整体和非限制实施下永久经理期权的最优实施策略
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摘要
经理期权简称为ESOs,是公司作为酬金发给经理或员工的一种美式看涨期权。自20世纪80年代中期以来,ESOs已成为美国和其他国家高管薪酬的重要组成部分。ESOs可看作为公司从经理或员工那里买到服务而支付的成本。由于ESOs的发行数量之多,其对应的公司发行成本也很可观。为了给出ESOs的合理价值,需要理性预测经理人未来的实施策略。由于公司高管是不能卖空公司股票的(非法的),因此经理人不能构造由公司股票、ESOs和无风险债券组成的组合来对冲ESOs的风险。事实上,这是一个不完全市场,这就使得无套利定价方法不能用来对ESOs进行定价。
本文主要研究与金融市场上的永久ESOs的最优实施策略有关的一个抛物型变分不等式的定解问题。我们通过严格的数学分析,研究了相应解的存在性、唯一性、正则性和自由边界的性质及其极限形态等问题。本文的研究分为两个方面：整体实施永久ESOs和非限制实施永久ESOs。整体实施是指,经理人要么不实施其持有的ESOs,要么一次性实施其持有的全部ESOs。而在非限制实施情况下,经理人在任意可实施时刻可以实施其持有的任意份ESOs。
首先,我们采用经理人的财富效用最大化方法来研究整体实施情况下的一个永久ESOs定价模型,这是一个带效用函数的永久美式看涨期权的自由边界问题,其值函数和最优实施策略不仅与股价有关,而且还与ESOs的数量有关。其值函数是一个退化的抛物型变分不等式的定解问题的解。对于整体实施情况下的ESOs问题,目前已有许多学者进行了研究,但大多数研究均是基于定性分析或数值计算,没有严格的理论结果。Kadam等(2005)[14]虽然在基于效用的模型中给出了永久ESOs价值的解析解,但他们只考虑一份ESOs,相应的值函数只与股价有关。因为效用函数是非线性函数,经理人所持有的ESOs的总价值不是ESOs数目的线性函数。因此,假设经理人只持有一份ESOs是不合理的。我们用财富效用最大化方法对整体实施情况下的永久ESOs问题建立了一个最优停时模型,其值函数是ESOs数目和股价的函数。利用最优停时理论,我们导出值函数满足一个退化的抛物型变分不等式的定解问题。然后通过切片法,即对变量τ(ESOs数目与风险厌恶系数的乘积)离散化的方法来研究该抛物型变分不等式的定解问题。我们得出了解的存在性、唯一性和正则性,证明了自由边界(最佳实施边界)的连续性、单调性和有关极限形态。
其次,我们研究非限制实施情况下的永久ESOs模型。Rogers和Scheinkman(2007)[49]通过经理人的财富效用最大化方法,对非限制实施情况下的具有有限到期日的ESOs问题,建立了一个随机最优控制模型,并获得了相应的抛物型变分不等式的定解问题。但他们对该定解问题没有给出严格的理论研究,只是通过数值分析给出经理人的最优实施策略及相应的一些性质。我们在Rogers和Scheinkman(2007)的模型的基础上,研究相应的永久ESOs模型。通过随机最优控制理论,我们得到其值函数满足一个退化抛物型变分不等式的定解问题。由于该变分不等式的障碍条件里含有值函数关于变量τ(ESOs数目与风险厌恶系数的乘积)的偏导数,这使得对该变分不等式定解问题的理论研究有很大难度。我们仍通过切片法来研究该问题,证明了解的存在性、正则性及解的其他一些性质。
最后,我们还对两种实施情况进行数值分析。通过偏微分方程数值解法对经理人的最佳实施边界进行分析和比较。
Executive stock options are called ESOs for short, which are American call optionsthat the company granted to the managers or employees as a compensation. Sincethe mid1980s, ESOs have become an important part of executive compensation inthe United States and other countries. ESOs can be seen as an expense to the firmbecause the firm is buying services from the executives. Since ESOs issued by a firmare so large, the cost to the firm is also very impressive. In order to give a reasonablevaluation for ESOs, it is necessary to have rational prediction of the future exercisestrategies of the executives. Because company executives can not short sell companystock (illegal), the executives can not construct the portfolio composed of companystocks, ESOs and risk-free bonds to hedge the risk of ESOs. In fact, it is an incompletemarket, which makes the no-arbitrage pricing method invalid for the pricing of ESOs.
