基于分数布朗运动的几何亚式—再装股票期权的保险精算定价及经理激励分析
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摘要
经理激励薪酬制度是现代公司治理机制中的重要内容,如何设计一套行之有效的经理激励制度也是企业面临的现实问题。本文讨论了再装股票期权在再装日和到期日按B-S定价模型执行时所产生的经理激励缺陷,提出了将有效期内股价的几何平均值作为再装期权结算价格的思想,建立了几何亚式-再装股票期权的定价模型。并利用保险精算方法,从评估实际损失和相应概率分布的角度,研究了几何亚式-再装股票期权的价值构成,获得了基于分数布朗运动下几何亚式-再装股票期权的保险精算定价公式。并将再装一次的几何亚式-再装股票期权推广到多次再装的情形,主要考虑了两次几何亚式-再装股票期权的定价模型,得到了相应的收益结构和定价公式。
本文还通过数值模拟分析比较了传统再装期权与几何亚式-再装股票期权在经理激励中的作用。比较了在同等价值下两种股票期权的敏感性参数(Delta值和Vega值)。发现与传统再装股票期权价值相比,几何亚式-再装股票期权的Delta值普遍要大一些、而Vega值普遍要小一些,这说明几何亚式-再装股票期权能更有效地激励经理人和防止经理采取冒险的行为。且通过数值分析发现,对于几何亚式-再装股票期权的定价来说,考虑股价的长期相关性即Hurst系数是十分重要的,采用保险精算定价方法也是十分必要的。
The compensation system of manager’s incentive is very important content in corporate governance mechanism, how to design an effective incentive system is also the real problems of manager.In this paper,we have discussed the implementation of Executive Compensation defects which was generated by BS model on the reloading time and expiration date, and the idea that the geometric mean of stock price in period of validity as a stock option’s settlement price was proposed to solve the Executive Compensation defects, and the geometric Asian - reload stock option pricing model was setted. An actuarial methods is proposed in view of evaluating actual losses and corresponding probability distribution to quantitatively cheek the price composition of the geometric asian-reload stock option, thus developing an option prcing model to deduce further the formula under the hypothesis of underlying asset price driven by fractional Brownian motion. And the once of the geometric Asian - reload stock options is extended to the case of multiple re-installed, the two geometric Asian - reload stock option pricing model is considered, and the corresponding income structure and pricing formulas are obtained.
This article also compares Reload Stock Option with Geometric Asian-Reload Stock Option in the manager's role by numerical simulation . In the same value , these sensitivity parameters (Delta value and Vega value )are Compared. Found that the Compare with the value of traditional reload stock options, the Delta value of the geometric Asian - reload stock option is generally bigger, but Vega value of the geometric Asian - reload stock option is generally smaller, indicating that the geometric Asian - reload stock option can more effectively motivate managers and prevent managers from taking more risky actions. For the geometric Asian - reload stock option pricing, the fact is obtained by the numerical analysis that the long term correlation of stock price(Hurst coefficient) is very important to the value of the geometric Asian - reload stock option, and the use of actuarial pricing methodology is also essential for.
本文还通过数值模拟分析比较了传统再装期权与几何亚式-再装股票期权在经理激励中的作用。比较了在同等价值下两种股票期权的敏感性参数(Delta值和Vega值)。发现与传统再装股票期权价值相比,几何亚式-再装股票期权的Delta值普遍要大一些、而Vega值普遍要小一些,这说明几何亚式-再装股票期权能更有效地激励经理人和防止经理采取冒险的行为。且通过数值分析发现,对于几何亚式-再装股票期权的定价来说,考虑股价的长期相关性即Hurst系数是十分重要的,采用保险精算定价方法也是十分必要的。
The compensation system of manager’s incentive is very important content in corporate governance mechanism, how to design an effective incentive system is also the real problems of manager.In this paper,we have discussed the implementation of Executive Compensation defects which was generated by BS model on the reloading time and expiration date, and the idea that the geometric mean of stock price in period of validity as a stock option’s settlement price was proposed to solve the Executive Compensation defects, and the geometric Asian - reload stock option pricing model was setted. An actuarial methods is proposed in view of evaluating actual losses and corresponding probability distribution to quantitatively cheek the price composition of the geometric asian-reload stock option, thus developing an option prcing model to deduce further the formula under the hypothesis of underlying asset price driven by fractional Brownian motion. And the once of the geometric Asian - reload stock options is extended to the case of multiple re-installed, the two geometric Asian - reload stock option pricing model is considered, and the corresponding income structure and pricing formulas are obtained.
