一类亚式期权的定价
|
摘要
期权是一种有效的风险管理工具。随着金融理论的发展和市场的需求,人们在标准期权的基础上设计出新型期权,称为奇异期权(Exotic Option)。常见的有亚式期权、障碍期权、再装期权、经理人员股票期权、回望期权、重置期权。奇异期权大多具备路径依赖特征。比如亚式期权的收益就不仅取决于到期日的标的资产价格还与标的资产在有效期内的平均价格有关。从而使得奇异期权的定价比标准期权复杂。
本文讨论亚式几何连续平均期权的定价。根据奇异期权各自的特点,将亚式期权与其它一些奇异期权组合,设计并得到了新型亚式几何连续期权,以满足市场和公司金融政策的具体需要。在标的资产服从指数O-U过程模型的假设下,利用Girsanov定理,构造了风险中性概率测度,应用随机分析的理论和工具,给出了一类亚式期权价格的显示表达式。
本文主要工作如下:
1、简化了亚式几何连续平均形式的表达式,用鞅方法推出了一般亚式期权定价公式,并讨论了看涨与看跌的平价关系。
2、将亚式期权与障碍期权的特点联系起来,创造出一种新型亚式障碍期权,并给出期权定价的显式表达式。
3、考虑到再装期权的特征,将亚式几何连续平均形式融入其中,并且进行了适当的变化,得到一种新型亚式再装期权,并推导出这种新型期权的定价公式。
4、根据市场的需求和普通经理人员股票的不足,设计了一种新型亚式经理股票期权,并给出相应期权的显示表达式。
Option is an effective tool of risk management.With the development of economy and demand of reset market,Some new type of options was designed from standard option,call it Exotic Options.We are Familiar with some exotic options ,such as Asian Options,Barrier Options,Reset Options,Executive Stock Option( ESO),Look-back Option,Reload Options.Most of all exotic options have characteristics of path-dependence.For example,the payoff of asian options depends not only on the maturity of the underlying asset price,but also some type average value of the underlying assets prices.So it is more complex to price the exotic options than standard options.
in this dissertation we discuss the pricing problem of geometric continue averagerate asian option.We created some new forms of asian option which associated with some other exotic option and asian option based on the character of themselves. Under the hypothesis of the underlying assets are driven by exponential O-U process model,we constructed risk-neutral probability measure by using Girsanov theorem and obtained the pricing formulas of new type option which talked above by application of the theory and tools of stochastic analysis.
In detail we have made works as follows:
1、By skillful handled the form of geometric continue average-rate asian option, the pricing formula of the asian option was driven by using martingale method and put-call parity relation is deduced.
2、Connected characteristic of asian option to barrier option,we created a new type of asian-barrier option and obtained pricing formula of the option.
3、Thinking about the trait of reload option,we syncretized form of geometry continue average-rate into reload option and get a new type exotic option,and changed properly.The pricing formula is deduced.
4、Based on demand of market and deficiency of executive stock option,we contrived a new type ESO which with asian option and obtained explicit expression of the new option.
本文讨论亚式几何连续平均期权的定价。根据奇异期权各自的特点,将亚式期权与其它一些奇异期权组合,设计并得到了新型亚式几何连续期权,以满足市场和公司金融政策的具体需要。在标的资产服从指数O-U过程模型的假设下,利用Girsanov定理,构造了风险中性概率测度,应用随机分析的理论和工具,给出了一类亚式期权价格的显示表达式。
本文主要工作如下:
1、简化了亚式几何连续平均形式的表达式,用鞅方法推出了一般亚式期权定价公式,并讨论了看涨与看跌的平价关系。
2、将亚式期权与障碍期权的特点联系起来,创造出一种新型亚式障碍期权,并给出期权定价的显式表达式。
3、考虑到再装期权的特征,将亚式几何连续平均形式融入其中,并且进行了适当的变化,得到一种新型亚式再装期权,并推导出这种新型期权的定价公式。
4、根据市场的需求和普通经理人员股票的不足,设计了一种新型亚式经理股票期权,并给出相应期权的显示表达式。
Option is an effective tool of risk management.With the development of economy and demand of reset market,Some new type of options was designed from standard option,call it Exotic Options.We are Familiar with some exotic options ,such as Asian Options,Barrier Options,Reset Options,Executive Stock Option( ESO),Look-back Option,Reload Options.Most of all exotic options have characteristics of path-dependence.For example,the payoff of asian options depends not only on the maturity of the underlying asset price,but also some type average value of the underlying assets prices.So it is more complex to price the exotic options than standard options.
in this dissertation we discuss the pricing problem of geometric continue averagerate asian option.We created some new forms of asian option which associated with some other exotic option and asian option based on the character of themselves. Under the hypothesis of the underlying assets are driven by exponential O-U process model,we constructed risk-neutral probability measure by using Girsanov theorem and obtained the pricing formulas of new type option which talked above by application of the theory and tools of stochastic analysis.
