几何平均亚式期权的定价研究
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摘要
近年来,随着金融市场需求复杂程度的提高,仅仅使用标准期权已很难满足市场的需要,为了满足市场及某些客户的特殊需求,许多金融公司除了提供人们广为熟悉的欧式、美式期权外,还设计了大量由标准期权衍生的通常在场外市场交易的新品种—非标准化的衍生证券,我们称其为新型期权(Exotic Options)。新型期权中大多都具备路径依赖特征,即期权价格不仅取决于其到期日标的资产价格,而且取决于标的资产价格在期权有效期内的变化路径,亚式期权(Asian Options)正是其中的一种代表性的产品,同时它也是当今金融衍生品市场中最为活跃的一种新型期权,由于具备路径依赖特征,使得亚式期权的定价模型与标准期权的定价模型相比呈现出比较大的差异,其定价问题远比欧式期权定价复杂。
本文主要研究新型期权中的亚式期权,它是强路径依赖型期权的典型代表,亚式期权与标准期权明显不同,它实质上是欧式期权的一种创新,因此它与标准欧式期权有着密切的联系,所以溯根求源,在B-S模型的基础上来展开本文的研究工作。
本文给出了一个强依赖期权统一的Black-Scholes模型。由于路径依赖期权的独有的路径依赖特征,我们引入一个路径因子,并利用It(?)定理与无套利原理推导出强路径依赖期权统一的B-S模型,对应不同的路径函数,只需在统一模型的基础上给出具体的路径函数,再结合其相应终值条件和所带有的一些边界条件,就可以通过求解方程得到定价公式。另外,利用强路径依赖期权统一的B-S模型得到了连续情形下固定敲定价格和浮动敲定价格的几何平均亚式期权的定价结果及在非风险中性世界中连续情况下具固定敲定价格的几何平均亚式期权的定价公式。
In recent years, along with the improvement of demand on the complexity in money market, it is difficult to satisfy the special needs of customers by only using the standard option. In order to satisfying the demand of markets and customers, many financial companies not only designed European options and American options but also devised a great deal of new breeds which derive from standard options and exchange out of the counter commonly. We call it exotic options. Most of exotic options possess the feature of path-dependent, that is to say, the option price not only depends on the option price at maturity; but also depends on varying of the underlying assets price. Asian options are just one of the typical products, and it is a kind of exotic options which is the most active one in financial derivative markets. Because of the property of path-dependent, there exists distinguished difference between Asian option pricing model and standard options.
The main goal of this paper is to study Asian options. It is a typical representative of strongly path-dependent options, Asian option is different from standard option obviously. It is innovation of European options. So it connects with the standard European options nearly. Black-Scholes Option Pricing Model is just one of the most effective means to solve main content of the article. Therefore we should understand Black-Scholes Option Pricing Model fully which concludes to our research.
This paper gives the uniform Black-Scholes Model of strong path-dependent options.Because of the unique character of path-dependent options, we import a path-factor, then deduce the uniform B-S Model of strong path-dependent option using Ito theory and no-arbitrage principle. For different path-factor, with a concrete path function, initial condition and boundary condition we can get the pricing formula on solving the equation. In addition, we get the price of geometric average Asian option with fixed strike price and floating strike price, making use of the uniform B-S Model of strong path-dependent option and the pricing formula of geometric average Asian options under no risk-neutral world.
本文主要研究新型期权中的亚式期权,它是强路径依赖型期权的典型代表,亚式期权与标准期权明显不同,它实质上是欧式期权的一种创新,因此它与标准欧式期权有着密切的联系,所以溯根求源,在B-S模型的基础上来展开本文的研究工作。
本文给出了一个强依赖期权统一的Black-Scholes模型。由于路径依赖期权的独有的路径依赖特征,我们引入一个路径因子,并利用It(?)定理与无套利原理推导出强路径依赖期权统一的B-S模型,对应不同的路径函数,只需在统一模型的基础上给出具体的路径函数,再结合其相应终值条件和所带有的一些边界条件,就可以通过求解方程得到定价公式。另外,利用强路径依赖期权统一的B-S模型得到了连续情形下固定敲定价格和浮动敲定价格的几何平均亚式期权的定价结果及在非风险中性世界中连续情况下具固定敲定价格的几何平均亚式期权的定价公式。
In recent years, along with the improvement of demand on the complexity in money market, it is difficult to satisfy the special needs of customers by only using the standard option. In order to satisfying the demand of markets and customers, many financial companies not only designed European options and American options but also devised a great deal of new breeds which derive from standard options and exchange out of the counter commonly. We call it exotic options. Most of exotic options possess the feature of path-dependent, that is to say, the option price not only depends on the option price at maturity; but also depends on varying of the underlying assets price. Asian options are just one of the typical products, and it is a kind of exotic options which is the most active one in financial derivative markets. Because of the property of path-dependent, there exists distinguished difference between Asian option pricing model and standard options.