In this paper, we mainly discuss the definite problem of a parabolic variational in-equality which comes from the study of the optimal exercise strategies for the perpetualESOs in financial market. The existence, uniqueness and regularity of the correspond-ing solution as well as some properties of the free boundary and its limit behaviorare established in a rigorous theoretical framework. Our research is divided into twoaspects: perpetual ESOs with block exercise and perpetual ESOs with unrestrictedexercise. Block exercise is that the executive can exercise none of the ESOs or exerciseall of the ESOs at the same time. While in the unrestricted exercise situation, theexecutive can exercise any copies of ESOs at arbitrary implementation moment.
First of all, by use of the executive’s wealth utility maximization method, weinvestigate the pricing model of the perpetual ESOs with block exercise. It is a freeboundary problem of the perpetual American call options with a utility function. Thecorresponding value function and the optimal exercise policy are not only related tothe stock price, but also to the number of ESOs. The value function is the solution tothe definite problem of a degenerate parabolic variational inequality. There has been a lot of interest in the problem of ESOs with block exercise. However, most of theliterature only give a qualitative analysis or numerical calculation without a rigoroustheoretical research. Kadam et.al(2005)[14]give the analytical solution to the value ofthe perpetual ESOs in the utility-based model. However, they only consider one ESOs,and the corresponding value function is only related to the stock price. Because theutility function is a nonlinear function, the total value of the ESOs that the executiveholds is not a linear function of the number of ESOs. Thus, the assumption that theexecutive only holds one ESOs is unreasonable. We establish a optimal stopping modelfor the perpetual ESOs with block exercise based on the wealth utility maximizationmethod. The corresponding value function is a binary function of the number of ESOsand the stock price. By the theory of optimal stopping, we derive that the valuefunction satisfies a degenerate parabolic variational inequality. And then by usingslicing method, i.e. the variable τ (the number of ESOs multiplied by the risk aversioncoefcient) discrete method, we study the definite problem of the parabolic variationalinequality. We get the existence, uniqueness and regularity of the solution and provethe continuity, monotonicity and limit behavior of the free boundary (the optimalexercise boundary).
Secondly, we study the model of the perpetual ESOs with unrestricted exercise.By use of the method of maximizing the expected utility of the executive’s wealth,Rogers and Scheinkman(2007)[49]have established a stochastic optimal control modelfor the ESOs with finite-horizon and obtained the definite problem of the correspondingparabolic variational inequality. However, they did not give a rigorous theoreticalstudy on this definite problem, and only provided the executive’s optimal strategyand the corresponding properties through numerical analysis. Based on the model ofRogers and Scheinkman (2007)[49], we study the corresponding model of the ESOs withinfinite-horizon. By stochastic optimal control theory, we get the value function whichsatisfies a definite problem of a degenerate parabolic variational inequality. Becausethe obstacle condition of the variational inequality contains the partial derivative of thevalue function to the variable τ (the number of ESOs multiplied by the risk aversioncoefcient), it makes the theoretical study of the definite problem of the the variationalinequality very difcult. We also turn to the slicing method to investigate this problem,and prove the existence, regularity and some other properties of the solution.
Finally, numerical analysis is given for both block exercise and unrestricted ex- ercise. By use of the numerical solution method of partial diferential equations, weanalyze and compare the optimal exercise strategies of the executive in two exercisecases.
