基于摄动方法的关卡期权定价及其误差分析
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摘要
关卡期权定价问题是现代金融学领域研究的热点之一,也是数理金融学的一个重要研究方向.本文在非线性Black-Scholes模型下,研究了关卡期权定价问题.首先,利用扰动理论中单参数摄动展开方法,给出了关卡期权的近似定价公式.其次,在给定的条件下,利用Feyman-Kac公式分析了近似定价公式的误差估计问题.最后,利用数值实验验证了理论分析的近似结果和误差估计的准确性.
The Barrier option pricing problem is one of hot topics in modern finance, and also one of important fields in Mathematical finance. In this paper, the pricing problems of barrier options are discussed under the nonlinear Black-Scholes model. Firstly, the author uses the perturbation method of single-parameter to obtain asymptomatic formulae of barrier options pricing problems. Secondly, error estimates of these asymptotic solutions are illustrated by using the Feymann-Kac formula under the given condition. Finally, numerical experiments confirm the correctness of the proposed theoretical results as well as error estimation.
The Barrier option pricing problem is one of hot topics in modern finance, and also one of important fields in Mathematical finance. In this paper, the pricing problems of barrier options are discussed under the nonlinear Black-Scholes model. Firstly, the author uses the perturbation method of single-parameter to obtain asymptomatic formulae of barrier options pricing problems. Secondly, error estimates of these asymptotic solutions are illustrated by using the Feymann-Kac formula under the given condition. Finally, numerical experiments confirm the correctness of the proposed theoretical results as well as error estimation.
引文
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[2]徐腾飞,曹小龙,胡云姣.离散障碍期权定价的蒙特卡罗模拟[J].北京化工大学学报(自然科学版),2013,40(3):123-127Xu T F,Cao X L,Hu Y J.On pricing discrete barrier options using the monte carlo method[J].Journal of Beijing University of Chemical Technology(Natural Science),2013,40(3):123-127
[3]孙玉东,师义民,吴敏.参数依赖股票价格情形下的障碍期权定价[J].数学物理学报,2013,33(5):912-925Sun Y D,Shi Y M,Wu M.Barrier options pricing when parameters dependent on stock price[J].Acta Mathematica Scientia,2013,33(5):912-925
[4]孙玉东,师义民,童红.基于摄动理论的障碍期权定价[J].应用数学学报,2015,38(1):67-79Sun Y D,Shi Y M,Tong H.The pricing for the barriers options based on the perturbation theory[J].Acta Mathematicae Applicatae Sinica,2015,38(1):67-79
[5]孙玉东,王秀芬,童红.非线性Black-Scholes模型下障碍期权定[J].系统科学与数学,2016,36(4):513-527Sun Y D,Wang X F,Tong H.Barrier options’pricing under the nonlinear Black-Scholes model[J].Journal of System Science and Mathematical Science,2016,36(4):513-527
[6]董艳.非线性Black-Scholes模型下Bala期权定价[J].高校应用数学学报,2016,31(1):9-20Dong Y.Bala options’pricing under the nonlinear Black-Scholes model[J].Applied Mathematics:a Journal of Chinese Universities,2016,31(1):9-20
[7]姜礼尚.期权定价的数学模型和方法(第2版)[M].北京:高等教育出版社,2008Jiang L S.The Mathematical Models and Method of Option Pricing(2nd Edition)[M].Beijing:Chinese Higher Education Press,2008
[8]郑连存,张欣欣.非线性偏微分方程近代分析方法[M].北京:科学出版社,2011Zheng L C,Zhang X X.The Modern Analysis Method of Nonlinear Partial Differential Equations[M].Beijing:Science Press,2011
[9]陈亚浙.二阶抛物型偏微分方程[M].北京:北京大学出版社,2002Chen Y Z.The Second Order Parabolic Partial Differential Equations[M].Beijing:Beijing University Press,2002
[10]金志明.随机分析基础[M].北京:国防工业出版社,2001Jin Z M.Basic of Stochastic Analysis[M].Beijing:National Defense Industry Press,2001