This article also compares Reload Stock Option with Geometric Asian-Reload Stock Option in the manager's role by numerical simulation . In the same value , these sensitivity parameters (Delta value and Vega value )are Compared. Found that the Compare with the value of traditional reload stock options, the Delta value of the geometric Asian - reload stock option is generally bigger, but Vega value of the geometric Asian - reload stock option is generally smaller, indicating that the geometric Asian - reload stock option can more effectively motivate managers and prevent managers from taking more risky actions. For the geometric Asian - reload stock option pricing, the fact is obtained by the numerical analysis that the long term correlation of stock price(Hurst coefficient) is very important to the value of the geometric Asian - reload stock option, and the use of actuarial pricing methodology is also essential for.
引文
[1] P.H.Dybvig and M.Loewenstein. Employee reload options:Pricing,hedging and optimal exercise[J]. Review of Financial Studies, 2003, 16(1):145-171.
[2] Johnson, S., Tian, Y. The value and incentive effects of nontraditional executive stockoption plans[J]. Journal of Financial Economics , 2000,57:3–34.
[3] Brickley,J.Bhagat,S.,Lease,R. The impact of long-range managerial compensation plans on the shareholder wealth[J]. Journal of Accounting and Ecomomics,1985,7:115-129.
[4] Gilson,S.Vetsuypens,M. CEO compensation in financially distressed firms an empirical analysis[J]. The Journal of Finance,1993,48:425-428.
[5] Hall,Brian J.and Leibman,Jeffrey B.“Are CEOs Really Paid Like Bureaucrats?”[J]. Quarterly Jounral of Economics,1998,113(3):653-691.
[6]许海峰,王凤荣,王德强.股票期权[M].北京:人民法院出版社,2005.
[7] Kulatilaka,N.,Mareus,A.J..Valuing employee stock options[J].Finaneial Ananlysts Journal,1994,50(6):46-56.
[8] Cuny,C.,Jorion,P.Valuing exeeutive stock options,with endogenous departure[J].Journal of Accounting and Eeonomics,1995,20:193-205.
[9] Lambert,R.A.,Larcker,D.F.,Verrecchia,D.F.Portfolio considerations in valuing executive compensation[J].Jounral of Accounting Research,1991,29:129-149.
[10] Carpenter,Jennifer.The exercise and valuation of executive stock options[J].Journal of Financial Economics,1998,48:127-158.
[11] Hall,Brian,and Thomas Knox.Managing Option Fragility,http://www.nber.org/,2002.
[12] Gay,C.Hard to Lose:’Reload’options promote stock ownership among executives; but critics say they’re a lot more costly than shareholders realize[J].Wall Street Journal, April 8,1999: R6.
[13] Hemmer, T., Matsunaga, S., Shevlin, T. Optimal exercise and the cost of granting employee stock options with a reload provision[J]. Journal of Accounting Research, 1998, 36:231–255.
[14] Huddart, S.. Employee stock options[J]. Journal of Accounting and Economics 1994,18: 207–231.
[15] Dybvig,Philip and Mark Loewenstein.Employee reload options: pricing, hedging, and optimal exercise [J].Review of Financial Studies ,2003,16:145-171.
[16]冯广波,刘再明,侯振挺.用二项式期权定价模型估价美式再装期权的价值[J].长沙铁道学院学报,2002,20(3):70-72.
[17]傅强,喻建龙.再装期权定价及其在经理激励中的应用[J].商业研究,2006,11(1):147-150.
[18]万建平,冯雅琴,冯文.再装股票期权的Esscher变换定价[J].经济数学,2007,24(2):139-146.
[19]刘坚,邓国和,杨向群.利率和股票价格遵循O-U过程的再装期权定价[J].工程数学学报,2007,24(2):238-241.
[20]王献东,杜雪樵.跳-扩散模型下的再装期权定价[J].经济数学,2007,24(3):275-281.
