亚式期权定价的两个问题探讨
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摘要
在金融市场中，路径依赖型期权因其路径依赖的特征而比标准期权更具有竞争优势，从而日益成为期权定价领域的研究热点．CEV模型是几何布朗运动的推广，近年来CEV模型下路径依赖型期权的定价研究主要集中在回望期权和障碍期权，而冷落了同样具有路径依赖特征的亚式期权．
本文主要考虑了两个问题：标的资产服从CEV过程时几何平均亚式期权的定价和B-S模型下具有浮动敲定价格的算术平均亚式期权的定价．
首先，阐述了标准几何亚式期权的涵义及其模型，介绍CED的涵义，借助Phelim P．Boyle和Yisong Tian为CEV模型下回望期权和障碍期权的定价技巧，利用二叉树逼近方法得到服从CEV过程且有离散红利支付几何亚式期权的定价。然后将几何平均的权数推广到更一般的加权几何平均，得到CEV过程下有离散红利支付的加权几何平均亚式期权的定价。
其次，考虑了B-S模型下具有浮动敲定价格的算术平均亚式期权的定价问题．为简化二叉树方法中繁琐的计算量，我们在已有的线性插值的基础上引入抛物插值方法。利用二叉树方法和抛物插值方法的结合得出了具有浮动敲定价格的算术平均亚式期权的近似定价。最后给出了实例分析，验证了二叉树方法的有效性及收敛性，同时证明了抛物插值具有比线性插值更快的收敛速度．
In the financial market, path-dependent option is more competitive than standardoption profit from its path-dependent character, and increasingly become the researcher’sfocus of the option pricing field. CEV model is the generalization of the Geometric Brown-ian motion, the option pricing research under CEV process mainly focuses on lookbackoptions and barrier options, but ignores the Asian options which is also path-dependent.
In this paper we consider two problems on Asian option pricing: pricing geometricAsian option on underlying assets obeying CEV process; pricing arithmetic average Asianoption with ?oating strike price under B-S model.
Firstly, standard geometric Asian option and its pricing model are considered, thenCEV is introduced. By virtue of the technique Phelim P.Boyle and Yisong Tian usedto price lookback options and barrier options under CEV process, the application of abinomial tree is put forward to get the price of the geometric Asian option with discretedividend paid under CEV process. Besides, we extend the geometric average weightnumber to the more commonly weighted geometric average.
Secondly, we consider the pricing of the arithmetic average Asian option with ?oatingstrike price under B-S model. In order to simplify the fussy computation in the binomialtree method, we introduce the Parabolic interpolation on the basis of the existed linearinterpolation. By the combine of the binomial tree method and the Parabolic interpo-lation, we obtained the approximate price of the arithmetic average Asian option with?oating strike price. In the end, we present the instance analysis, validate the validityand astringency of the binomial tree method, at the same time proved that the parabolicinterpolation is more e?cient than the linear interpolation.
本文主要考虑了两个问题：标的资产服从CEV过程时几何平均亚式期权的定价和B-S模型下具有浮动敲定价格的算术平均亚式期权的定价．
首先，阐述了标准几何亚式期权的涵义及其模型，介绍CED的涵义，借助Phelim P．Boyle和Yisong Tian为CEV模型下回望期权和障碍期权的定价技巧，利用二叉树逼近方法得到服从CEV过程且有离散红利支付几何亚式期权的定价。然后将几何平均的权数推广到更一般的加权几何平均，得到CEV过程下有离散红利支付的加权几何平均亚式期权的定价。
其次，考虑了B-S模型下具有浮动敲定价格的算术平均亚式期权的定价问题．为简化二叉树方法中繁琐的计算量，我们在已有的线性插值的基础上引入抛物插值方法。利用二叉树方法和抛物插值方法的结合得出了具有浮动敲定价格的算术平均亚式期权的近似定价。最后给出了实例分析，验证了二叉树方法的有效性及收敛性，同时证明了抛物插值具有比线性插值更快的收敛速度．
In the financial market, path-dependent option is more competitive than standardoption profit from its path-dependent character, and increasingly become the researcher’sfocus of the option pricing field. CEV model is the generalization of the Geometric Brown-ian motion, the option pricing research under CEV process mainly focuses on lookbackoptions and barrier options, but ignores the Asian options which is also path-dependent.
