
Affiliation:
Center for Financial Engineering, Soochow University, Suzhou 215006, People's Republic of China email: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we establish the existence and uniqueness of a classical solution of a degenerate parabolic variational inequality of which a strong solution was shown to exist by Song and Yu [21]. The problem arises from optimal stochastic control of exercising continuously perpetual executive stock options (ESOs). We also characterize the basic graph, continuity, and monotonicity properties of the free boundary from which the optimal control strategy can be described precisely.

Keywords

Variational inequalityfree boundaryperpetual executive stock optionstochastic control
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 2 , April 2015 , pp. 193 - 213
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Carpenter, J. N. (1998) The exercise and valuation of executive stock options. J. Financ. Econ. 48, 127–158.Google Scholar
[2]Carpenter, J. N., Stanton, R. & Wallace, N. (2010) Optimal exercise of executive stock options and implications for firm cost. J. Financ. Econ. 98, 315–337.Google Scholar
[3]Chen, X. & Dai, M. (2013) Characterization of optimal strategy for multi-asset investment and consumption with transaction costs. SIAM J. Math. Financ. 4, 857–883.Google Scholar
[4]Chen, X., Hu, B. & Liang, J.Finite difference scheme convergence rate for approximated free boundaries of American options, preprint.Google Scholar
[5]Chen, Y. (2003) Second Order Parabolic Partial Differential Equations (in Chinese), Peking University Press, Beijing.Google Scholar
[6]Fleming, W. H. & Soner, H. M. (2006) Controlled Markov processes and viscosity solutions, Springer, New York.Google Scholar
[7]Friedman, A. (1982) Variational Principles and Free Boundary Problems, John Wiley & Sons, New York.Google Scholar
[8]Friedman, A. (2008) Partial Differential Equations of Parabolic Type, Dover Publications, Mineola, New York.Google Scholar
[9]Grasselli, M. & Henderson, V. (2009) Risk aversion and block exercise of executive stock options. J. Econ. Dyn. Control 33, 109–127.Google Scholar
[10]Hall, B. J. & Murphy, K. J. (2002) Stock option for undiversified executives. J. Account. Econ. 33, 3–42.CrossRefGoogle Scholar
[11]Ingersoll, J. (2006) The subjective and objective evaluation of incentive stock options. J. Business 79, 453–487.Google Scholar
[12]Jain, A. & Subramanian, A. (2004) The intertemporal exercise and valuation of employee options. Account. Rev. 79, 705–743.Google Scholar
[13]Jiang, L. (2005) Mathematical Modeling and Methods of Option Pricing, World Scientific Publication, Hackensack.Google Scholar
[14]Lai, X., Chen, X., Wang, M., Qin, C. & Yu, W. A variational inequality arising from optimal exercise perpetual executive stock options, submitted.Google Scholar
[15]Lambert, R., Larcker, D. & Verrecchia, R. (1991) Portfolio considerations in valuing executive compensation. Journal of Accounting Research 29, 129–149.CrossRefGoogle Scholar
[16]Leung, T. & Sircar, R. (2009) Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options. Math. Financ. 19, 99–128.CrossRefGoogle Scholar
[17]Pham, H. (2009) Continuous-time Stochastic Control and Optimization with Financial Applications, Springer, Berlin.Google Scholar
[18]Qin, C., Chen, X., Lai, X. & Yu, W. The optimal exercise strategy for perpetual ESOs under continuous-exercise model and its applications, submitted.Google Scholar
[19]Qin, C., Chen, X., Lai, X. & Yu, W. Regularity of free boundary arising from optimal exercise perpetual executive stock options, submitted.Google Scholar
[20]Rogers, L. C. G. & Scheinkman, J. (2007) Optimal exercise of executive stock options. Financ. Stoch. 11, 357–372.Google Scholar
[21]Song, L. & Yu, W. (2011) A parabolic variational inequality related to the perpetual American executive stock options. Nonlinear Anal. 74, 6583–6600.Google Scholar
[22]Song, L. & Yu, W. (2013) A free boundary problem coming from the perpetual American call options with utility. Eur. J. Appl. Math. 24, 231–271.Google Scholar