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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2021, Vol. 38 ›› Issue (6): 879-900.doi: 10.3969/j.issn.1005-3085.2021.06.010

• • 上一篇    

一类新的高效数值方法定价美式零息债券期权（英）

甘小艇1,   易  华2   

  1. 1. 楚雄师范学院数学与计算机科学学院，云南 楚雄 675000
    2. 井冈山大学数理学院，江西 吉安 343009
  • 出版日期:2021-12-15 发布日期:2022-02-15
  • 通讯作者: 易 华 E-mail: 876145777@qq.com
  • 基金资助:
    国家自然科学基金 (61463002)；云南省教育厅科学研究基金 (2019J0396)；江西省教育厅科技计划项目 (GJJ201009)；云南省地方本科高校基础研究联合专项面上项目 (2019FH001-079).

A New Efficient Numerical Method for Pricing American Options on Zero-coupon Bonds

GAN Xiaoting1,   YI Hua2   

  1. 1. School of Mathematics and Computer Science, Chuxiong Normal University, Chuxiong, Yunnan 675000 
    2. Department of Mathematics, Jinggangshan University, Ji'an, Jiangxi 343009
  • Online:2021-12-15 Published:2022-02-15
  • Contact: H. Yi. E-mail address: 876145777@qq.com
  • Supported by:
    The National Natural Science Foundation of China (61463002); the Scientific Research Fund of Yunnan Provincial Education Department (2019J0396); the Scientific Research Foundation of the Education Bureau of Jiangxi Province (GJJ201009); the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities' Association (2019FH001-079).

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摘要：

与欧式期权定价不同，由于提前行使的特性，美式期权一般不存在封闭形式的解．因此，往往采用数值方法进行求解．本文讨论了一类新的高效数值方法定价美式债券期权模型．针对该偏微分互补问题中的偏微分方程，空间方向采用经典的有限体积离散，时间方向构造稳定的全隐式格式．针对离散得到的线性互补问题，引入高效的模系矩阵分裂迭代法求解，并建立了 $H_{+}$-离散矩阵下的收敛性定理．数值实验验证了新方法的精确性、高效性和稳健性．

关键词: 美式债券期权, 有限体积法, 线性互补问题, 模系矩阵分裂迭代法

Abstract:

Unlike the European options pricing, the closed-form solution generally does not exist due to the early exercise feature of American options. Hence, numerical approximation methods are normally employed to solve them. Presented in this paper is a new numerical method to price American bond options. To numerically solve the resulting partial differential complementarity problem (PDCP), we develop a class of finite volume method for the spatial discretization, coupled with the stable fully implicit time stepping scheme of the partial differential equation (PDE). Then, the resulting linear complementarity problems (LCPs) are solved by using an efficient iterative method, the modulus-based matrix splitting iteration method, where the $H_{+}$-matrix property of the system matrix guarantees its convergence. Numerical experiments are implemented to verify the accuracy, efficiency and robustness of the new method.

Key words: American bond option, finite volume method, linear complementarity problems, modulus-based matrix splitting iteration method

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