In this paper we consider two problems on Asian option pricing: pricing geometricAsian option on underlying assets obeying CEV process; pricing arithmetic average Asianoption with ?oating strike price under B-S model.
Firstly, standard geometric Asian option and its pricing model are considered, thenCEV is introduced. By virtue of the technique Phelim P.Boyle and Yisong Tian usedto price lookback options and barrier options under CEV process, the application of abinomial tree is put forward to get the price of the geometric Asian option with discretedividend paid under CEV process. Besides, we extend the geometric average weightnumber to the more commonly weighted geometric average.
Secondly, we consider the pricing of the arithmetic average Asian option with ?oatingstrike price under B-S model. In order to simplify the fussy computation in the binomialtree method, we introduce the Parabolic interpolation on the basis of the existed linearinterpolation. By the combine of the binomial tree method and the Parabolic interpo-lation, we obtained the approximate price of the arithmetic average Asian option with?oating strike price. In the end, we present the instance analysis, validate the validityand astringency of the binomial tree method, at the same time proved that the parabolicinterpolation is more e?cient than the linear interpolation.
引文
[1] Bachelier Louis. Theorie de la speculation[J]. Annales de LEcole Normale-Superieure,1900,17:21～86
[2] Sprenkle Case M. Warrant prices as indicators of expectations and preferences, in therandom character of stock market prices[M]. ed. Paul H. Cootner, Cambridge, MIT Press,1964,412～474
[3] Boness A.J. Elements of a Theory of stock option value[J]. Journal of Political Economy,1964,72:163～175
[4] Samuelson P.A. Rational Theory of Warrant Pricing[J]. Industrial Management Review,1965,6:13～31
[5] Black F, Scholes M. The pricing of options and corporate liablities[J]. Journal of PoliticalEconomy, 1973,81:637～659
[6] Merton R.C. Theory of rational pricing[J]. Bell J of Econ and Magement Sci,1973,4:141～183
[7] Cox John C, Ross Stephen A. The valuation of option for alternative stochastic process[J].Journal of Finacial Economics, 1976,3:145～166
[8] Cox John C, Ross Stephen A,Stein Mark R. Option pricing:a simplified approach[J]. Jour-nal of Financial Economics, 1979,7(5):229～263
[9] Kamard,Ritchken. Multinomial Approximating Models for Options with k-State Vari-ables[J]. Management Science, 1991,12(37):1640～1652
[10] Merton R.C. Option Pricing When Underlying Stock Returns Are Discontinuous[J]. Jour-nal of Financial Economics, 3,Mar.,1976,125～144
[11] John. C. Hull and White A. The Pricing o? Options on Assets with Stochastic Volatili-ties[J]. Journal of Finance 42 Jun, 1987, 281～300
[12] Dupire B. Pricing with a Smile[J]. RISK, Feb.,1994, 18～20
[13] Leland,H.E. Option Pricing and Replication with Transaction Cost[J]. Journal of Finance,Vol.40, Dec.,1985,1283～1301