本文主要研究与金融市场上的永久ESOs的最优实施策略有关的一个抛物型变分不等式的定解问题。我们通过严格的数学分析,研究了相应解的存在性、唯一性、正则性和自由边界的性质及其极限形态等问题。本文的研究分为两个方面：整体实施永久ESOs和非限制实施永久ESOs。整体实施是指,经理人要么不实施其持有的ESOs,要么一次性实施其持有的全部ESOs。而在非限制实施情况下,经理人在任意可实施时刻可以实施其持有的任意份ESOs。
首先,我们采用经理人的财富效用最大化方法来研究整体实施情况下的一个永久ESOs定价模型,这是一个带效用函数的永久美式看涨期权的自由边界问题,其值函数和最优实施策略不仅与股价有关,而且还与ESOs的数量有关。其值函数是一个退化的抛物型变分不等式的定解问题的解。对于整体实施情况下的ESOs问题,目前已有许多学者进行了研究,但大多数研究均是基于定性分析或数值计算,没有严格的理论结果。Kadam等(2005)[14]虽然在基于效用的模型中给出了永久ESOs价值的解析解,但他们只考虑一份ESOs,相应的值函数只与股价有关。因为效用函数是非线性函数,经理人所持有的ESOs的总价值不是ESOs数目的线性函数。因此,假设经理人只持有一份ESOs是不合理的。我们用财富效用最大化方法对整体实施情况下的永久ESOs问题建立了一个最优停时模型,其值函数是ESOs数目和股价的函数。利用最优停时理论,我们导出值函数满足一个退化的抛物型变分不等式的定解问题。然后通过切片法,即对变量τ(ESOs数目与风险厌恶系数的乘积)离散化的方法来研究该抛物型变分不等式的定解问题。我们得出了解的存在性、唯一性和正则性,证明了自由边界(最佳实施边界)的连续性、单调性和有关极限形态。
其次,我们研究非限制实施情况下的永久ESOs模型。Rogers和Scheinkman(2007)[49]通过经理人的财富效用最大化方法,对非限制实施情况下的具有有限到期日的ESOs问题,建立了一个随机最优控制模型,并获得了相应的抛物型变分不等式的定解问题。但他们对该定解问题没有给出严格的理论研究,只是通过数值分析给出经理人的最优实施策略及相应的一些性质。我们在Rogers和Scheinkman(2007)的模型的基础上,研究相应的永久ESOs模型。通过随机最优控制理论,我们得到其值函数满足一个退化抛物型变分不等式的定解问题。由于该变分不等式的障碍条件里含有值函数关于变量τ(ESOs数目与风险厌恶系数的乘积)的偏导数,这使得对该变分不等式定解问题的理论研究有很大难度。我们仍通过切片法来研究该问题,证明了解的存在性、正则性及解的其他一些性质。
最后,我们还对两种实施情况进行数值分析。通过偏微分方程数值解法对经理人的最佳实施边界进行分析和比较。
Executive stock options are called ESOs for short, which are American call optionsthat the company granted to the managers or employees as a compensation. Sincethe mid1980s, ESOs have become an important part of executive compensation inthe United States and other countries. ESOs can be seen as an expense to the firmbecause the firm is buying services from the executives. Since ESOs issued by a firmare so large, the cost to the firm is also very impressive. In order to give a reasonablevaluation for ESOs, it is necessary to have rational prediction of the future exercisestrategies of the executives. Because company executives can not short sell companystock (illegal), the executives can not construct the portfolio composed of companystocks, ESOs and risk-free bonds to hedge the risk of ESOs. In fact, it is an incompletemarket, which makes the no-arbitrage pricing method invalid for the pricing of ESOs.
In this paper, we mainly discuss the definite problem of a parabolic variational in-equality which comes from the study of the optimal exercise strategies for the perpetualESOs in financial market. The existence, uniqueness and regularity of the correspond-ing solution as well as some properties of the free boundary and its limit behaviorare established in a rigorous theoretical framework. Our research is divided into twoaspects: perpetual ESOs with block exercise and perpetual ESOs with unrestrictedexercise. Block exercise is that the executive can exercise none of the ESOs or exerciseall of the ESOs at the same time. While in the unrestricted exercise situation, theexecutive can exercise any copies of ESOs at arbitrary implementation moment.
First of all, by use of the executive’s wealth utility maximization method, weinvestigate the pricing model of the perpetual ESOs with block exercise. It is a freeboundary problem of the perpetual American call options with a utility function. Thecorresponding value function and the optimal exercise policy are not only related tothe stock price, but also to the number of ESOs. The value function is the solution tothe definite problem of a degenerate parabolic variational inequality. There has been a lot of interest in the problem of ESOs with block exercise. However, most of theliterature only give a qualitative analysis or numerical calculation without a rigoroustheoretical research. Kadam et.al(2005)[14]give the analytical solution to the value ofthe perpetual ESOs in the utility-based model. However, they only consider one ESOs,and the corresponding value function is only related to the stock price. Because theutility function is a nonlinear function, the total value of the ESOs that the executiveholds is not a linear function of the number of ESOs. Thus, the assumption that theexecutive only holds one ESOs is unreasonable. We establish a optimal stopping modelfor the perpetual ESOs with block exercise based on the wealth utility maximizationmethod. The corresponding value function is a binary function of the number of ESOsand the stock price. By the theory of optimal stopping, we derive that the valuefunction satisfies a degenerate parabolic variational inequality. And then by usingslicing method, i.e. the variable τ (the number of ESOs multiplied by the risk aversioncoefcient) discrete method, we study the definite problem of the parabolic variationalinequality. We get the existence, uniqueness and regularity of the solution and provethe continuity, monotonicity and limit behavior of the free boundary (the optimalexercise boundary).
