美式期权定价的数值方法
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摘要
期权是最重要的金融衍生工具之一,它作为一种金融创新工具,在防范和规避风险以及投机中起着非常重要的作用。而如何通过合理的数学模型来确定期权的价格就成了投资者应用期权规避金融风险的关键性问题,所以在金融领域中,期权的定价问题成为理论和应用研究的一个重要领域。对于欧式期权,Black & Sholes早已给出解析形式的定价公式。然而,对于美式期权的定价,并不存在这样的解析公式,也无法求得精确解。而现实世界中,交易所中交易的大多数期权为美式期权。因此,发展各种计算美式期权价格的数值方法具有重要的理论和实际意义。美式期权定价问题的数学模型一般可归结为自由边值问题。
本文绪论部分对金融衍生工具及其定价理论作了概括性的回顾;第二部分阐述了衍生证券价格所服从的Black—Scholes偏微分方程;第三部分对传统的二叉树、三叉树模型进行一些改进。即在适当的区域加密树图网格,而其他的区域不予变化,以消除粗糙树图网格中存在的“非线性误差”问题,而更能反映实际情况,使得只增加很少的计算量,就可以达到原本需要更高密度树图才能达到的计算效果。第四部分基于B-S微分方程,对有限差分方法中的一些参数设置方法进行改进,并将传统的内含有限差分法和外推有限差分法进行一定的结合,获得更好的结果。第五部分介绍了多项标的资产欧式期权所派生出来的标准形式,并使用基于对流扩散微分方程的基本解方法(MFS方法)求解派生出来的偏微分方程标准形式。在考虑了美式期权的特点与MFS方法的特性之后,将MFS方法推广到了美式期权的求解。第三、四、五每部分均采用了数值算例验证了该章方法的有效性和实用性。
Option is one of the core tools of financial derivatives, which plays an important role in the effective management of risk and speculation. Risk management is depended on the right evaluation of option in a certain extent .The critical thing is how to value its fair price. A wide variety of the options traded in exchanges are American options. It is thus important to find appropriate ways to price American options. Thus, unlike European options, no explicit closed-form formulas have been found for American options, approximation methods have to be used in practice. Generally the classic Black-Scholes model for an American option leads to a free boundary problem with a degenerate partial differential operator.
The part of this text introduction has done the reviewing of generality to the financial derivative and pricing theory. At the second chapter, the article expatiates the instauration of the Black-Scholes Differential Equation. At the third chapter of the article, nonlinearity error has been sharply reduced by grafting one or more small sections of fine high-resolution lattice onto the traditional Binomial Tree and Trinomial Tree with coarser time and price steps on the proper area. In the fourth chapter, the traditional differential scheme, based on the B-S differential equation, has been improved by changing some parameters presented, and collecting their both advantages. The following chapter presents the procedure for deriving canonical forms of European options of multi-assets is introduced. Then, the MFS based on the convection-diffusion fundamental solution is proposed to solve the resulted canonical forms. The MFS method has been expanded to the American option of single asset after considering the characteristics of MFS and American option. Numerical example were showed the efficiency and practicability of the algorithm at the end of the chapter 3, 4 & 5.
本文绪论部分对金融衍生工具及其定价理论作了概括性的回顾;第二部分阐述了衍生证券价格所服从的Black—Scholes偏微分方程;第三部分对传统的二叉树、三叉树模型进行一些改进。即在适当的区域加密树图网格,而其他的区域不予变化,以消除粗糙树图网格中存在的“非线性误差”问题,而更能反映实际情况,使得只增加很少的计算量,就可以达到原本需要更高密度树图才能达到的计算效果。第四部分基于B-S微分方程,对有限差分方法中的一些参数设置方法进行改进,并将传统的内含有限差分法和外推有限差分法进行一定的结合,获得更好的结果。第五部分介绍了多项标的资产欧式期权所派生出来的标准形式,并使用基于对流扩散微分方程的基本解方法(MFS方法)求解派生出来的偏微分方程标准形式。在考虑了美式期权的特点与MFS方法的特性之后,将MFS方法推广到了美式期权的求解。第三、四、五每部分均采用了数值算例验证了该章方法的有效性和实用性。
Option is one of the core tools of financial derivatives, which plays an important role in the effective management of risk and speculation. Risk management is depended on the right evaluation of option in a certain extent .The critical thing is how to value its fair price. A wide variety of the options traded in exchanges are American options. It is thus important to find appropriate ways to price American options. Thus, unlike European options, no explicit closed-form formulas have been found for American options, approximation methods have to be used in practice. Generally the classic Black-Scholes model for an American option leads to a free boundary problem with a degenerate partial differential operator.