In detail we have made works as follows:
1、By skillful handled the form of geometric continue average-rate asian option, the pricing formula of the asian option was driven by using martingale method and put-call parity relation is deduced.
2、Connected characteristic of asian option to barrier option,we created a new type of asian-barrier option and obtained pricing formula of the option.
3、Thinking about the trait of reload option,we syncretized form of geometry continue average-rate into reload option and get a new type exotic option,and changed properly.The pricing formula is deduced.
4、Based on demand of market and deficiency of executive stock option,we contrived a new type ESO which with asian option and obtained explicit expression of the new option.
引文
[1]Black F,Scholes M.The pricing of options and corporate liabilities.Journal of Political Economy,1973,81(3):637-654.
[2]雍炯敏,刘道百.数学金融学[M].上海:上海人民出版社,2003:239-240.
[3]姜礼尚.期权定价的数学模型和方法[M].北京:高等教育出版社,2003.
[4]钱敏平,龚光鲁著.随机过程论[M].北京:北京大学出版社,1997.
[5]Karatzas I,Shreve S E.Brownian motion and Stochastic Calculus[M].Second Edition,New York:Springer-Verlag,1991.
[6]杜雪樵,唐玲.亚式期权定价中的鞅方法[J].合肥工业大学学报,2005,2,206-208.
[7]肖文宁,王杨,张寄洲.几何平均亚式期权定价方法的探析[J].应用数学,2005,18(2):253-259.
[8]章珂,周文彪,沈荣芳.几何平均亚式期权的定价方法[J].同济大学学报.2001.29(8):924-926.
[9]欧辉等.重置期权的创新及其在随机利率下的定价[J].湖南文理学院学报,2004,16(3):6-10.
[10]王莉君,张曙光.随机利率下重置期权的定价问题[J].高校应用数学学报A 辑,2002,17(4).
[11]许永庆,李时银.跳跃扩散型重设卖出期权的定价公式[J].厦门大学学报,2005,1:20-23.
[12]袁国军,杜雪樵.跳-扩散模型中回望期权的定价研究[J].合肥工业大学学报,2006,10:1302-1305.
[13]田蓉,柴俊.等价鞅测度模型在外汇期权定价中的应用[J].华东师范大学学报.2003:27-31.
[14]袁国军,杜雪樵.有交易成本的回望期权定价研究[J].运筹与管理,2006,6:141-143.
[15]刘国买,邹捷中.双边敲出障碍期权定价模型[J].经济数学,2003,20(3):31-37.
[16]叶小青,蹇明,吴永红.亚式期权的保险精算法[J].华中科技大学学报,2005,3:91-93.
[17]薛红,彭玉成.鞅在未定权益中的应用[J].工程数学学报.2000,17(3):135-138.
[18]Margrabe W.The value of an option to exchange one asset for another[J].Journal of Finance,1978,33:177-186.
[19]Peter G,Zhang.Exotic options.World Scientific Publishing co.Ptelet Singapore,1997.
[20]S M Ross.著,陈典发等译.数理金融初步[M].原书第二版.北京:机械工业出版社.2005.
[21]钱晓松.跳扩散模型中亚式期权的定价[J].应用数学.2003,16(4):161-164.
[22]钱晓松.跳扩散模型中的测度变换与期权定价[J].应用概率统计,2004,20(1):91-99.
[23]邓国和.市场结构风险下双指数跳扩散模型期权定价与最具投资消费.博士学位论文,湖南师范大学,2006.
[24]Merton R C.Theory of rational option pricing[J].Bell Journal of Economics and Management Sciense,1973,4:141-183.
[25]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[J].经济数学与应用数学,2004,20(1):79-83.
[26]姚落根,王雄,杨向群.Vasicek利率模型下几何亚式期权的定价[J].湘潭大学自然科学学报.2004,26(3):20-23.
[27]V.P.Knopow,T.V.Petelyaeva.Stochastic models of financial mathematics.Cybernetics and Systems Analysis,2001,37(3):427-433.
[28]Yong-Jin Kim,Naoto Kunitomo.Pricing options under stochastic interest rates.A new research[J].Asia-Pacific Financial Markets.1999,6(1):49-70.
[29]Bernt Oksendal.Stochastic Differential Equations[M].Sixth Edition,NewYork:Springer-Verlag,2005.
[30]闽海峰,刘三阳.带有Posisson跳的股票价格模型的期权定价[J].工程数学学报.2003,20(2):35-40.