Secondly, we study the model of the perpetual ESOs with unrestricted exercise.By use of the method of maximizing the expected utility of the executive’s wealth,Rogers and Scheinkman(2007)[49]have established a stochastic optimal control modelfor the ESOs with finite-horizon and obtained the definite problem of the correspondingparabolic variational inequality. However, they did not give a rigorous theoreticalstudy on this definite problem, and only provided the executive’s optimal strategyand the corresponding properties through numerical analysis. Based on the model ofRogers and Scheinkman (2007)[49], we study the corresponding model of the ESOs withinfinite-horizon. By stochastic optimal control theory, we get the value function whichsatisfies a definite problem of a degenerate parabolic variational inequality. Becausethe obstacle condition of the variational inequality contains the partial derivative of thevalue function to the variable τ (the number of ESOs multiplied by the risk aversioncoefcient), it makes the theoretical study of the definite problem of the the variationalinequality very difcult. We also turn to the slicing method to investigate this problem,and prove the existence, regularity and some other properties of the solution.
Finally, numerical analysis is given for both block exercise and unrestricted ex- ercise. By use of the numerical solution method of partial diferential equations, weanalyze and compare the optimal exercise strategies of the executive in two exercisecases.
引文
[1] Rubinstein, M.. On the accounting valuation of employee stock options. Journalof Derivatives,1995,3:8-24.
[2] Murphy, K., J.. Executive compensation, in O. Ashenfelter and D. Card (eds.),Handbook of Labor Economics, vol. III. North Holland,1999:2485-2563.
[3] Hall, B., Murphy, K., J.. Stock options for undiversified employees. Journal ofAccounting and Economics,2002,33:3-42.
[4] Coopers, Lybrand. Stock options: Accounting, valuation and management issues,New York, NY,1993.
[5] Hall, B., Murphy, K., J.. The trouble with stock options. Journal of EconomicPerspectives,2003,17:125-136.
[6] Financial Accounting Standards Board. Accounting for stock-based compensation.FASB Statement No.123,1995.
[7] Financial Accounting Standards Board. Share-based payment. FASB StatementNo.123(revised2004),2004.
[8] International Accounting Standards Board. Share-based payment. InternationalFinancial Reporting Standards2,2004.
[9] Securities and Exchange Commission. Staf Accounting Bulletin No.107,2005.
[10] Foster, T., Koogler, P., Vickrey, D.. Valuation of executive stock options and theFASB proposal. The Accounting Review,1991,66:595-610.
[11] Jennergren, L., Naslund, B.. A comment on”Valuation of executive stock optionsand the FASB proposal”. The Accounting Review,1993,68:179-183.
[12] Carpenter, J.. The exercise and valuation of executive stock options. Journal ofFinancial Economics,1998,48:127-158.
[13] Abowd, J., Kaplan, D.. Executive compensation: Six questions that need answer-ing. Journal of Economic Perspectives,1999,13(4):145-168.
[14] Kadam, A, Lakner, P. Srinivasan, A.. Perpetual call options with non-tradeability,Optimal Control Applications and Methods,2005,26(3):107-127.
[15] Black, F., Scholes, M.. The Pricing of options and corporate liabilities. Journalof Political Economy,1973,81(3):637-654.
[16] Merton, R.. Theory of rational options pricing. Bell Journal of Economics andManagement Science,1973,4(1):141-183.
[17] Smith, C., Zimmerman, J.. Valuing employee stock options plans using optionspricing models. Journal of Accounting Research,1976,14:64-357.
[18] Antle, R., Smith, A.. An empirical investigation of the relative performance eval-uation of corporate executives. Journal of Accounting Research,1985,24:1-39.
[19] Yermack, D.. Do corporations award CEO stock options efectively? Journal ofFinancial Economics,1995,39:237-269.
[20] Brenner, H., Eldor, R., Hauser, S.. The price of options illiquidity. Journal ofFinance,2001(2):789-1085.
[21] Brian, H., Murphy, K.. Stock options for undiversified executive. Journal of Ac-counting and Economics,2002(33):3-43.
[22] Marquardt, C. A.. The cost of employee stock options grants: an empirical anal-ysis. Journal of Accounting Research,2002,40(4):1192-1217.
[23] Cuny, C.J., Jorion, P.. Valuing executive stock options with an endogenous de-parture decision. Journal of Accounting and Economics,1995,20:193-205.
[24] Meulbroek, L. K.. The efciency of equity-linked compensation: understandingthe full cost of awarding exeeutive stock options. Finaneial Management,2001,30:5-3.
