AAM  >> Vol. 2 No. 4 (November 2013)

    支付红利下Black-Scholes方程的交替分段C-N格式解法
    Alternating Segment C-N Algorithm for Black-Scholes Equation with Dividend Paying

  • 全文下载: PDF(1585KB) HTML    PP.152-158   DOI: 10.12677/AAM.2013.24020  
  • 下载量: 2,249  浏览量: 7,732   国家自然科学基金支持

作者:  

吴立飞,杨晓忠:华北电力大学理学院,北京

关键词:
支付红利的Black-Scholes方程交替分段Crank-Nicolson格式(ASC-N格式)并行计算 数值试验The Payment of Dividend Black-Scholes Equation; Alternating Segment Crank-Nicolson Scheme; Parallel Computing; Numerical Experiment

摘要:

Black-Scholes方程是金融数学中期权定价的重要模型,研究它的数值解法具有非常重要的理论意义和实际应用价值。本文对支付红利下Black-Scholes方程构造了一种具有并行本性的交替分段Crank-Nicolson格式(ASC-N格式),给出格式解的存在唯一性、稳定性和收敛性分析;理论分析和数值试验表明ASC-N格式与经典格式C-N计算精度相当,但是其计算效率(计算时间)要比经典C-N节省近40%;数值试验验证了理论分析,表明本文ASC-N格式对求解支付红利下Black-Scholes方程是有效的。

Black-Scholes equation is an important model in option pricing theory of financial mathematics, which is very practical in the application of numerical computation. This paper constructs a kind of parallel alternating segment Crank-Nicolson (ASC-N) scheme for solving the payment of dividend Black-Scholes equation. Secondly, it gives the existence and uniqueness of solution, stability and convergence analysis of the scheme. The theoretical analysis and numerical examples demonstrate that ASC-N scheme has same computational accuracy with C-N scheme’s, but its computational efficiency (computational time) can save nearly 40% compared with C-N scheme. Numerical experiment verifies the theoretical analysis, and it shows that ASC-N scheme is effective for solving Black-Scholes equation with dividend paying.

文章引用:
吴立飞, 杨晓忠. 支付红利下Black-Scholes方程的交替分段C-N格式解法[J]. 应用数学进展, 2013, 2(4): 152-158. http://dx.doi.org/10.12677/AAM.2013.24020

参考文献

[1] Kwork, Y.K. (2011) Mathematical models of financial derivatives. The World Book Publishing Company, Beijing.
[2] 姜礼尚 (2008) 期权定价的数学模型和方法(第2版).高等教育出版社, 北京.
[3] 赵胜民 (2008) 衍生金融工具定价.中国财政经济出版社, 北京.
[4] Ballester, C., Company, R. and Jodar, L. (2008) An efficient method for option pricing with discrete dividend payment. Computers and Mathe- matics with Applications, 56, 822-835.
[5] Company, R., Navarro, E., Pintos, J.R. et al. (2008) Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Com- puters and Mathematics with Applications, 56, 813-821.
[6] Yang, X.Z., Liu, Y.G. and Wang, G.H. (2007) A study on a new kind of universal difference schemes for solving Black-Scholes equation. International Journal of Information and Systems Sciences, 3, 251-260.
[7] 唐耀宗, 金朝嵩 (2006) 有红利美式看跌期权定价的Crank-Nicolson有限差分法. 经济数学, 4, 349-352.
[8] 吴立飞, 杨晓忠 (2011) 支付红利下Black-Scholes方程的显隐和隐显差分格式解法. 中国科技论文在线精品论文, 13, 1207-1212.
[9] Evans, D.J. and Sahimi, M.S. (1989) The numerical solution of Burgers’ equations by the alternating group explicit (AGE) method. Interna- tional Journal of Computer Mathematics, 29, 39-64.
[10] 张宝琳 等 (1999) 数值并行计算原理与方法. 国防工业出版社, 北京.
[11] 陆金甫, 张宝琳, 徐涛 (1998) 求解对流-扩散方程的交替分段显–隐式方法. 数值计算与计算机应用, 3, 161-167.
[12] 王文洽 (2002) 对流–扩散方程的一类交替分组方法. 高等学校计算数学学报, 4, 289-297.
[13] 张锁春 (2010) 抛物型方程定解问题的有限差分数值计算. 科学出版社, 北京.