[14] Beckers.S. The constant Elasticity of Variance Model and Its Implications for OptionPricing[J]. Journal of Finance, 1980,35:661～673
[15] Emanuel D,J MacBeth. Further Results on the Constant Elasticity of Variance CallOption Pricing Model[J]. Journalof Finance and Quantitative Analysis, 1982,17,533～554
[16] Schroder M. Computing the Constant Elasticity of Variance Option Pricing Formula[J].Journal of Finance, 1989,44,Mar.,211~219
[17] Lo C F,Hui C H,YuenP H. Pricing barrier options with squre root process[J]. InternatinalJournal of Theoretical and Applied Finance, 2001,4(5),515~518
[18] Davy D, Linetsky V. Pricing and hedging path Dependent Options Under the CEVprocess[J]. Management Science, 2001,Vol.47,No.7,949~965
[19] MacBeth J D,Merville L L. An empirical examination of the Black-Scholes call optionpricing model[J]. Journal of Finance, 1979,34,1173~1186
[20] MacBeth J D,Merville L L. Test of the Black-Scholes and Cox call option valuation mod-els[J]. Journal of Finance, 1980,35,285~300
[21] Emanuel D C,MacBeth J D. Further results on the constant elasticity of variance calloption pricing model[J]. Journal of Financial and Quantitative Analysis, 1982,17,533~554
[22] Laurent Bouaziz,Eric Briys,Michel Crouly. The pricing of forward-starting Asian op-tions[J]. Journal of Banking and Finance,Vol.18(5),October 1994,823~839
[23]吴云，何建敏．服从cEV的几何亚式期权的定价研究[J]．系统工程理论与实践，No．4,2003,32—36
[24]Phelimp.Boyle,Yisong Tian.Pricing lookback and barrier options under the CEV process[J].Journal of Financial and Quantitative Analysis,1999,34(2),242~264
[25]魏正元．欧式加权几何平均价格亚式期权定价[J]．重庆工学院学报，18(1)，Feb．2004
[26]郑小迎，陈金贤．关于路径依赖型期权定价模型的研究[J].河北师范大学学报，2000,36(2)，15—21
[27]郑小迎，陈金贤．关于新型期权及其定价模型的研究[J].管理工程学报，2000,14(3)，56—60
[28]戴民．路径依赖期权二叉树方法的数值方法[J].[博士学位论文],上海：复旦大学，2000，2，28
[29]章珂等．几何平均亚式期权的定价方法[J].．同济大学学报，2001，29(8)，924—927
[30]孙坚强，李时银．离散算术平均亚式期权近似定价[J].厦门大学学报，V01．42,No．4．Jul．2003[3l]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，Vol．28，No．2，Feb．2005
[32]叶小青，蹇明，吴永红．亚式期权的保险精算定价[J].华中科技大学学报，V01．33,No．3,Mar．2005
[33]李庆扬，王能超，易大义．数值分析[M]．清华大学出版社，200l，23—26
[34]连颖颖．新型二叉树参数模型在亚式期权定价中的应用[J]．中国优秀博硕士学位论文全文数据库，2004
[35]．John c，Hull．期权、期货和其他衍生产品[M]，张陶伟译，北京：华夏出版社，2000,424-425
[36] Zhang P．Exotic Options[M]．World Scientific Publishing,1997
[37]杨少华．亚式期权定价模型及其实证研究[J]中国优秀博硕士学位论文全文数据库，2006
[38] Wilmott P,Dewynne J N,Howison s D. Option pricing, Mathematical computation[M].Oxford Financial press, Oxford 1993
[39] Rogers L,Shi Z. The value of an Asian option[J]. Journal of Applied Probability, 1995,1077~1088
[38] Wilmott P,Dewynne J N,Howison s D. Option pricing, Mathematical computation[M].Oxford Financial press, Oxford 1993
[39] Rogers L,Shi Z. The value of an Asian option[J]. Journal of Applied Probability, 1995,1077~1088
[40]姜礼尚．期权定价的数学模型与方法[M]．高等教育出版社，2003，277~281
[41] Ritchken P,Sankarasubramanian L, Viljh A M. The valuation of path-dependent contractson average[M]. Management Sicence,39,Oct.,1993,1202~1213
[42] Carverhill A,Clewlow L. Flexible convolution[J]. RISK, Apr.1990, 25~29