[25] Lambert, R., Larcker, D., Verrechia, R.E.. Portfolio considerations in valuingexecutive compensation. Journal of Accounting Research,1991,29:129-149.
[26] Huddart, S.. Employee stock options. Journal of Accounting and Economics,1994,18:207-231.
[27] Kulatilaka, N., Marcus, A. J.. Valuing employee stock options. Financial AnalystsJournal,1994,50(6):46-56.
[28] Aboody, D.. Market valuation of employee stock options. Journal of Accountingand Economics,1996,22:357-391.
[29] Detemple, J., Sundaresan, S.. Nontraded asset valuation with portfolio con-straints: a binomial approach. Review of Financial Studies,1999,12:835-872.
[30] Carpenter, J.. Does options compensation increase managerial risk appetite?Journal of Finance,2000,55:2311-2331.
[31] Tian, Y. S.. Too much of a good incentive? the case of executive stock options.Journal of Banking and Finance,2004,28:1225-1245.
[32] Chance, D., Yang, D. H.. The utility-based valuation and cost of executive stockoptions in a binomial framework: Issues and methodologies. Journal of Derivativeand Accounting,2005,2:165-188.
[33] Bettis, J. C., Bizjak, J. M., Lemmon, M. L.. Exercise behavior, valuation and theincentive efects of employee stock options. Journal of Financial Economics,2005,76:445-470.
[34] Ingersoll, J. E.. The subjective and objective valuation of incentive stock options.Journal of Business,2006,79(2):453-487.
[35] Carpenter, J. N., Stanton, R., Wallace, N.. Optimal exercise of executive stockoptions and implications for firm cost, Working paper, Haas School of Business,2009.
[36] Carpenter, J. N., Stanton, R., Wallace, N.. Estimation of employee stock optionsexercise rates and firm cost, Working paper, Haas School of Business,2011.
[37] Agrawal, A., Mandelker, G.. Managerial incentives and corporate investment andfinancing decisions. Journal of Finance,1987,42:823-837.
[38] Lambert, R., Lanen, W., Larcker, D.. Executive stock options plans and corporatedividend policy. Journal of Financial and Quantitative Analysis,1989,24:409-424.
[39] DeFusco, R., Johnson, R., Zorn, T.. The efect of executive stock options planson stockholders and bondholders. Journal of Finance,1990,45:617-627.
[40] Becherer, D.. Utility indiference hedging and valuation via reaction difusionsystems. Stochast. Anal. Appl. Math. Finance,2004,460:27-51.
[41] Henderson, V.. The impact of the market portfolio on the valuation, incentivesand optimality of executive stock options. Quantitive Finance,2005,5:1-13.
[42] Henderson, V., Hobson, D.. Utility indiference pricing: An overview, in: ReneCarmona (Ed.), Volume on Indiference Pricing, Princeton University Press,Princeton,2007.
[43] Henderson, V. and Sun, J.. Employee Stock Options, working paper, Universityof Oxford,2010.
[44] Huddart, S., Lang, M.. Employee stock options exercises: An empirical analysis.Journal of Accounting and Economics,1996,21:5-43.
[45] Jain, A., Subramanian, A.. The intertemporal exercise and valuation of employeeoptions. Accounting Review,2004,79:705-744.
[46] Cox, J., Ross, S., Rubinstein, M.. options pricing: A simplified approach. Journalof Financial Economics,1979,7:229-264.
[47] Grasselli, M.. Nonlinearity, correlation and the valuation of employee options.Working paper, McMaster University,2005.
[48] Musiela, M., Zariphopoulou, T.. An example of indiference pricing under expo-nential preferences. Finance and Stochastic,2004,8:229-239.
[49] Rogers, L. C. G., Scheinkman, J.. Optimal exercise of executive stock options.Finance and stochastic,2007,11:357-372.
[50] Grasselli, M., Henderson, V.. Risk aversion and block exercise of executive stockoptions. Journal of Economic Dynamics and Control,2009,33(1):109-127.
[51]Henderson, V., Hobson, D.. Horizon-unbiased utility functions. Stochastic Pro-cesses and Their Applications,2007,117(11):1621-1641.
[52]Leung, T., Sircar, R.. Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options. Mathematical Finance,2009,19(1):99-128.
[53]Saly, P.J.. Repricing executive stock options in a down market. Journal of Ac-counting and Economics,1994,18:325-356.
[54]Brenner, M., Sundaram, R., Yermack, D.. Altering the terms of executive stock options. Journal of Financial Economics,2000,57:103-128.
