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随着金融市场的不断发展, 期权作为一种能够规避风险的金融衍生产品越来越引起投资者的青睐, 成交量呈逐年上升的趋势, 期权定价问题已经成为金融数学领域中一个重要的研究课题. 本文主要研究Black-Scholes模型下美式回望期权定价问题的数值解法. 美式回望期权定价问题是一个二维非线性抛物问题, 难以直接应用数值方法进行求解. 通过分析该问题的求解难点, 本文给出解决该困难的有效方法. 首先利用计价单位变换将定价问题转换为一维自由边值问题, 并采用Landau's变换将求解区域规范化; 而后针对问题的非线性特点,利用有限体积法和Newton法交替迭代求解期权价格和最佳实施边界, 并对数值解的非负性进行了分析. 最后, 通过与二叉树方法进行比较, 验证了本文方法的正确性和有效性, 为实际应用提供了理论基础.
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关键词:
- 经济物理学 /
- 美式回望期权 /
- 有限体积法 /
- Newton 迭代法
Due to the characteristic of risk aversion, option has become one of the most fashionable derivatives in the financial field. More and more investigators are attracted to devote themselves to exploring the option pricing problem. In this paper, we are concerned with the valuation of American lookback options in terms of the Black-Scholes model. It is well known that the American lookback option satisfies a two-dimensional nonlinear partial differential equation in an unbounded domain, which couldn't be numerically solved directly. Based on the analysis of the issues for solving this problem, this paper introduces an approach to settle it. First, we transform the problem into a one-dimensional form by the numeraire transformation. And then, the Landau's transformation is applied to normalize the defined domain. For the nonlinear feature of the resulting problem, we propose a finite volume method coupled with Newton iterative method to obtain the optional value and the optimal exercise boundary simultaneously. We also give a proof on the nonnegativity of the numerical solutions under some appropriate assumptions. Finally, some numerical simulations are presented using the proposed method in this paper. Comparing with the binomial method, we can conclude that the proposed method is an effective one, which provides a theoretical basis for practical applications.[1] Vogel E E, Saravia G 2014 Eur. Phys. J. B 87 177
[2] Ibuki T, Suzuki S, Inoue J 2013 Econophysics of systemic risk and network dynamics (Milan:Springer) pp239-259
[3] Mimkes J 2012 Continuum Mech. Thermodyn. 24 731
[4] Chakraborti A, Muni T I, Patriarca M, Abergel F 2011 Fuant. Financ. 11 1013
[5] Sousa T, Domingos T 2006 Phys. A 371 492
[6] Fang H 2014 Acta Phys. Sin. 63 038902 (in Chinese) [范宏 2014 63 038902]
[7] Da C, Fan H Y 2014 Acta Phys. Sin. 63 098901 (in Chinese) [笪诚, 范洪义 2014 63 098901]
[8] Yu Z R 2000 Physics 29 662 (in Chinese) [于祖荣 2000 物理 29 662]
[9] Kwok Y K 2008 Mathematical models of financial derivatives (Vol. 2) (Berlin:Springer) pp201-211
[10] Zhang R, Song H M, Luan N N 2014 Front. Math. China 9 455
[11] Kim K Ik, Park H S, Qian X S 2011 J. Comput. Appl. Math. 235 5140
[12] Zhang T, Zhang S H, Zhu D M 2009 J. Comput. Math. 27 484
[13] Li G, Zhu B X, Zhang Q, Song H M 2014 Journal of Jilin University(Science Edition) 52 698 (in Chinese) [李庚, 朱本喜, 张琪, 宋海明 2014 吉林大学学报(理学版) 52 698]
[14] Wang H, Basu T S 2012 SIAM J. Sci. Comput. 34 A2444
[15] Han Q G, Ma H A, Xiao H Y, Li R, Zhang C, Li Z C, Tian Y, Jia X P 2010 Acta Phys. Sin. 59 1923 (in Chinese) [韩奇钢, 马红安, 肖宏宇, 李瑞, 张聪, 李战厂, 田宇, 贾晓鹏 2010 59 1923]