[31]陈松男.金融工程学[M].上海:复旦大学出版社.2002.
[32]洛伦兹,格利茨著,唐旭等译.金融工程学[M].修订版.北京:经济科学出版社.1999.
[33]林元烈.应用随机过程[M].北京:清华大学出版社,2002.
[34]严士健,刘秀芳.测度与概率[M].北京:北京师范大学出版.1994.
[35]黄志远.随机分析学基础[M].第二版.北京:科学出版社,2001.
[2]雍炯敏,刘道百.数学金融学[M].上海:上海人民出版社,2003:239-240.
[3]姜礼尚.期权定价的数学模型和方法[M].北京:高等教育出版社,2003.
[4]钱敏平,龚光鲁著.随机过程论[M].北京:北京大学出版社,1997.
[5]Karatzas I,Shreve S E.Brownian motion and Stochastic Calculus[M].Second Edition,New York:Springer-Verlag,1991.
[6]杜雪樵,唐玲.亚式期权定价中的鞅方法[J].合肥工业大学学报,2005,2,206-208.
[7]肖文宁,王杨,张寄洲.几何平均亚式期权定价方法的探析[J].应用数学,2005,18(2):253-259.
[8]章珂,周文彪,沈荣芳.几何平均亚式期权的定价方法[J].同济大学学报.2001.29(8):924-926.
[9]欧辉等.重置期权的创新及其在随机利率下的定价[J].湖南文理学院学报,2004,16(3):6-10.
[10]王莉君,张曙光.随机利率下重置期权的定价问题[J].高校应用数学学报A 辑,2002,17(4).
[11]许永庆,李时银.跳跃扩散型重设卖出期权的定价公式[J].厦门大学学报,2005,1:20-23.
[12]袁国军,杜雪樵.跳-扩散模型中回望期权的定价研究[J].合肥工业大学学报,2006,10:1302-1305.
[13]田蓉,柴俊.等价鞅测度模型在外汇期权定价中的应用[J].华东师范大学学报.2003:27-31.
[14]袁国军,杜雪樵.有交易成本的回望期权定价研究[J].运筹与管理,2006,6:141-143.
[15]刘国买,邹捷中.双边敲出障碍期权定价模型[J].经济数学,2003,20(3):31-37.
[16]叶小青,蹇明,吴永红.亚式期权的保险精算法[J].华中科技大学学报,2005,3:91-93.
[17]薛红,彭玉成.鞅在未定权益中的应用[J].工程数学学报.2000,17(3):135-138.
[18]Margrabe W.The value of an option to exchange one asset for another[J].Journal of Finance,1978,33:177-186.
[19]Peter G,Zhang.Exotic options.World Scientific Publishing co.Ptelet Singapore,1997.
[20]S M Ross.著,陈典发等译.数理金融初步[M].原书第二版.北京:机械工业出版社.2005.
[21]钱晓松.跳扩散模型中亚式期权的定价[J].应用数学.2003,16(4):161-164.
[22]钱晓松.跳扩散模型中的测度变换与期权定价[J].应用概率统计,2004,20(1):91-99.
[23]邓国和.市场结构风险下双指数跳扩散模型期权定价与最具投资消费.博士学位论文,湖南师范大学,2006.
[24]Merton R C.Theory of rational option pricing[J].Bell Journal of Economics and Management Sciense,1973,4:141-183.
[25]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[J].经济数学与应用数学,2004,20(1):79-83.
[26]姚落根,王雄,杨向群.Vasicek利率模型下几何亚式期权的定价[J].湘潭大学自然科学学报.2004,26(3):20-23.
[27]V.P.Knopow,T.V.Petelyaeva.Stochastic models of financial mathematics.Cybernetics and Systems Analysis,2001,37(3):427-433.
[28]Yong-Jin Kim,Naoto Kunitomo.Pricing options under stochastic interest rates.A new research[J].Asia-Pacific Financial Markets.1999,6(1):49-70.
[29]Bernt Oksendal.Stochastic Differential Equations[M].Sixth Edition,NewYork:Springer-Verlag,2005.
[30]闽海峰,刘三阳.带有Posisson跳的股票价格模型的期权定价[J].工程数学学报.2003,20(2):35-40.
[31]陈松男.金融工程学[M].上海:复旦大学出版社.2002.
[32]洛伦兹,格利茨著,唐旭等译.金融工程学[M].修订版.北京:经济科学出版社.1999.
[33]林元烈.应用随机过程[M].北京:清华大学出版社,2002.
[34]严士健,刘秀芳.测度与概率[M].北京:北京师范大学出版.1994.
[35]黄志远.随机分析学基础[M].第二版.北京:科学出版社,2001.