[43] Levy E. Asian arithmetic[J]. RISK, May.l990,7~8
[44] Levy E. Pricing European average rate currency options[J]. J. Internat, Money Fi-nance,1992
[45] Levy E,Turnbull S. Average intelligence[J]. RISK Magazine,Feb.l992
[46] Ruttiens A. Classical replica[J]. RISK,Feb.l992,5~9
[47] Turnbull A,Wakeman L. A quick algorithm for pricing European average options[J]. J.Financial and Quantitative Analysis,1991,26(3),377~390
[48] Yor M. On some exponential functionals of Brownian Motion[J]. Adv,Appl.prob,1992
[49] Geman H,Yor M. Bessel processes,asian options and perpetuities. In FORC conference,Warwick,UK,1992
[50] Yunzhong Liu, Huiyu Xuan. Asian option pricing based on genetic algorithms[J]. J.System Sicence and Informatin,2004,2(1),109~117
[2] Sprenkle Case M. Warrant prices as indicators of expectations and preferences, in therandom character of stock market prices[M]. ed. Paul H. Cootner, Cambridge, MIT Press,1964,412～474
[3] Boness A.J. Elements of a Theory of stock option value[J]. Journal of Political Economy,1964,72:163～175
[4] Samuelson P.A. Rational Theory of Warrant Pricing[J]. Industrial Management Review,1965,6:13～31
[5] Black F, Scholes M. The pricing of options and corporate liablities[J]. Journal of PoliticalEconomy, 1973,81:637～659
[6] Merton R.C. Theory of rational pricing[J]. Bell J of Econ and Magement Sci,1973,4:141～183
[7] Cox John C, Ross Stephen A. The valuation of option for alternative stochastic process[J].Journal of Finacial Economics, 1976,3:145～166
[8] Cox John C, Ross Stephen A,Stein Mark R. Option pricing:a simplified approach[J]. Jour-nal of Financial Economics, 1979,7(5):229～263
[9] Kamard,Ritchken. Multinomial Approximating Models for Options with k-State Vari-ables[J]. Management Science, 1991,12(37):1640～1652
[10] Merton R.C. Option Pricing When Underlying Stock Returns Are Discontinuous[J]. Jour-nal of Financial Economics, 3,Mar.,1976,125～144
[11] John. C. Hull and White A. The Pricing o? Options on Assets with Stochastic Volatili-ties[J]. Journal of Finance 42 Jun, 1987, 281～300
[12] Dupire B. Pricing with a Smile[J]. RISK, Feb.,1994, 18～20
[13] Leland,H.E. Option Pricing and Replication with Transaction Cost[J]. Journal of Finance,Vol.40, Dec.,1985,1283～1301
[14] Beckers.S. The constant Elasticity of Variance Model and Its Implications for OptionPricing[J]. Journal of Finance, 1980,35:661～673
[15] Emanuel D,J MacBeth. Further Results on the Constant Elasticity of Variance CallOption Pricing Model[J]. Journalof Finance and Quantitative Analysis, 1982,17,533～554
[16] Schroder M. Computing the Constant Elasticity of Variance Option Pricing Formula[J].Journal of Finance, 1989,44,Mar.,211~219
[17] Lo C F,Hui C H,YuenP H. Pricing barrier options with squre root process[J]. InternatinalJournal of Theoretical and Applied Finance, 2001,4(5),515~518
[18] Davy D, Linetsky V. Pricing and hedging path Dependent Options Under the CEVprocess[J]. Management Science, 2001,Vol.47,No.7,949~965
