IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v207y2010i3p1368-1379.html
   My bibliography  Save this article

Valuing executive stock options: A quadratic approximation

Author

Listed:
  • Kimura, Toshikazu

Abstract

This paper develops a continuous-time model for valuing executive stock options (ESOs) with features of early exercise, delayed vesting and forfeiture. Applying the quadratic approximation established for valuing American options into ESOs, we obtain an explicit formula for the fair ESO value at its grant date. We show that the approximation formula is consistent with the exact results for two special cases either with no dividend or infinite maturity, and also that the perpetual value for the latter case gives an upper bound of the ESO value. To see the performance of the formula, we numerically examine it with benchmark results generated by a binomial-tree model for some particular cases. Numerical experiments show that there is a complementary relation between the vesting and trading periods with respect to exit rate of ESO holders.

Suggested Citation

  • Kimura, Toshikazu, 2010. "Valuing executive stock options: A quadratic approximation," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1368-1379, December.
    • Handle: RePEc:eee:ejores:v:207:y:2010:i:3:p:1368-1379
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377-2217(10)00480-7
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hall, Brian J. & Murphy, Kevin J., 2002. "Stock options for undiversified executives," Journal of Accounting and Economics, Elsevier, vol. 33(1), pages 3-42, February.
    2. Jing-Zhi Huang & Marti G. Subrahmanyam & G. George Yu, 1999. "Pricing And Hedging American Options: A Recursive Integration Method," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 8, pages 219-239, World Scientific Publishing Co. Pte. Ltd..
    3. Jakša Cvitanić & Zvi Wiener & Fernando Zapatero, 2008. "Analytic Pricing of Employee Stock Options," The Review of Financial Studies, Society for Financial Studies, vol. 21(2), pages 683-724, April.
    4. Elettra Agliardi & Rainer Andergassen, 2005. "Incentives of Stock Option Based Compensation," Review of Quantitative Finance and Accounting, Springer, vol. 25(1), pages 21-32, August.
    5. Smith, Cw & Zimmerman, Jl, 1976. "Valuing Employee Stock Option Plans Using Option Pricing Models," Journal of Accounting Research, Wiley Blackwell, vol. 14(2), pages 357-364.
    6. Aboody, David, 1996. "Market valuation of employee stock options," Journal of Accounting and Economics, Elsevier, vol. 22(1-3), pages 357-391, October.
    7. Detemple, Jerome & Sundaresan, Suresh, 1999. "Nontraded Asset Valuation with Portfolio Constraints: A Binomial Approach," The Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 835-872.
    8. Cuny, Charles J. & Jorion, Philippe, 1995. "Valuing executive stock options with endogenous departure," Journal of Accounting and Economics, Elsevier, vol. 20(2), pages 193-205, September.
    9. Peter Carr & Vadim Linetsky, 2000. "The Valuation of Executive Stock Options in an Intensity-Based Framework," Review of Finance, European Finance Association, vol. 4(3), pages 211-230.
    10. Carr, Peter, 1998. "Randomization and the American Put," The Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 597-626.
    11. Geske, Robert, 1979. "The valuation of compound options," Journal of Financial Economics, Elsevier, vol. 7(1), pages 63-81, March.
    12. Sircar, Ronnie & Xiong, Wei, 2007. "A general framework for evaluating executive stock options," Journal of Economic Dynamics and Control, Elsevier, vol. 31(7), pages 2317-2349, July.
    13. Murphy, Kevin J., 1985. "Corporate performance and managerial remuneration : An empirical analysis," Journal of Accounting and Economics, Elsevier, vol. 7(1-3), pages 11-42, April.
    14. Johnson, Shane A. & Tian, Yisong S., 2000. "Indexed executive stock options," Journal of Financial Economics, Elsevier, vol. 57(1), pages 35-64, July.
    15. Brenner, Menachem & Sundaram, Rangarajan K. & Yermack, David, 2000. "Altering the terms of executive stock options," Journal of Financial Economics, Elsevier, vol. 57(1), pages 103-128, July.
    16. Johnson, Shane A. & Tian, Yisong S., 2000. "The value and incentive effects of nontraditional executive stock option plans," Journal of Financial Economics, Elsevier, vol. 57(1), pages 3-34, July.
    17. L. Rogers & José Scheinkman, 2007. "Optimal exercise of executive stock options," Finance and Stochastics, Springer, vol. 11(3), pages 357-372, July.
    18. Geske, Robert & Johnson, Herb E, 1984. "The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-1524, December.
    19. Muurling, Rutger & Lehnert, Thorsten, 2004. "Option-based compensation: a survey," The International Journal of Accounting, Elsevier, vol. 39(4), pages 365-401.
    20. Lambert, Ra & Larcker, Df & Verrecchia, Re, 1991. "Portfolio Considerations In Valuing Executive-Compensation," Journal of Accounting Research, Wiley Blackwell, vol. 29(1), pages 129-149.
    21. Barone-Adesi, Giovanni & Whaley, Robert E, 1987. "Efficient Analytic Approximation of American Option Values," Journal of Finance, American Finance Association, vol. 42(2), pages 301-320, June.
    22. Bunch, David S & Johnson, Herb, 1992. "A Simple and Numerically Efficient Valuation Method for American Puts Using a Modified Geske-Johnson Approach," Journal of Finance, American Finance Association, vol. 47(2), pages 809-816, June.