The part of this text introduction has done the reviewing of generality to the financial derivative and pricing theory. At the second chapter, the article expatiates the instauration of the Black-Scholes Differential Equation. At the third chapter of the article, nonlinearity error has been sharply reduced by grafting one or more small sections of fine high-resolution lattice onto the traditional Binomial Tree and Trinomial Tree with coarser time and price steps on the proper area. In the fourth chapter, the traditional differential scheme, based on the B-S differential equation, has been improved by changing some parameters presented, and collecting their both advantages. The following chapter presents the procedure for deriving canonical forms of European options of multi-assets is introduced. Then, the MFS based on the convection-diffusion fundamental solution is proposed to solve the resulted canonical forms. The MFS method has been expanded to the American option of single asset after considering the characteristics of MFS and American option. Numerical example were showed the efficiency and practicability of the algorithm at the end of the chapter 3, 4 & 5.
引文
[1]郑振龙.金融工程[M].北京:高等教育出版社,2003:121~128,174.
[2] John C.Hull.张陶伟(译).期权、期货和其它衍生品[M].北京:华夏出版社,1999:332~342.
[3]李玉立,金朝嵩.美式看跌期权的差分格式[J].重庆建筑大学学报,2004,26(4),100~114.
[4]张铁.美式期权定价问题的数值方法[J].应用数学学报,2002,25(1)113-122
[5]刘棠,张盘铭.美式期权定价中非局部问题的有限元方法[J].数学的实践与认识,2003,33(11)72-81
[6]刘棠,张盘铭.期权定价问题的数值方法[J].系统科学与数学,2004,24(1)10-16
[7]马俊海,张维.金融衍生工具丁家忠门特卡罗方法的近期应用分析[J].管理工程学报,2000,14(2)47-50
[8]张敏,王键.Monte-Carlo模拟法在期权定价中的应用[J].湖南商学院学报,2003,10(4)76-78
[9]李莉英等.一种基于支付红利股票的美式期权定价方法[J].重庆交通学院学报,2005,24(2)148-150
[10]孙春燕,陈耀辉.基于最小二乘法的美式期权定价的最优停时分析[J].系统工程,2005,21(6)104-108
[11]张天永,彭隆泽.期权价格的拟Monte Carlo仿真计算[J].重庆建筑大学学报,2005,27(4)111-114
[12]孙春燕,陈耀辉,李楚霖.对美式期权定价中一类蒙特卡洛过程收敛速率的研究[J].系统工程,2004,22(6)95-98
[13]李东,金朝嵩.美式看跌期权定价中的小波方法[J].经济数学,2003,20(4)25-30
[14]张铁.一个新型的期权定价二叉树参数模型[J].系统工程与实践,2000,(11)90-93
[15]白丹宇,吕方,陈彩霞.美式看跌期权定价的新型树图参数模型[J].2006,33(3)213-215
[16] Broadie M, Glasserman P. Pricing American style Securities Using Simulation [J]. Journal of Economics Dynamics and Control, 1997, 21: 1323-1352.
[17] Boyle P, Evinne J, Gibbs S. Numerical Evaluation of Multivariate Contingent Claims [J]. Review of Financial Analysis, 1989, 2: 241- 250.
[18] Breen R. The Accelerated Binomial Option Pricing Model [J]. J.F.Q.A ,1991,26:153～164.
[19] Bernnan M, Schwartz E. Finite Difference Methods and Processes Arising in the Pricing of Contingent Claims: A Synthesis [J]. J.F.Q.A, 1978, 13:461-474.
[20] Broadie M, Detemple J. American Option Valuation: New Bound, Approximations, and a Comparison existing Method [J]. Review of Financial Studies, 1996,9: 1211-1250.
[21] Carr P, Faguet D. Fast Accurate Valuation of American Options[C]. Working Paper, Cornell University, 1994.
[22] Carr P, Jarrow R, Mynent R. Alternative Characterization of American Put Option [J]. Mathematical Finance, 1992, 2: 87-106.
[23] Cox J. Ross S, Rubinstein M. Option Pricing: A Simplified Approach [J]. Journal of Financial Economics, 1979, 7: 229-263.
[24] Grant D, Vora G, W eek D. Path2Dependent Option: Extending the Monte Carlo Simulation Approach [J]. Management Science, 1997, 43 (1): 1589-1602.
[25] Kamrad B, Ritchken P. Multinomial Approximating Models for Options with k State Variables [J]. Management Science, 1991, 37: 1640-1652.