The main goal of this paper is to study Asian options. It is a typical representative of strongly path-dependent options, Asian option is different from standard option obviously. It is innovation of European options. So it connects with the standard European options nearly. Black-Scholes Option Pricing Model is just one of the most effective means to solve main content of the article. Therefore we should understand Black-Scholes Option Pricing Model fully which concludes to our research.
This paper gives the uniform Black-Scholes Model of strong path-dependent options.Because of the unique character of path-dependent options, we import a path-factor, then deduce the uniform B-S Model of strong path-dependent option using Ito theory and no-arbitrage principle. For different path-factor, with a concrete path function, initial condition and boundary condition we can get the pricing formula on solving the equation. In addition, we get the price of geometric average Asian option with fixed strike price and floating strike price, making use of the uniform B-S Model of strong path-dependent option and the pricing formula of geometric average Asian options under no risk-neutral world.
引文
1.Black F,Scholes M.The pricing of options and corporate liabilities.Journal of political Economy.1973(81):637-659.
2.Merton R.C.Application of Option Pricing:Theory Twenty Five Years Later.The American Economic Review,1998.88(3).
3.Cox J,Ross S.The valuation of options for alternative stochastic processes[J].Jounal of Financial Economics,1976.3(2).
4.张顺明,杨新民.Black-Scholes期权定价公式.重庆师范学院学报(自然科学版),2000.3,17(1):1-7.
5.刘海龙,吴冲锋.期权定价方法综述.管理科学学报,2002.4,5(2):63-73.
6.项立群.概率方法在一种期权定价中的应用.安徽工程科技学院学报,2003.6,18(2):66-68.
7.俞迎达,俞苗.Black-Shcoles期权定价模型的简化推导.数学的实践和认识,2001.11,31(6):754-758.
8.张彩玉.期权定价的博弈论分析.西南交通大学学报,2003.6,38(3).
9.陈占锋,章珂.期权定价原理的数理逻辑探析.中国管理科学,2001,9(2):10-15
10.Cox J,Ross S,Rubinstein M.Option Pricing:A simplified approach[J].Jounal of Financial Economics,1979.
11.关莉,李耀堂.修正的Black-Scholes期权定价模型.云南大学学报(自然科学版),2001,23(2):84-56.
12.王琳.期权定价模型及其改进.武汉金融高等专科学校学报,2002.11:29-34.
13.Huang Zhenghong.Modified Black-Scholes Option Pricing Model and Formula.Mathematical Theory and Applications,2003.9,23(3):18-21.
14.陈万义.非风险中性定价意义下的欧式期权定价公式.数学实践与认识,2004.1,34(1):76-79.
15.郑小迎,陈金贤.有交易成本的期权定价研究.管理工程学报,2001.15(3):377-381.
16.李秉祥.对欧式期权B-S模型的推广.西安理工大学学报,2003.19(4):377-381.
17.王峰,徐小平,赵炜.布朗运动和泊松过程共同驱动下的欧式期权定价.系统工程学报,2004.20(1):45-46.
18.闰海峰,刘三阳.股票价格遵循Ornstein-Uhlenback过程的期权定价.系统工程学报,2003.12,18(6):547-551.
19.郑小迎,陈金贤.关于路径依赖型期权定价模型的研究.河北师范大学学报(自然科学版),2002.36(2):15-21.
20.郑小迎,陈金贤.关于新型期权及其定价模型的研究.管理工程学报,2004.14(3):56-60.
21.戴民.路径依赖期权二叉树方法的数值方法.复旦大学博士学位论文,2000.2,28.
22.郑小迎,陈金贤.关于亚式期权及其定价模型的研究.系统工程,2000.3,18(2):22-26.
23.章坷等.几何平均亚式期权的定价方法.同济大学学报,2001,29(8):924-927.
24.姜礼尚.期权定价的数学模型和方法.高等教育出版社,2003.1.
25.冯广波.前向打靶格方法计算变异期权的价格.数学理论与应用,2003.6,23(2):23-26.
26.孙坚强等.离散算术平均亚式期权近似定价.厦门大学学报(自然科学版),2003,(4):435-438.
27.金春红,陈德燕.亚式期权定价模型的研究.辽宁大学学报(自然科学版),2003,30(2):100-101.
28.胡日东.关于亚式股票期权及其定价方法的研究.数量经济技术经济研究,1998,(2):72-75.
29.曲军恒,沈尧天,姚仰新.有交易费的几何平均亚式期权的定价公式.华南理工大学学报(自然科学报),2004.5,32(5):84-86.