    23. Brennan, Michael J & Schwartz, Eduardo S, 1977. "The Valuation of American Put Options," Journal of Finance, American Finance Association, vol. 32(2), pages 449-462, May.
    24. Carpenter, Jennifer N., 1998. "The exercise and valuation of executive stock options," Journal of Financial Economics, Elsevier, vol. 48(2), pages 127-158, May.
    25. Bettis, J. Carr & Bizjak, John M. & Lemmon, Michael L., 2005. "Exercise behavior, valuation, and the incentive effects of employee stock options," Journal of Financial Economics, Elsevier, vol. 76(2), pages 445-470, May.
    26. Huddart, Steven & Lang, Mark, 1996. "Employee stock option exercises an empirical analysis," Journal of Accounting and Economics, Elsevier, vol. 21(1), pages 5-43, February.
    27. Ju, Nengjiu, 1998. "Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function," The Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 627-646.
    28. Huddart, Steven, 1994. "Employee stock options," Journal of Accounting and Economics, Elsevier, vol. 18(2), pages 207-231, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Carmona, Julio & León, Angel & Vaello-Sebastià, Antoni, 2012. "Does stock return predictability affect ESO fair value?," European Journal of Operational Research, Elsevier, vol. 223(1), pages 188-202.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    2. repec:dau:papers:123456789/9550 is not listed on IDEAS
    3. repec:dau:papers:123456789/13098 is not listed on IDEAS
    4. Grasselli, Matheus & Henderson, Vicky, 2009. "Risk aversion and block exercise of executive stock options," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 109-127, January.
    5. Carmona, Julio & León, Angel & Vaello-Sebastià, Antoni, 2011. "Pricing executive stock options under employment shocks," Journal of Economic Dynamics and Control, Elsevier, vol. 35(1), pages 97-114, January.
    6. Carpenter, Jennifer N. & Stanton, Richard & Wallace, Nancy, 2010. "Optimal exercise of executive stock options and implications for firm cost," Journal of Financial Economics, Elsevier, vol. 98(2), pages 315-337, November.
    7. Tang, Chun-Hua, 2012. "Revisiting the incentive effects of executive stock options," Journal of Banking & Finance, Elsevier, vol. 36(2), pages 564-574.
    8. Tim Leung & Ronnie Sircar, 2009. "Accounting For Risk Aversion, Vesting, Job Termination Risk And Multiple Exercises In Valuation Of Employee Stock Options," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 99-128, January.
    9. Len, Angel & Vaello-Sebasti, Antoni, 2009. "American GARCH employee stock option valuation," Journal of Banking & Finance, Elsevier, vol. 33(6), pages 1129-1143, June.
    10. Susana Álvarez-Díez & J. Baixauli-Soler & María Belda-Ruiz, 2014. "Are we using the wrong letters? An analysis of executive stock option Greeks," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 22(2), pages 237-262, June.
    11. Philip Brown & Alex Szimayer, 2008. "Valuing executive stock options: performance hurdles, early exercise and stochastic volatility," Accounting and Finance, Accounting and Finance Association of Australia and New Zealand, vol. 48(3), pages 363-389, September.
    12. Wei Xiong & Ronnie Sircar, 2004. "Evaluating Incentive Options," Econometric Society 2004 North American Winter Meetings 253, Econometric Society.
    13. Sircar, Ronnie & Xiong, Wei, 2007. "A general framework for evaluating executive stock options," Journal of Economic Dynamics and Control, Elsevier, vol. 31(7), pages 2317-2349, July.
    14. Weihan Li & Jin E. Zhang & Xinfeng Ruan & Pakorn Aschakulporn, 2024. "An empirical study on the early exercise premium of American options: Evidence from OEX and XEO options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(7), pages 1117-1153, July.
    15. Hodder, James E. & Jackwerth, Jens Carsten, 2011. "Managerial responses to incentives: Control of firm risk, derivative pricing implications, and outside wealth management," Journal of Banking & Finance, Elsevier, vol. 35(6), pages 1507-1518, June.
    16. Brian J. Hall & Thomas A. Knox, 2002. "Managing Option Fragility," NBER Working Papers 9059, National Bureau of Economic Research, Inc.
    17. Vicky Henderson, 2005. "The impact of the market portfolio on the valuation, incentives and optimality of executive stock options," Quantitative Finance, Taylor & Francis Journals, vol. 5(1), pages 35-47.
    18. Manuel Moreno & Javier Navas, 2003. "On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives," Review of Derivatives Research, Springer, vol. 6(2), pages 107-128, May.
    19. Kwangil Bae & Jangkoo Kang & Hwa‐Sung Kim, 2018. "Call options with concave payoffs: An application to executive stock options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(8), pages 943-957, August.
    20. Klein, Daniel, 2018. "Executive turnover and the valuation of stock options," Journal of Corporate Finance, Elsevier, vol. 48(C), pages 76-93.
    21. Minqiang Li, 2010. "A quasi-analytical interpolation method for pricing American options under general multi-dimensional diffusion processes," Review of Derivatives Research, Springer, vol. 13(2), pages 177-217, July.
    22. Ruas, João Pedro & Dias, José Carlos & Vidal Nunes, João Pedro, 2013. "Pricing and static hedging of American-style options under the jump to default extended CEV model," Journal of Banking & Finance, Elsevier, vol. 37(11), pages 4059-4072.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:207:y:2010:i:3:p:1368-1379. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.