[26] Marchuk G, Shaidurov V. Difference Method and Their Extrapolation [M]. New York: Springer, 1983.
[27] Rubinstein R. Discrete Event System [M]. Wiley, New York, 1993.
[28] Tilly J. Valuing American Option in a Path Simulation Model [J]. Trans of the Society of Actuaries, 1993, 45: 83-104.
[29] John C.Hull.Options, Futures, and Other Derivatives (Fourth Edition)[M].Tsinghua University Press,Peking,2001.
[30] Tilly J. Valuing American Option in a Path Simulation Model [J]. Trans of the Society of Actuaries, 1993, 45: 83-104.
[31]金浩,刘新平,朱文武.三种数值方法在期权定价中的定量研究[J].云南师范大学学报,2006,26(2),9-13
[32]梅立泉,李瑞,李智.三元期权定价问题的偏微分方程数值解[J].西安交通大学学报,2006,40(4),484-487
[33]蒙彦宇,谭硕,孙威,郭鹏飞,崔阳.有限元方法的重要补充[J].辽宁工学院学报,2006,26(3),178-182
[34]龙述尧,陈胜铭.利用点插值函数作为试函数的最小二乘配点法[J].湖南大学学报(自然科学版),2005,32(4),29-33
[35]王燕昌,李进.无单元方法的研究进展[J].宁夏大学学报(自然科学版),2003,24(1),42-46
[36]张雄,宋康祖,陆明万.无网格法研究进展及其应用[J].计算力学学报,2003,20(6),730-742
[37]李九红,程玉民.无网格方法的研究进展与展望[J].力学季刊,2006,27(1),143-152
[38]叶翔,丁成辉.关于无网格法的几点研究[J].南昌大学学报·工科版,2005,27(3),38-40
[39]张延军,王恩志.新型无网格法(IMLS方形域法)的研究及其应用[J].岩土工程学报,2006,28(7),886-890
[40]刘海龙,吴冲锋.期权定价方法综述[J].管理科学学报, 2002, 5(2): 67-72
[41]刘海龙,吴冲锋等.现代金融理论的进展综述[J].系统工程理论与实践,2001,1(1):14~40.
[42] Friedman A. Partial Differential Equations of Parabolic Type[J].Robert E. Krieger Publishing, Huntington, NY, 1983.
[43] Geske R. and Johnson H.E. The American Put Options Valued Analytical[J]. Fin., 1984, 39: 1511-1524.
[44] Geske R. and Shastri K. Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques[J]. Fin. Quan. Anal., 1985, 20: 45-71.
[45]丁丽娟.数值计算方法[C].第一版.北京理工大学出版社. 1997.
[46]李立康.微分方程数值解法[C].上海:复旦大学出版社, 1999
[47] D.E. Koulouriotis, I.E. Diakoulakis, Development of dynamic cognitive networks as complex systems approximators: validation in financial time series[J].Applied Soft Computing. 2005, 5 : 157–179
[48] Schwartz E. S. The Valuation of Warrants: Implementing a New Approach[J].Fin. Econ., 1979, 4: 79-93
[49] Wilmott P., Dewynne J. and Howlson S. Option Pricing: Mathematical Models and Computation[J].Oxford Financial Press, Oxford, UK, 1993.
[50] Stephen Figlewski, Bin Gao. The adaptive mesh model: a new approach to efficient option pricing[J].Journal of Financial Economics. 1999, 53: 313-351
[51] Ulrika Patterson, Elisabeth Larsson .Improved radial basis function methods for multi-dimensional option pricing[J]. Journal of Computational and Applied Mathematics.2006.
[52] Yumi Goto, Zhai Fe. Options valuation by using radial basis function approximation. Engineering Analysis with Boundary Elements[J].2007,31:836–843
[53] Patrick Smolinski, Timothy Palmer .Procedures for multi-time step integration of element-free Galerkin methods for diffusion problems. [J].Computers and Structures. 2000,77: 171-183
[54] C.C. Tsai, D.L.Young .The method of fundamental solutions for solving options pricing models[J].Applied Mathematics and Computation .2006, 181 : 390–401
[55]姜礼尚.期权定价的数学模型和方法[M].北京.高等教育出版社. 2003.
[56] Y. C. Hon. Option Pricing: A Quasi-Radial Basis Functions Method for American Options Pricing[J].Computers and Mathematics with Applications 2002,43:513-524.
[57] Zongmin Wu,Y. C. Hon. Convergence error estimate in solving free boundary diffusionproblem by radial basis functions method. Engineering Analysis with Boundary Elements [J].2003,27 : 73–79.