[55]Acharya V.V., John K., Sundaram R.K.. On the optimality of resetting executive stock options. Journal of Financial Economics,2000,57:65-101.
[56]Chance, D. M., Kumar, R., Todd, R. B.. The 'repricing' of executive stock options. Journal of Financial Economics,2000,57:129-154.
[57]Corrado, C.J., Jordan, B.D., Miller Jr., T.W., Stansfeld, J. J.. Repricing and employee stock options valuation. Journal of Banking and Finance,2001,25:1059-1082.
[58]Chen, M.. Executive options repricing, incentives, and retention. Journal of Fi-nance,2004,59(3):1167-1199.
[59]Johnson, S.A., Tian, Y.S.. Indexed executive stock options. Journal of Financial Economics,2000,57(1):35-64.
[60]Hemmer, T., Matsunaga, S., Shevlin, T.. Optimal exercise and the cost of granting employee stock options with a reload provision. Journal of Accounting Research,1998,36(2):231-255.
[61]Ingersoll, Jonathan E., Jr. Valuing reload options. Working paper, Yale School of Management,2001.
[62]Dybvig, P., Lowenstein, M.. Employee reload options:pricing, hedging gand optimal exercise. Review of Financial Studies,2003,12:145-171.
[63]Egloff, D., Farkas, W., Leippold, M.. American options with stopping time con-straints to executive stock options, Working paper, QuantAlea GmbH,2005.
[64]Oksendal, B.. Stochastic differential equations(5th edition), Springer-Verlag, Berlin,2000.
[65]Pham, H.. Continuous-time stochastic control and optimization with financial application, Springer-Verlag, Berlin,2009.
[66]Yong, J. M., Zhou, X. Y.. Stochastic control:Hamiltonian systems and HJB equations, Springer-Verlag, Berlin,1999.
[67]Jiang, L.. Mathematical modeling and methods of options pricing, World Scientific Publication Co. Pte. Ltd., Singapore,2005.
[68]Kim, I.. The analytic valuation of American options. Review of Financial Studies,1990,3:547-572.
[69]Jaillet, P., Lamberton, D., Lapeyre, B.. Variational inequalities and pricing of American options. Acta Appl. Math.,1990,21:263-289.
[70]Myneni, R.. The pricing of the American options. The Annals of Applied Proba-bility,1992,2(1):1-23.
[71]Dewynne, J. N., Howison, S. D., Rupf, I., Wilmott, P.. Some mathematical results in the pricing of American options. Europ. J. Appl. Math.,1993,4:381-398.
[72]Wilmott, P., Dewynne, J. N., Howison, S. D.. options pricing:mathematical models and computation, Oxford Financial Press, Oxford,1993.
[73]余王辉著.金融数学中的自由边界问题讲义.
[74]王颖.整体实施永久经理期权模型的若干问题,苏州大学硕士学位论文,2013.
[75]Friedman, A.. Variational principles and free-boundary problems, John Wiley&Sons Ltd., Chichester,1988.
[2] Murphy, K., J.. Executive compensation, in O. Ashenfelter and D. Card (eds.),Handbook of Labor Economics, vol. III. North Holland,1999:2485-2563.
[3] Hall, B., Murphy, K., J.. Stock options for undiversified employees. Journal ofAccounting and Economics,2002,33:3-42.
[4] Coopers, Lybrand. Stock options: Accounting, valuation and management issues,New York, NY,1993.
[5] Hall, B., Murphy, K., J.. The trouble with stock options. Journal of EconomicPerspectives,2003,17:125-136.
[6] Financial Accounting Standards Board. Accounting for stock-based compensation.FASB Statement No.123,1995.
[7] Financial Accounting Standards Board. Share-based payment. FASB StatementNo.123(revised2004),2004.
[8] International Accounting Standards Board. Share-based payment. InternationalFinancial Reporting Standards2,2004.
[9] Securities and Exchange Commission. Staf Accounting Bulletin No.107,2005.
[10] Foster, T., Koogler, P., Vickrey, D.. Valuation of executive stock options and theFASB proposal. The Accounting Review,1991,66:595-610.
[11] Jennergren, L., Naslund, B.. A comment on”Valuation of executive stock optionsand the FASB proposal”. The Accounting Review,1993,68:179-183.
[12] Carpenter, J.. The exercise and valuation of executive stock options. Journal ofFinancial Economics,1998,48:127-158.
[13] Abowd, J., Kaplan, D.. Executive compensation: Six questions that need answer-ing. Journal of Economic Perspectives,1999,13(4):145-168.
