基于奇点分离法的美式期权定价方法研究
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摘要
Fischer Black和Myron Scholes在1973年推导出了基于无红利支付股票期权的Black-Scholes期权定价方程(B-S方程),并精确地给出了欧式期权定价的解析表达式,但对于美式期权,由于存在提前执行的可能性,所以只能找到方程近似解析解或是数值解。有限元差分方法是美式期权定价中较为常见的数值计算方法,它对离散后的方程在有限的区域进行差分。由于差分区域的不规则性,在差分区域的边界往往存在相对较大的截断误差。本文中利用美式期权和欧式期权的关系,通过适当的变量代换,简化含自由边界的抛物线偏微分方程的美式期权定价模型,使得在数值计算中进行有限差分的区域变为规则形状,消除初始条件的奇点,进而减少数值计算的截断误差。
B-S方程是期权定价的基础。对于标的物支付连续红利的欧式期权,在假设标的物价格服从几何布朗运动的前提下,我们详细推导了B-S方程。基于B-S方程,我们分析了美式期权的定价问题,其本质是一个障碍问题。以美式看跌期权为例,对于任意一个到期日之前的时间,存在一个相应的标的物价格,在此价格的左侧,美式看跌期权的价值始终会等于相应的价值函数,而当标的物价格高于此价格时,期权价格会大于相应的价值函数。这个临界的标的物价格是依赖于时间的,这就导致了一个自由边界,从而使得难以找到方程的精确的解析解,寻找美式期权数值解便成为研究其解特性的主要方法。我们详细介绍了奇点分离法的主要思想,分析了它在数值计算方法中的应用。由于欧式期权可以准确得到,利用美式和欧式期权的关系,我们考虑两者之间的差而不是直接求解美式期权的价格,这样能减少相对误差。由于美式期权定价方程中的边界条不够光滑,我们通过引入适当的变量代换,使得方程的边界条件足够的光滑,进而减少数值计算的截断误差。二叉树法及投影SOR法是两种常用的美式期权定价的数值方法,通过数值实验与并这的两种数值方法进行比较,发现奇点分离法能明显改善计算精度和速度。
In 1973, based on non-dividend-paying stocks option, Fischer Black and Myron Scholes derived Black-Scholes option pricing formula and gave a precise analytical solution for European option. However, due to the possibility of early exercise of American option, we could only find the approximated analytical solution or, mostly the numerical solution in stead of analytical one. Finite difference method which solves discrete equation on a finite area, is one of the popular methods in American option pricing. The irregularity of the difference area may lead to relatively big truncation error. By making use of the relation between the American option and the corresponding European option and introducing some appropriate variable transformations, in this paper, we simplify the pricing model for American option which involves a parabolic equation with free boundary condition. These simplifications convert the original area where to take difference into a regular shape and eliminate the singularity in the initial condition. Moreover, the separation of singularity decreases the truncation error of numerical computation.
B-S equation is the pillar of the option pricing. In this paper, assuming that the price of the stock follows the geometric Brownian motion, we deduce the B-S equation in details. Based on this equation, we study the pricing model for American option which essentially is an obstacle problem. Take the American put option as an example, for any particular time before expiration, there exists a corresponding price for which the value of the American put option equals the corresponding payoff function if the real price is less than the fixed price otherwise the value of the put option is greater than the payoff function. This particular price depends on time, which leads to a so called free boundary. Due to the uncertainty of this boundary, it is hard to find the analytical solution of the American option. Therefore, numerical methods become the main approaches to study the pricing of American option. The fundamental idea of singularity separation method is discussed in details. Because the Europe option price could be obtained precisely, we consider the difference of the price of American and the corresponding Europe option instead of the price of the American option itself, which could decrease the relative error. Moreover, the bound of the equation of the American option pricing is not smooth enough. Therefore, we introduce some appropriate variable transformation in order to control the truncation error. The binomial method and Projected SOR method are two normal method in American option pricing, we compare the singularity separation method with the these two and find that singularity separation method efficiently improves the accuracy and the speed of computation.