[58] Y.C. Hon, X.Z. Mao, A radial basis function method for solving options pricing model [J].Financial Engineering 8 (1999) 31–49.
[59] M.D. Marcozzi, S. Choi, C.S. Chen, On the use of boundary conditions for variation formulations arising in financial mathematics[J].Applied Mathematics and Computation 124 (2001) 197–214.
[60] R.B. Holmes, On random correlation matrices[J]. SIAM Journal on Matrix Analysis and Applications 12 (1991) 239–272.
[2] John C.Hull.张陶伟(译).期权、期货和其它衍生品[M].北京:华夏出版社,1999:332~342.
[3]李玉立,金朝嵩.美式看跌期权的差分格式[J].重庆建筑大学学报,2004,26(4),100~114.
[4]张铁.美式期权定价问题的数值方法[J].应用数学学报,2002,25(1)113-122
[5]刘棠,张盘铭.美式期权定价中非局部问题的有限元方法[J].数学的实践与认识,2003,33(11)72-81
[6]刘棠,张盘铭.期权定价问题的数值方法[J].系统科学与数学,2004,24(1)10-16
[7]马俊海,张维.金融衍生工具丁家忠门特卡罗方法的近期应用分析[J].管理工程学报,2000,14(2)47-50
[8]张敏,王键.Monte-Carlo模拟法在期权定价中的应用[J].湖南商学院学报,2003,10(4)76-78
[9]李莉英等.一种基于支付红利股票的美式期权定价方法[J].重庆交通学院学报,2005,24(2)148-150
[10]孙春燕,陈耀辉.基于最小二乘法的美式期权定价的最优停时分析[J].系统工程,2005,21(6)104-108
[11]张天永,彭隆泽.期权价格的拟Monte Carlo仿真计算[J].重庆建筑大学学报,2005,27(4)111-114
[12]孙春燕,陈耀辉,李楚霖.对美式期权定价中一类蒙特卡洛过程收敛速率的研究[J].系统工程,2004,22(6)95-98
[13]李东,金朝嵩.美式看跌期权定价中的小波方法[J].经济数学,2003,20(4)25-30
[14]张铁.一个新型的期权定价二叉树参数模型[J].系统工程与实践,2000,(11)90-93
[15]白丹宇,吕方,陈彩霞.美式看跌期权定价的新型树图参数模型[J].2006,33(3)213-215
[16] Broadie M, Glasserman P. Pricing American style Securities Using Simulation [J]. Journal of Economics Dynamics and Control, 1997, 21: 1323-1352.
[17] Boyle P, Evinne J, Gibbs S. Numerical Evaluation of Multivariate Contingent Claims [J]. Review of Financial Analysis, 1989, 2: 241- 250.
[18] Breen R. The Accelerated Binomial Option Pricing Model [J]. J.F.Q.A ,1991,26:153～164.
[19] Bernnan M, Schwartz E. Finite Difference Methods and Processes Arising in the Pricing of Contingent Claims: A Synthesis [J]. J.F.Q.A, 1978, 13:461-474.
[20] Broadie M, Detemple J. American Option Valuation: New Bound, Approximations, and a Comparison existing Method [J]. Review of Financial Studies, 1996,9: 1211-1250.
[21] Carr P, Faguet D. Fast Accurate Valuation of American Options[C]. Working Paper, Cornell University, 1994.
[22] Carr P, Jarrow R, Mynent R. Alternative Characterization of American Put Option [J]. Mathematical Finance, 1992, 2: 87-106.
[23] Cox J. Ross S, Rubinstein M. Option Pricing: A Simplified Approach [J]. Journal of Financial Economics, 1979, 7: 229-263.
[24] Grant D, Vora G, W eek D. Path2Dependent Option: Extending the Monte Carlo Simulation Approach [J]. Management Science, 1997, 43 (1): 1589-1602.
[25] Kamrad B, Ritchken P. Multinomial Approximating Models for Options with k State Variables [J]. Management Science, 1991, 37: 1640-1652.
[26] Marchuk G, Shaidurov V. Difference Method and Their Extrapolation [M]. New York: Springer, 1983.
[27] Rubinstein R. Discrete Event System [M]. Wiley, New York, 1993.
[28] Tilly J. Valuing American Option in a Path Simulation Model [J]. Trans of the Society of Actuaries, 1993, 45: 83-104.
[29] John C.Hull.Options, Futures, and Other Derivatives (Fourth Edition)[M].Tsinghua University Press,Peking,2001.