[14] Kadam, A, Lakner, P. Srinivasan, A.. Perpetual call options with non-tradeability,Optimal Control Applications and Methods,2005,26(3):107-127.
[15] Black, F., Scholes, M.. The Pricing of options and corporate liabilities. Journalof Political Economy,1973,81(3):637-654.
[16] Merton, R.. Theory of rational options pricing. Bell Journal of Economics andManagement Science,1973,4(1):141-183.
[17] Smith, C., Zimmerman, J.. Valuing employee stock options plans using optionspricing models. Journal of Accounting Research,1976,14:64-357.
[18] Antle, R., Smith, A.. An empirical investigation of the relative performance eval-uation of corporate executives. Journal of Accounting Research,1985,24:1-39.
[19] Yermack, D.. Do corporations award CEO stock options efectively? Journal ofFinancial Economics,1995,39:237-269.
[20] Brenner, H., Eldor, R., Hauser, S.. The price of options illiquidity. Journal ofFinance,2001(2):789-1085.
[21] Brian, H., Murphy, K.. Stock options for undiversified executive. Journal of Ac-counting and Economics,2002(33):3-43.
[22] Marquardt, C. A.. The cost of employee stock options grants: an empirical anal-ysis. Journal of Accounting Research,2002,40(4):1192-1217.
[23] Cuny, C.J., Jorion, P.. Valuing executive stock options with an endogenous de-parture decision. Journal of Accounting and Economics,1995,20:193-205.
[24] Meulbroek, L. K.. The efciency of equity-linked compensation: understandingthe full cost of awarding exeeutive stock options. Finaneial Management,2001,30:5-3.
[25] Lambert, R., Larcker, D., Verrechia, R.E.. Portfolio considerations in valuingexecutive compensation. Journal of Accounting Research,1991,29:129-149.
[26] Huddart, S.. Employee stock options. Journal of Accounting and Economics,1994,18:207-231.
[27] Kulatilaka, N., Marcus, A. J.. Valuing employee stock options. Financial AnalystsJournal,1994,50(6):46-56.
[28] Aboody, D.. Market valuation of employee stock options. Journal of Accountingand Economics,1996,22:357-391.
[29] Detemple, J., Sundaresan, S.. Nontraded asset valuation with portfolio con-straints: a binomial approach. Review of Financial Studies,1999,12:835-872.
[30] Carpenter, J.. Does options compensation increase managerial risk appetite?Journal of Finance,2000,55:2311-2331.
[31] Tian, Y. S.. Too much of a good incentive? the case of executive stock options.Journal of Banking and Finance,2004,28:1225-1245.
[32] Chance, D., Yang, D. H.. The utility-based valuation and cost of executive stockoptions in a binomial framework: Issues and methodologies. Journal of Derivativeand Accounting,2005,2:165-188.
[33] Bettis, J. C., Bizjak, J. M., Lemmon, M. L.. Exercise behavior, valuation and theincentive efects of employee stock options. Journal of Financial Economics,2005,76:445-470.
[34] Ingersoll, J. E.. The subjective and objective valuation of incentive stock options.Journal of Business,2006,79(2):453-487.
[35] Carpenter, J. N., Stanton, R., Wallace, N.. Optimal exercise of executive stockoptions and implications for firm cost, Working paper, Haas School of Business,2009.
[36] Carpenter, J. N., Stanton, R., Wallace, N.. Estimation of employee stock optionsexercise rates and firm cost, Working paper, Haas School of Business,2011.
[37] Agrawal, A., Mandelker, G.. Managerial incentives and corporate investment andfinancing decisions. Journal of Finance,1987,42:823-837.
[38] Lambert, R., Lanen, W., Larcker, D.. Executive stock options plans and corporatedividend policy. Journal of Financial and Quantitative Analysis,1989,24:409-424.
[39] DeFusco, R., Johnson, R., Zorn, T.. The efect of executive stock options planson stockholders and bondholders. Journal of Finance,1990,45:617-627.
[40] Becherer, D.. Utility indiference hedging and valuation via reaction difusionsystems. Stochast. Anal. Appl. Math. Finance,2004,460:27-51.
[41] Henderson, V.. The impact of the market portfolio on the valuation, incentivesand optimality of executive stock options. Quantitive Finance,2005,5:1-13.
[42] Henderson, V., Hobson, D.. Utility indiference pricing: An overview, in: ReneCarmona (Ed.), Volume on Indiference Pricing, Princeton University Press,Princeton,2007.
[43] Henderson, V. and Sun, J.. Employee Stock Options, working paper, Universityof Oxford,2010.