B-S方程是期权定价的基础。对于标的物支付连续红利的欧式期权,在假设标的物价格服从几何布朗运动的前提下,我们详细推导了B-S方程。基于B-S方程,我们分析了美式期权的定价问题,其本质是一个障碍问题。以美式看跌期权为例,对于任意一个到期日之前的时间,存在一个相应的标的物价格,在此价格的左侧,美式看跌期权的价值始终会等于相应的价值函数,而当标的物价格高于此价格时,期权价格会大于相应的价值函数。这个临界的标的物价格是依赖于时间的,这就导致了一个自由边界,从而使得难以找到方程的精确的解析解,寻找美式期权数值解便成为研究其解特性的主要方法。我们详细介绍了奇点分离法的主要思想,分析了它在数值计算方法中的应用。由于欧式期权可以准确得到,利用美式和欧式期权的关系,我们考虑两者之间的差而不是直接求解美式期权的价格,这样能减少相对误差。由于美式期权定价方程中的边界条不够光滑,我们通过引入适当的变量代换,使得方程的边界条件足够的光滑,进而减少数值计算的截断误差。二叉树法及投影SOR法是两种常用的美式期权定价的数值方法,通过数值实验与并这的两种数值方法进行比较,发现奇点分离法能明显改善计算精度和速度。
In 1973, based on non-dividend-paying stocks option, Fischer Black and Myron Scholes derived Black-Scholes option pricing formula and gave a precise analytical solution for European option. However, due to the possibility of early exercise of American option, we could only find the approximated analytical solution or, mostly the numerical solution in stead of analytical one. Finite difference method which solves discrete equation on a finite area, is one of the popular methods in American option pricing. The irregularity of the difference area may lead to relatively big truncation error. By making use of the relation between the American option and the corresponding European option and introducing some appropriate variable transformations, in this paper, we simplify the pricing model for American option which involves a parabolic equation with free boundary condition. These simplifications convert the original area where to take difference into a regular shape and eliminate the singularity in the initial condition. Moreover, the separation of singularity decreases the truncation error of numerical computation.
B-S equation is the pillar of the option pricing. In this paper, assuming that the price of the stock follows the geometric Brownian motion, we deduce the B-S equation in details. Based on this equation, we study the pricing model for American option which essentially is an obstacle problem. Take the American put option as an example, for any particular time before expiration, there exists a corresponding price for which the value of the American put option equals the corresponding payoff function if the real price is less than the fixed price otherwise the value of the put option is greater than the payoff function. This particular price depends on time, which leads to a so called free boundary. Due to the uncertainty of this boundary, it is hard to find the analytical solution of the American option. Therefore, numerical methods become the main approaches to study the pricing of American option. The fundamental idea of singularity separation method is discussed in details. Because the Europe option price could be obtained precisely, we consider the difference of the price of American and the corresponding Europe option instead of the price of the American option itself, which could decrease the relative error. Moreover, the bound of the equation of the American option pricing is not smooth enough. Therefore, we introduce some appropriate variable transformation in order to control the truncation error. The binomial method and Projected SOR method are two normal method in American option pricing, we compare the singularity separation method with the these two and find that singularity separation method efficiently improves the accuracy and the speed of computation.
引文
[1]Louis Bachelier. Theorie de la speculation.Annales de l'Ecole Normale Superieure[M],1900.
[2]C.M. Sprenkle. Warrant Prices as Indicators of Expectations and Preferences[M]. Yale Economic Essays,1961.
[3]Boness. Elements of a Theory of Stock Option Value. Journal of Political Economy[M],1964, 72.
[4]P. A. Samnelson. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science[M],1965,4.
[5]杨垚,胡兵.支付红利的美式看涨期权定价的数值方法[J].四川大学学报(自然科学版)2008(4)：757-760.
[6]马俊海,张维,刘凤琴.期权定价的蒙特卡罗模拟综合性方差减少技术[J].管理科学学报.2005(4)：125-129.
[7]张顺明,杨新民.Black-Scholes期权定价公式[J].重庆师范学院学报(自然科学版),2000(1)：1-8.
[8]宋逢明.金融工程原理—无套利均衡分析[M].清华大学出版社,1999.
[9]茅宁.期权分析-理论与应用[M].南京： 南京大学出版社,2000.