[16] Kwon Y H, Lee Y 2011 SIAM J. Numer. Anal. 49 2598
[17] Mattsson K, Carpenter M H 2010 SIAM J. Sci. Comput. 32 2298
[18] Forsyth P A, Vetzal K R, Zvan R 1999 Appl. Math. Finance 6 87
[19] Li H Y, Ma H P, Sun W W 2013 SIAM J. Numer. Anal. 51 353
[20] Pindza E, Patidar K C, Ngounda E 2014 Numer. Methods Partial Differential Equations 30 1169
[21] Chen Y P, Huang F L, Yi N Y, Liu W B 2011 SIAM J. Numer. Anal. 49 1625
[22] Bessemoulin C M, Filbet F 2012 SIAM J. Sci. Comput. 34 B559
[23] Zhang K, Wang S 2008 Appl. Math. Comput. 201 398
[24] Berton J, Eymard R 2006 MSAIN Math. Model. Numer. Anal. 40 311
[25] Angermann L, Wang S 2007 Numer. Math. 106 1
[26] Jiang L S 2007 Mathematical modeling and methods of option pricing (Vol. 2) (Beijing:Higher Education Press) p303 (in Chinese) [姜礼尚 2007 期权定价的数学模拟和方法(第二版)(北京:高等教育出版社)第303页]
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[1] Vogel E E, Saravia G 2014 Eur. Phys. J. B 87 177
[2] Ibuki T, Suzuki S, Inoue J 2013 Econophysics of systemic risk and network dynamics (Milan:Springer) pp239-259
[3] Mimkes J 2012 Continuum Mech. Thermodyn. 24 731
[4] Chakraborti A, Muni T I, Patriarca M, Abergel F 2011 Fuant. Financ. 11 1013
[5] Sousa T, Domingos T 2006 Phys. A 371 492
[6] Fang H 2014 Acta Phys. Sin. 63 038902 (in Chinese) [范宏 2014 63 038902]
[7] Da C, Fan H Y 2014 Acta Phys. Sin. 63 098901 (in Chinese) [笪诚, 范洪义 2014 63 098901]
[8] Yu Z R 2000 Physics 29 662 (in Chinese) [于祖荣 2000 物理 29 662]
[9] Kwok Y K 2008 Mathematical models of financial derivatives (Vol. 2) (Berlin:Springer) pp201-211
[10] Zhang R, Song H M, Luan N N 2014 Front. Math. China 9 455
[11] Kim K Ik, Park H S, Qian X S 2011 J. Comput. Appl. Math. 235 5140
[12] Zhang T, Zhang S H, Zhu D M 2009 J. Comput. Math. 27 484
[13] Li G, Zhu B X, Zhang Q, Song H M 2014 Journal of Jilin University(Science Edition) 52 698 (in Chinese) [李庚, 朱本喜, 张琪, 宋海明 2014 吉林大学学报(理学版) 52 698]
[14] Wang H, Basu T S 2012 SIAM J. Sci. Comput. 34 A2444
[15] Han Q G, Ma H A, Xiao H Y, Li R, Zhang C, Li Z C, Tian Y, Jia X P 2010 Acta Phys. Sin. 59 1923 (in Chinese) [韩奇钢, 马红安, 肖宏宇, 李瑞, 张聪, 李战厂, 田宇, 贾晓鹏 2010 59 1923]
[16] Kwon Y H, Lee Y 2011 SIAM J. Numer. Anal. 49 2598
[17] Mattsson K, Carpenter M H 2010 SIAM J. Sci. Comput. 32 2298
[18] Forsyth P A, Vetzal K R, Zvan R 1999 Appl. Math. Finance 6 87
[19] Li H Y, Ma H P, Sun W W 2013 SIAM J. Numer. Anal. 51 353
[20] Pindza E, Patidar K C, Ngounda E 2014 Numer. Methods Partial Differential Equations 30 1169
[21] Chen Y P, Huang F L, Yi N Y, Liu W B 2011 SIAM J. Numer. Anal. 49 1625
[22] Bessemoulin C M, Filbet F 2012 SIAM J. Sci. Comput. 34 B559
[23] Zhang K, Wang S 2008 Appl. Math. Comput. 201 398
[24] Berton J, Eymard R 2006 MSAIN Math. Model. Numer. Anal. 40 311
[25] Angermann L, Wang S 2007 Numer. Math. 106 1
[26] Jiang L S 2007 Mathematical modeling and methods of option pricing (Vol. 2) (Beijing:Higher Education Press) p303 (in Chinese) [姜礼尚 2007 期权定价的数学模拟和方法(第二版)(北京:高等教育出版社)第303页]
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