[30] Tilly J. Valuing American Option in a Path Simulation Model [J]. Trans of the Society of Actuaries, 1993, 45: 83-104.
[31]金浩,刘新平,朱文武.三种数值方法在期权定价中的定量研究[J].云南师范大学学报,2006,26(2),9-13
[32]梅立泉,李瑞,李智.三元期权定价问题的偏微分方程数值解[J].西安交通大学学报,2006,40(4),484-487
[33]蒙彦宇,谭硕,孙威,郭鹏飞,崔阳.有限元方法的重要补充[J].辽宁工学院学报,2006,26(3),178-182
[34]龙述尧,陈胜铭.利用点插值函数作为试函数的最小二乘配点法[J].湖南大学学报(自然科学版),2005,32(4),29-33
[35]王燕昌,李进.无单元方法的研究进展[J].宁夏大学学报(自然科学版),2003,24(1),42-46
[36]张雄,宋康祖,陆明万.无网格法研究进展及其应用[J].计算力学学报,2003,20(6),730-742
[37]李九红,程玉民.无网格方法的研究进展与展望[J].力学季刊,2006,27(1),143-152
[38]叶翔,丁成辉.关于无网格法的几点研究[J].南昌大学学报·工科版,2005,27(3),38-40
[39]张延军,王恩志.新型无网格法(IMLS方形域法)的研究及其应用[J].岩土工程学报,2006,28(7),886-890
[40]刘海龙,吴冲锋.期权定价方法综述[J].管理科学学报, 2002, 5(2): 67-72
[41]刘海龙,吴冲锋等.现代金融理论的进展综述[J].系统工程理论与实践,2001,1(1):14~40.
[42] Friedman A. Partial Differential Equations of Parabolic Type[J].Robert E. Krieger Publishing, Huntington, NY, 1983.
[43] Geske R. and Johnson H.E. The American Put Options Valued Analytical[J]. Fin., 1984, 39: 1511-1524.
[44] Geske R. and Shastri K. Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques[J]. Fin. Quan. Anal., 1985, 20: 45-71.
[45]丁丽娟.数值计算方法[C].第一版.北京理工大学出版社. 1997.
[46]李立康.微分方程数值解法[C].上海:复旦大学出版社, 1999
[47] D.E. Koulouriotis, I.E. Diakoulakis, Development of dynamic cognitive networks as complex systems approximators: validation in financial time series[J].Applied Soft Computing. 2005, 5 : 157–179
[48] Schwartz E. S. The Valuation of Warrants: Implementing a New Approach[J].Fin. Econ., 1979, 4: 79-93
[49] Wilmott P., Dewynne J. and Howlson S. Option Pricing: Mathematical Models and Computation[J].Oxford Financial Press, Oxford, UK, 1993.
[50] Stephen Figlewski, Bin Gao. The adaptive mesh model: a new approach to efficient option pricing[J].Journal of Financial Economics. 1999, 53: 313-351
[51] Ulrika Patterson, Elisabeth Larsson .Improved radial basis function methods for multi-dimensional option pricing[J]. Journal of Computational and Applied Mathematics.2006.
[52] Yumi Goto, Zhai Fe. Options valuation by using radial basis function approximation. Engineering Analysis with Boundary Elements[J].2007,31:836–843
[53] Patrick Smolinski, Timothy Palmer .Procedures for multi-time step integration of element-free Galerkin methods for diffusion problems. [J].Computers and Structures. 2000,77: 171-183
[54] C.C. Tsai, D.L.Young .The method of fundamental solutions for solving options pricing models[J].Applied Mathematics and Computation .2006, 181 : 390–401
[55]姜礼尚.期权定价的数学模型和方法[M].北京.高等教育出版社. 2003.
[56] Y. C. Hon. Option Pricing: A Quasi-Radial Basis Functions Method for American Options Pricing[J].Computers and Mathematics with Applications 2002,43:513-524.
[57] Zongmin Wu,Y. C. Hon. Convergence error estimate in solving free boundary diffusionproblem by radial basis functions method. Engineering Analysis with Boundary Elements [J].2003,27 : 73–79.
[58] Y.C. Hon, X.Z. Mao, A radial basis function method for solving options pricing model [J].Financial Engineering 8 (1999) 31–49.
[59] M.D. Marcozzi, S. Choi, C.S. Chen, On the use of boundary conditions for variation formulations arising in financial mathematics[J].Applied Mathematics and Computation 124 (2001) 197–214.
[60] R.B. Holmes, On random correlation matrices[J]. SIAM Journal on Matrix Analysis and Applications 12 (1991) 239–272.
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