[10]刘海龙,吴冲锋.期权定价方法综述[J].系统工程理论与实践,2001,1(1)：14-40.
[11]A Bensoussan, On the Theory of Option Pricing[J]. Journal of Acta Applicandae Mathematicae,1984,2,139-158.
[12]S. D. Jacka. Optimal Stopping and the American Put[J]. Journal of Mathematical Finance, 1991,1,1-14.
[13]Johnson, H. An Analytic Approximation of the American Put Price[J]. J. Finan.Quant.Anal. 1983,18,141-148.
[14]Karatzas, I. On the Pricing American Options[J]. Appl. Math. Optim.1988,17,37-60.
[15]Kuske R. A., Keller, J. B. Optimal Exercise Boundary for an American Put Option[J]. Applied Mathematical Finance,1998,5,107-116.
[16]Geske, R., Johnson, H. E. The American Put Option Valued Analytically [J]. J. Finance,1984, 39,1511-1524.
[17]McKean, H. P. Appendix:A Free Boundary Problem for the Heat Equation Arising from a Problem in Mathematical Economics[J]. Industrial Management Review,1965,6,32-39.
[18]I. Karatzas. On the Pricing of American Options[J]. Applied Mathematics and Optimization 1988,17,37-60.
[19]Fischer Black and John Cox, Valuing Corporate Securities:Some Effects of Bond Indenture Provisions[J]. Journal of Finance,1976,351-367.
[20]B. F. Nielsen, O. Skavhaug and A. Tveito. Penalty and Front Fixing Methods for the Numerical Solution of American Option Problems[J]. J. Comput. Finance,2002,5,69-97.
[21]冯广波.期权定价有关问题的探讨.中南大学博士学位论文[D],2002.
[22]Cox, J., Ross. S., Rubinstein, M. Option Pricing:A Simplified Approach[J]. Journal of Financial Economics,1979,7,229-263.
[23]Breen, R. The Accelerated Binomial Option Pricing Model. Journal of Financial and Quantitative Analysis[J],1991,26,153-164.
[24]Parkinson, M. Option Pricing:The American Put[J]. Journal of Business.1977,50,21-36.
[25]Broadie, M., Detemple. J. American Option Valuation:New Bounds, Approximations and a Comparison of Existing Methods[J]. Review of Financial Studies.1996,9,1211-1250.
[26]Johnson, H. An Analytical Approximation for the American Put Price[J]. Journal of Financial and Quantitative Analysis,1983,18,141-148.
[27]MacMillan, L. Analytic Approximation for the American Put Option. Advances in Futures and Options Research[M],1986,9,119-139
[28]Omberg, E. The Valuation of American Puts with Exponential Exercise Policies[J]. Advances in Futures and Options Research.1987,2,117-142.
[29]Broadie, M., J. Detemple. Option Pricing:Valuation and Models and Applications[J]. Management Science,2004,50(9):1145-1177.
[30]Jingzhi Huang, Marti G. Subrahmanyam and G. George Yu. The Review of Financial Studies[M],1996,9,277-300.
[31]Belomestny D.Milstein. Monte Carlo Evaluation of American Options Using Consumption Processes[J]. International journal of theoretical and applied finance,2006,9,455-481.
[32]Okten G, EastmanW. Randomized Quasi-Monte Carlo Methods in Pricing Securities[M], 2004,12,2399-2426.
[33]Zhu Youlan. Some New Developments of the Singularity-Separating Difference Method[M], Springer.1982.
[34]You-lan Zhu, Bin-mu Chen. Application of the Singularity-Separating Method to American Exotic Option Pricing[J]. Advances in Computational Mathematics,2003,19,147-158.
[35]Zhongdi Cen, Anbo Le. A Robust Finite Difference Scheme for Pricing American Put Options with Singularity-Separating method[J]. Numerical Algorithms,2009,7,1017-1398.
[36]Kuske R. A., Keller, J. B. Optimal Exercise Boundary for an American Put Option[J]. Applied Mathematical Finance,1998,5,107-116.
[37]S. D. Jacka. Optimal Stopping and the American Put. Mathematical Finance[M],1991,11(2), 1-14.
[38]Goran Peskir, Albert Shiryaev. Optimal Stopping and Free-Boundary Problems. Birkhauser Verlag[M],2000.