[44] Huddart, S., Lang, M.. Employee stock options exercises: An empirical analysis.Journal of Accounting and Economics,1996,21:5-43.
[45] Jain, A., Subramanian, A.. The intertemporal exercise and valuation of employeeoptions. Accounting Review,2004,79:705-744.
[46] Cox, J., Ross, S., Rubinstein, M.. options pricing: A simplified approach. Journalof Financial Economics,1979,7:229-264.
[47] Grasselli, M.. Nonlinearity, correlation and the valuation of employee options.Working paper, McMaster University,2005.
[48] Musiela, M., Zariphopoulou, T.. An example of indiference pricing under expo-nential preferences. Finance and Stochastic,2004,8:229-239.
[49] Rogers, L. C. G., Scheinkman, J.. Optimal exercise of executive stock options.Finance and stochastic,2007,11:357-372.
[50] Grasselli, M., Henderson, V.. Risk aversion and block exercise of executive stockoptions. Journal of Economic Dynamics and Control,2009,33(1):109-127.
[51]Henderson, V., Hobson, D.. Horizon-unbiased utility functions. Stochastic Pro-cesses and Their Applications,2007,117(11):1621-1641.
[52]Leung, T., Sircar, R.. Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options. Mathematical Finance,2009,19(1):99-128.
[53]Saly, P.J.. Repricing executive stock options in a down market. Journal of Ac-counting and Economics,1994,18:325-356.
[54]Brenner, M., Sundaram, R., Yermack, D.. Altering the terms of executive stock options. Journal of Financial Economics,2000,57:103-128.
[55]Acharya V.V., John K., Sundaram R.K.. On the optimality of resetting executive stock options. Journal of Financial Economics,2000,57:65-101.
[56]Chance, D. M., Kumar, R., Todd, R. B.. The 'repricing' of executive stock options. Journal of Financial Economics,2000,57:129-154.
[57]Corrado, C.J., Jordan, B.D., Miller Jr., T.W., Stansfeld, J. J.. Repricing and employee stock options valuation. Journal of Banking and Finance,2001,25:1059-1082.
[58]Chen, M.. Executive options repricing, incentives, and retention. Journal of Fi-nance,2004,59(3):1167-1199.
[59]Johnson, S.A., Tian, Y.S.. Indexed executive stock options. Journal of Financial Economics,2000,57(1):35-64.
[60]Hemmer, T., Matsunaga, S., Shevlin, T.. Optimal exercise and the cost of granting employee stock options with a reload provision. Journal of Accounting Research,1998,36(2):231-255.
[61]Ingersoll, Jonathan E., Jr. Valuing reload options. Working paper, Yale School of Management,2001.
[62]Dybvig, P., Lowenstein, M.. Employee reload options:pricing, hedging gand optimal exercise. Review of Financial Studies,2003,12:145-171.
[63]Egloff, D., Farkas, W., Leippold, M.. American options with stopping time con-straints to executive stock options, Working paper, QuantAlea GmbH,2005.
[64]Oksendal, B.. Stochastic differential equations(5th edition), Springer-Verlag, Berlin,2000.
[65]Pham, H.. Continuous-time stochastic control and optimization with financial application, Springer-Verlag, Berlin,2009.
[66]Yong, J. M., Zhou, X. Y.. Stochastic control:Hamiltonian systems and HJB equations, Springer-Verlag, Berlin,1999.
[67]Jiang, L.. Mathematical modeling and methods of options pricing, World Scientific Publication Co. Pte. Ltd., Singapore,2005.
[68]Kim, I.. The analytic valuation of American options. Review of Financial Studies,1990,3:547-572.
[69]Jaillet, P., Lamberton, D., Lapeyre, B.. Variational inequalities and pricing of American options. Acta Appl. Math.,1990,21:263-289.
[70]Myneni, R.. The pricing of the American options. The Annals of Applied Proba-bility,1992,2(1):1-23.
[71]Dewynne, J. N., Howison, S. D., Rupf, I., Wilmott, P.. Some mathematical results in the pricing of American options. Europ. J. Appl. Math.,1993,4:381-398.
[72]Wilmott, P., Dewynne, J. N., Howison, S. D.. options pricing:mathematical models and computation, Oxford Financial Press, Oxford,1993.
[73]余王辉著.金融数学中的自由边界问题讲义.
[74]王颖.整体实施永久经理期权模型的若干问题,苏州大学硕士学位论文,2013.
[75]Friedman, A.. Variational principles and free-boundary problems, John Wiley&Sons Ltd., Chichester,1988.