30.许端,蔡金绪.亚式期权估价的最新进展.预测(理论与方法研究),1999,(6):40-43.
31.Kwok YK.Mathematical Models of Financial Derivatives[M].Springer,Singapore,1998.
32.袁震东.近代概率引论—测度、鞍和随机微分方程.北京:科学出版社,1991.
33.Conze A.and Viswanathan.Path Dependent Options:the Case of Lookback Options.Journal of Finance,1991,46.
34.Wilmott P,Howsion S,Dewynne J.Option Pricing mathematical models and computation[M].Oxford:Oxford Financial Press,1997.
2.Merton R.C.Application of Option Pricing:Theory Twenty Five Years Later.The American Economic Review,1998.88(3).
3.Cox J,Ross S.The valuation of options for alternative stochastic processes[J].Jounal of Financial Economics,1976.3(2).
4.张顺明,杨新民.Black-Scholes期权定价公式.重庆师范学院学报(自然科学版),2000.3,17(1):1-7.
5.刘海龙,吴冲锋.期权定价方法综述.管理科学学报,2002.4,5(2):63-73.
6.项立群.概率方法在一种期权定价中的应用.安徽工程科技学院学报,2003.6,18(2):66-68.
7.俞迎达,俞苗.Black-Shcoles期权定价模型的简化推导.数学的实践和认识,2001.11,31(6):754-758.
8.张彩玉.期权定价的博弈论分析.西南交通大学学报,2003.6,38(3).
9.陈占锋,章珂.期权定价原理的数理逻辑探析.中国管理科学,2001,9(2):10-15
10.Cox J,Ross S,Rubinstein M.Option Pricing:A simplified approach[J].Jounal of Financial Economics,1979.
11.关莉,李耀堂.修正的Black-Scholes期权定价模型.云南大学学报(自然科学版),2001,23(2):84-56.
12.王琳.期权定价模型及其改进.武汉金融高等专科学校学报,2002.11:29-34.
13.Huang Zhenghong.Modified Black-Scholes Option Pricing Model and Formula.Mathematical Theory and Applications,2003.9,23(3):18-21.
14.陈万义.非风险中性定价意义下的欧式期权定价公式.数学实践与认识,2004.1,34(1):76-79.
15.郑小迎,陈金贤.有交易成本的期权定价研究.管理工程学报,2001.15(3):377-381.
16.李秉祥.对欧式期权B-S模型的推广.西安理工大学学报,2003.19(4):377-381.
17.王峰,徐小平,赵炜.布朗运动和泊松过程共同驱动下的欧式期权定价.系统工程学报,2004.20(1):45-46.
18.闰海峰,刘三阳.股票价格遵循Ornstein-Uhlenback过程的期权定价.系统工程学报,2003.12,18(6):547-551.
19.郑小迎,陈金贤.关于路径依赖型期权定价模型的研究.河北师范大学学报(自然科学版),2002.36(2):15-21.
20.郑小迎,陈金贤.关于新型期权及其定价模型的研究.管理工程学报,2004.14(3):56-60.
21.戴民.路径依赖期权二叉树方法的数值方法.复旦大学博士学位论文,2000.2,28.
22.郑小迎,陈金贤.关于亚式期权及其定价模型的研究.系统工程,2000.3,18(2):22-26.
23.章坷等.几何平均亚式期权的定价方法.同济大学学报,2001,29(8):924-927.
24.姜礼尚.期权定价的数学模型和方法.高等教育出版社,2003.1.
25.冯广波.前向打靶格方法计算变异期权的价格.数学理论与应用,2003.6,23(2):23-26.
26.孙坚强等.离散算术平均亚式期权近似定价.厦门大学学报(自然科学版),2003,(4):435-438.
27.金春红,陈德燕.亚式期权定价模型的研究.辽宁大学学报(自然科学版),2003,30(2):100-101.
28.胡日东.关于亚式股票期权及其定价方法的研究.数量经济技术经济研究,1998,(2):72-75.
29.曲军恒,沈尧天,姚仰新.有交易费的几何平均亚式期权的定价公式.华南理工大学学报(自然科学报),2004.5,32(5):84-86.
30.许端,蔡金绪.亚式期权估价的最新进展.预测(理论与方法研究),1999,(6):40-43.
31.Kwok YK.Mathematical Models of Financial Derivatives[M].Springer,Singapore,1998.
32.袁震东.近代概率引论—测度、鞍和随机微分方程.北京:科学出版社,1991.
33.Conze A.and Viswanathan.Path Dependent Options:the Case of Lookback Options.Journal of Finance,1991,46.
34.Wilmott P,Howsion S,Dewynne J.Option Pricing mathematical models and computation[M].Oxford:Oxford Financial Press,1997.