[39]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.
[40]何颖俞.美式期权的三叉树定价模型[J].黑龙江大学自然科学学报,2008,25(2)：71-76.
[41]张铁.美式期权定价问题的数值方法[J].应用数学学报,2001,25(1)：113-122.
[42]叶永刚.衍生金融工具概论[M].武汉：武汉大学出版社,2000.143-147,211-215.
[43]李玉立.美式期权定价的自由边界问题及数值方法[D].重庆大学硕士论文,2006.
[44]冯广波.期权定价有关问题的探讨.中南大学博士学位论文[D],2002.
[45]Joseph Stampfli, Victor Goodman.金融数学,蔡明超(译)[M].机械工业出版社,2006.
[46]Merton, R. C. Option Pricing When Underlying Stock Returns are Discontinuous[J]. Journal of Financial Economics,1976,3,125-144.
[47]John C Cox. Option Pricing:A Simplified Approach[J]. Journal of Financial Economics.1976.
[48]Fischer Black and John Cox, Valuing Corporate Securities:Some Effects of Bond Indenture Provisions[J]. Journal of Finance,1976,351-367.
[49]John C. Cox and Stephen A. Ross, The Valuation of Options for Alternative Stochastic Processes[J]. Journal of Financial Economics.1976,3,145-166.
[50]Geske, Robert. The Valuation of Compound Options[J]. Journal of Financial Economics. 1979,12,63-81.
[51]Kwok, Y. K. Mathematical Models of Financial Derivatives. New York:Springer Verlag. 1998.
[2]C.M. Sprenkle. Warrant Prices as Indicators of Expectations and Preferences[M]. Yale Economic Essays,1961.
[3]Boness. Elements of a Theory of Stock Option Value. Journal of Political Economy[M],1964, 72.
[4]P. A. Samnelson. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science[M],1965,4.
[5]杨垚,胡兵.支付红利的美式看涨期权定价的数值方法[J].四川大学学报(自然科学版)2008(4)：757-760.
[6]马俊海,张维,刘凤琴.期权定价的蒙特卡罗模拟综合性方差减少技术[J].管理科学学报.2005(4)：125-129.
[7]张顺明,杨新民.Black-Scholes期权定价公式[J].重庆师范学院学报(自然科学版),2000(1)：1-8.
[8]宋逢明.金融工程原理—无套利均衡分析[M].清华大学出版社,1999.
[9]茅宁.期权分析-理论与应用[M].南京： 南京大学出版社,2000.
[10]刘海龙,吴冲锋.期权定价方法综述[J].系统工程理论与实践,2001,1(1)：14-40.
[11]A Bensoussan, On the Theory of Option Pricing[J]. Journal of Acta Applicandae Mathematicae,1984,2,139-158.
[12]S. D. Jacka. Optimal Stopping and the American Put[J]. Journal of Mathematical Finance, 1991,1,1-14.
[13]Johnson, H. An Analytic Approximation of the American Put Price[J]. J. Finan.Quant.Anal. 1983,18,141-148.
[14]Karatzas, I. On the Pricing American Options[J]. Appl. Math. Optim.1988,17,37-60.
[15]Kuske R. A., Keller, J. B. Optimal Exercise Boundary for an American Put Option[J]. Applied Mathematical Finance,1998,5,107-116.
[16]Geske, R., Johnson, H. E. The American Put Option Valued Analytically [J]. J. Finance,1984, 39,1511-1524.
[17]McKean, H. P. Appendix:A Free Boundary Problem for the Heat Equation Arising from a Problem in Mathematical Economics[J]. Industrial Management Review,1965,6,32-39.
[18]I. Karatzas. On the Pricing of American Options[J]. Applied Mathematics and Optimization 1988,17,37-60.
[19]Fischer Black and John Cox, Valuing Corporate Securities:Some Effects of Bond Indenture Provisions[J]. Journal of Finance,1976,351-367.
[20]B. F. Nielsen, O. Skavhaug and A. Tveito. Penalty and Front Fixing Methods for the Numerical Solution of American Option Problems[J]. J. Comput. Finance,2002,5,69-97.