[21]张慧,陈晓兰,聂秀山.不确定环境下再装股票期权的稳健定价模型[J].中国管理科学,2008,16(1):25-31.
[22] M.DAI and Y.K.KWOK. Valuing employee reload options under time vesting requirement[J]. Quantitative Finance,2005, 5(1):61-69.
[23]李超杰,何建敏.再装股票期权执行价格最低水平的决定[J].系统工程理论方法应用,2004,13(6):509-511.
[24]傅强,郜琳琳.具有上下障碍的再装期权定价模型与计算[J].重庆大学学报(自然科学版),2007,30(4):145-147.
[25] A.C. B′elanger, P.A. Forsyth. Infinite reload options: Pricing and analysis[J].Journal of. Computational and Applied Mathematics, 2008, 222(1):54–81.
[26] Min Dai, Yue Kuen Kwok. Optimal multiple stopping models of reload options and shout options[J].Journal of Economic Dynamics & Control ,2008,32(7):2269–2290.
[27]陈松男.金融工程学[M].上海:复旦大学出版社,2002.
[28]胡必锦等.鞅分析及其应用[M].武汉:武汉科技大学出版社,2001.
[29]严加安,彭实戈,方诗赞,吴黎明.随机分析选讲[M].北京:科学出版社,1997.
[30]王梓坤.随机过程论[M].北京:科学出版社,1978.
[31]黄志远.随机分析学基础[M].北京:科学出版社,2001.
[32] Black, F, Scholes, M. The pricing of options and corporate liabilities[J].Journal of Political Economy,1973,81(3):637-654.
[33] T.E.Duncan,Y.Hu,Pasik-Duncan.Stochatic calculus for factional Brownian motion[J].Theory SIAMJ.Control Opti,2000,38:582-612.
[34] Yaozhong Hu, Bernt ?ksendal. Fractional white calculus and application to finance[J].Pure Review, 1997,10:422-437.
[35] E.Alos,O.Mazet,D.Nualart. Stochastic calculus with respect to fractional Brown motion with Hurst parameter lesser than 1/2[J].Stochastic Process Appl, 2000,86:121-139.
[36] Elliott,R.J,Van Der Hoek. A general fractional White Noise theory and applications to finance[J].Mathematical Finance, 2003, 2(13):301-33.
[37] Yaozhong Hu, Bernt ?ksendal. Fractional white noise calculus and applications to finance[J]. Inf. Dim. Anal, Quantum Probab ,2003, 1(6):1-32.
[38] Bachelier L..Theorie de la speculation[J].English Translation in the Random Character of Stock Market Prices,1900,23(4):153-165.
[39] Bladt,M.,Rydberg,H.T.An actuarial approach to option pricing under the physical measure and without market assumptions[J].Insurance:Mathematics and conomics.1998,22(1):65-73.
[2] Johnson, S., Tian, Y. The value and incentive effects of nontraditional executive stockoption plans[J]. Journal of Financial Economics , 2000,57:3–34.
[3] Brickley,J.Bhagat,S.,Lease,R. The impact of long-range managerial compensation plans on the shareholder wealth[J]. Journal of Accounting and Ecomomics,1985,7:115-129.
[4] Gilson,S.Vetsuypens,M. CEO compensation in financially distressed firms an empirical analysis[J]. The Journal of Finance,1993,48:425-428.
[5] Hall,Brian J.and Leibman,Jeffrey B.“Are CEOs Really Paid Like Bureaucrats?”[J]. Quarterly Jounral of Economics,1998,113(3):653-691.
[6]许海峰,王凤荣,王德强.股票期权[M].北京:人民法院出版社,2005.
[7] Kulatilaka,N.,Mareus,A.J..Valuing employee stock options[J].Finaneial Ananlysts Journal,1994,50(6):46-56.
[8] Cuny,C.,Jorion,P.Valuing exeeutive stock options,with endogenous departure[J].Journal of Accounting and Eeonomics,1995,20:193-205.
[9] Lambert,R.A.,Larcker,D.F.,Verrecchia,D.F.Portfolio considerations in valuing executive compensation[J].Jounral of Accounting Research,1991,29:129-149.
[10] Carpenter,Jennifer.The exercise and valuation of executive stock options[J].Journal of Financial Economics,1998,48:127-158.