[19] MacBeth J D,Merville L L. An empirical examination of the Black-Scholes call optionpricing model[J]. Journal of Finance, 1979,34,1173~1186
[20] MacBeth J D,Merville L L. Test of the Black-Scholes and Cox call option valuation mod-els[J]. Journal of Finance, 1980,35,285~300
[21] Emanuel D C,MacBeth J D. Further results on the constant elasticity of variance calloption pricing model[J]. Journal of Financial and Quantitative Analysis, 1982,17,533~554
[22] Laurent Bouaziz,Eric Briys,Michel Crouly. The pricing of forward-starting Asian op-tions[J]. Journal of Banking and Finance,Vol.18(5),October 1994,823~839
[23]吴云，何建敏．服从cEV的几何亚式期权的定价研究[J]．系统工程理论与实践，No．4,2003,32—36
[24]Phelimp.Boyle,Yisong Tian.Pricing lookback and barrier options under the CEV process[J].Journal of Financial and Quantitative Analysis,1999,34(2),242~264
[25]魏正元．欧式加权几何平均价格亚式期权定价[J]．重庆工学院学报，18(1)，Feb．2004
[26]郑小迎，陈金贤．关于路径依赖型期权定价模型的研究[J].河北师范大学学报，2000,36(2)，15—21
[27]郑小迎，陈金贤．关于新型期权及其定价模型的研究[J].管理工程学报，2000,14(3)，56—60
[28]戴民．路径依赖期权二叉树方法的数值方法[J].[博士学位论文],上海：复旦大学，2000，2，28
[29]章珂等．几何平均亚式期权的定价方法[J].．同济大学学报，2001，29(8)，924—927
[30]孙坚强，李时银．离散算术平均亚式期权近似定价[J].厦门大学学报，V01．42,No．4．Jul．2003[3l]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，Vol．28，No．2，Feb．2005
[32]叶小青，蹇明，吴永红．亚式期权的保险精算定价[J].华中科技大学学报，V01．33,No．3,Mar．2005
[33]李庆扬，王能超，易大义．数值分析[M]．清华大学出版社，200l，23—26
[34]连颖颖．新型二叉树参数模型在亚式期权定价中的应用[J]．中国优秀博硕士学位论文全文数据库，2004
[35]．John c，Hull．期权、期货和其他衍生产品[M]，张陶伟译，北京：华夏出版社，2000,424-425
[36] Zhang P．Exotic Options[M]．World Scientific Publishing,1997
[37]杨少华．亚式期权定价模型及其实证研究[J]中国优秀博硕士学位论文全文数据库，2006
[38] Wilmott P,Dewynne J N,Howison s D. Option pricing, Mathematical computation[M].Oxford Financial press, Oxford 1993
[39] Rogers L,Shi Z. The value of an Asian option[J]. Journal of Applied Probability, 1995,1077~1088
[38] Wilmott P,Dewynne J N,Howison s D. Option pricing, Mathematical computation[M].Oxford Financial press, Oxford 1993
[39] Rogers L,Shi Z. The value of an Asian option[J]. Journal of Applied Probability, 1995,1077~1088
[40]姜礼尚．期权定价的数学模型与方法[M]．高等教育出版社，2003，277~281
[41] Ritchken P,Sankarasubramanian L, Viljh A M. The valuation of path-dependent contractson average[M]. Management Sicence,39,Oct.,1993,1202~1213
[42] Carverhill A,Clewlow L. Flexible convolution[J]. RISK, Apr.1990, 25~29
[43] Levy E. Asian arithmetic[J]. RISK, May.l990,7~8
[44] Levy E. Pricing European average rate currency options[J]. J. Internat, Money Fi-nance,1992
[45] Levy E,Turnbull S. Average intelligence[J]. RISK Magazine,Feb.l992
[46] Ruttiens A. Classical replica[J]. RISK,Feb.l992,5~9
[47] Turnbull A,Wakeman L. A quick algorithm for pricing European average options[J]. J.Financial and Quantitative Analysis,1991,26(3),377~390
[48] Yor M. On some exponential functionals of Brownian Motion[J]. Adv,Appl.prob,1992
[49] Geman H,Yor M. Bessel processes,asian options and perpetuities. In FORC conference,Warwick,UK,1992
[50] Yunzhong Liu, Huiyu Xuan. Asian option pricing based on genetic algorithms[J]. J.System Sicence and Informatin,2004,2(1),109~117