[21]冯广波.期权定价有关问题的探讨.中南大学博士学位论文[D],2002.
[22]Cox, J., Ross. S., Rubinstein, M. Option Pricing:A Simplified Approach[J]. Journal of Financial Economics,1979,7,229-263.
[23]Breen, R. The Accelerated Binomial Option Pricing Model. Journal of Financial and Quantitative Analysis[J],1991,26,153-164.
[24]Parkinson, M. Option Pricing:The American Put[J]. Journal of Business.1977,50,21-36.
[25]Broadie, M., Detemple. J. American Option Valuation:New Bounds, Approximations and a Comparison of Existing Methods[J]. Review of Financial Studies.1996,9,1211-1250.
[26]Johnson, H. An Analytical Approximation for the American Put Price[J]. Journal of Financial and Quantitative Analysis,1983,18,141-148.
[27]MacMillan, L. Analytic Approximation for the American Put Option. Advances in Futures and Options Research[M],1986,9,119-139
[28]Omberg, E. The Valuation of American Puts with Exponential Exercise Policies[J]. Advances in Futures and Options Research.1987,2,117-142.
[29]Broadie, M., J. Detemple. Option Pricing:Valuation and Models and Applications[J]. Management Science,2004,50(9):1145-1177.
[30]Jingzhi Huang, Marti G. Subrahmanyam and G. George Yu. The Review of Financial Studies[M],1996,9,277-300.
[31]Belomestny D.Milstein. Monte Carlo Evaluation of American Options Using Consumption Processes[J]. International journal of theoretical and applied finance,2006,9,455-481.
[32]Okten G, EastmanW. Randomized Quasi-Monte Carlo Methods in Pricing Securities[M], 2004,12,2399-2426.
[33]Zhu Youlan. Some New Developments of the Singularity-Separating Difference Method[M], Springer.1982.
[34]You-lan Zhu, Bin-mu Chen. Application of the Singularity-Separating Method to American Exotic Option Pricing[J]. Advances in Computational Mathematics,2003,19,147-158.
[35]Zhongdi Cen, Anbo Le. A Robust Finite Difference Scheme for Pricing American Put Options with Singularity-Separating method[J]. Numerical Algorithms,2009,7,1017-1398.
[36]Kuske R. A., Keller, J. B. Optimal Exercise Boundary for an American Put Option[J]. Applied Mathematical Finance,1998,5,107-116.
[37]S. D. Jacka. Optimal Stopping and the American Put. Mathematical Finance[M],1991,11(2), 1-14.
[38]Goran Peskir, Albert Shiryaev. Optimal Stopping and Free-Boundary Problems. Birkhauser Verlag[M],2000.
[39]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.
[40]何颖俞.美式期权的三叉树定价模型[J].黑龙江大学自然科学学报,2008,25(2)：71-76.
[41]张铁.美式期权定价问题的数值方法[J].应用数学学报,2001,25(1)：113-122.
[42]叶永刚.衍生金融工具概论[M].武汉：武汉大学出版社,2000.143-147,211-215.
[43]李玉立.美式期权定价的自由边界问题及数值方法[D].重庆大学硕士论文,2006.
[44]冯广波.期权定价有关问题的探讨.中南大学博士学位论文[D],2002.
[45]Joseph Stampfli, Victor Goodman.金融数学,蔡明超(译)[M].机械工业出版社,2006.
[46]Merton, R. C. Option Pricing When Underlying Stock Returns are Discontinuous[J]. Journal of Financial Economics,1976,3,125-144.
[47]John C Cox. Option Pricing:A Simplified Approach[J]. Journal of Financial Economics.1976.
[48]Fischer Black and John Cox, Valuing Corporate Securities:Some Effects of Bond Indenture Provisions[J]. Journal of Finance,1976,351-367.
[49]John C. Cox and Stephen A. Ross, The Valuation of Options for Alternative Stochastic Processes[J]. Journal of Financial Economics.1976,3,145-166.
[50]Geske, Robert. The Valuation of Compound Options[J]. Journal of Financial Economics. 1979,12,63-81.
[51]Kwok, Y. K. Mathematical Models of Financial Derivatives. New York:Springer Verlag. 1998.