[11] Hall,Brian,and Thomas Knox.Managing Option Fragility,http://www.nber.org/,2002.
[12] Gay,C.Hard to Lose:’Reload’options promote stock ownership among executives; but critics say they’re a lot more costly than shareholders realize[J].Wall Street Journal, April 8,1999: R6.
[13] Hemmer, T., Matsunaga, S., Shevlin, T. Optimal exercise and the cost of granting employee stock options with a reload provision[J]. Journal of Accounting Research, 1998, 36:231–255.
[14] Huddart, S.. Employee stock options[J]. Journal of Accounting and Economics 1994,18: 207–231.
[15] Dybvig,Philip and Mark Loewenstein.Employee reload options: pricing, hedging, and optimal exercise [J].Review of Financial Studies ,2003,16:145-171.
[16]冯广波,刘再明,侯振挺.用二项式期权定价模型估价美式再装期权的价值[J].长沙铁道学院学报,2002,20(3):70-72.
[17]傅强,喻建龙.再装期权定价及其在经理激励中的应用[J].商业研究,2006,11(1):147-150.
[18]万建平,冯雅琴,冯文.再装股票期权的Esscher变换定价[J].经济数学,2007,24(2):139-146.
[19]刘坚,邓国和,杨向群.利率和股票价格遵循O-U过程的再装期权定价[J].工程数学学报,2007,24(2):238-241.
[20]王献东,杜雪樵.跳-扩散模型下的再装期权定价[J].经济数学,2007,24(3):275-281.
[21]张慧,陈晓兰,聂秀山.不确定环境下再装股票期权的稳健定价模型[J].中国管理科学,2008,16(1):25-31.
[22] M.DAI and Y.K.KWOK. Valuing employee reload options under time vesting requirement[J]. Quantitative Finance,2005, 5(1):61-69.
[23]李超杰,何建敏.再装股票期权执行价格最低水平的决定[J].系统工程理论方法应用,2004,13(6):509-511.
[24]傅强,郜琳琳.具有上下障碍的再装期权定价模型与计算[J].重庆大学学报(自然科学版),2007,30(4):145-147.
[25] A.C. B′elanger, P.A. Forsyth. Infinite reload options: Pricing and analysis[J].Journal of. Computational and Applied Mathematics, 2008, 222(1):54–81.
[26] Min Dai, Yue Kuen Kwok. Optimal multiple stopping models of reload options and shout options[J].Journal of Economic Dynamics & Control ,2008,32(7):2269–2290.
[27]陈松男.金融工程学[M].上海:复旦大学出版社,2002.
[28]胡必锦等.鞅分析及其应用[M].武汉:武汉科技大学出版社,2001.
[29]严加安,彭实戈,方诗赞,吴黎明.随机分析选讲[M].北京:科学出版社,1997.
[30]王梓坤.随机过程论[M].北京:科学出版社,1978.
[31]黄志远.随机分析学基础[M].北京:科学出版社,2001.
[32] Black, F, Scholes, M. The pricing of options and corporate liabilities[J].Journal of Political Economy,1973,81(3):637-654.
[33] T.E.Duncan,Y.Hu,Pasik-Duncan.Stochatic calculus for factional Brownian motion[J].Theory SIAMJ.Control Opti,2000,38:582-612.
[34] Yaozhong Hu, Bernt ?ksendal. Fractional white calculus and application to finance[J].Pure Review, 1997,10:422-437.
[35] E.Alos,O.Mazet,D.Nualart. Stochastic calculus with respect to fractional Brown motion with Hurst parameter lesser than 1/2[J].Stochastic Process Appl, 2000,86:121-139.
[36] Elliott,R.J,Van Der Hoek. A general fractional White Noise theory and applications to finance[J].Mathematical Finance, 2003, 2(13):301-33.
[37] Yaozhong Hu, Bernt ?ksendal. Fractional white noise calculus and applications to finance[J]. Inf. Dim. Anal, Quantum Probab ,2003, 1(6):1-32.
[38] Bachelier L..Theorie de la speculation[J].English Translation in the Random Character of Stock Market Prices,1900,23(4):153-165.
[39] Bladt,M.,Rydberg,H.T.An actuarial approach to option pricing under the physical measure and without market assumptions[J].Insurance:Mathematics and conomics.1998,22(1):65-73.