分形市场下的期权定价及其风险管理研究
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摘要
在经济全球化的背景下,金融在中国经济中的核心地位进一步凸现。金融问题,将是各国经济社会发展、改革、稳定必须谨慎处理的首要战略性问题,也将是世界大国间竞争博弈、展现实力的战略性新领域。期权作为金融市场中的衍生证券之一,其理论的研究,对于金融市场的发展和金融风险的防范有着极其重要的应用,在理论和实践上都有重要的意义。
首先,文章对期权定价理论、金融风险的主要成果以及目前的研究热点进行了详尽的介绍,深入阐述了课题的现实意义。接着,介绍了期权的有关概念及相关性质,从有效市场理论优缺点出发,引出了基于非线性市场结构的分形市场理论,并与有效市场理论进行比较,介绍了布朗运动与分数布朗运动的定义及相关性质。
其次,文章研究了标的资产服从分数布朗运动的分数随机微分方程解的一般形式及其基本性质、分数Black-Scholes公式、分数Black-Scholes微分方程以及其对应的推广形式,重点研究了基于分数布朗运动下的障碍期权定价和权证定价公式。
再次,文章基于风险思想和分形市场的期权定价理论,综述性的介绍了风险价值(VaR)法和重标极差法(R/S)的基本原理、计算方法以及有关应用。
最后,文章特别地结合当前形势,针对由美国次贷危机引发的尚未见底、仍在蔓延的国际金融危机。在汲取众多学者的优秀思想的基础上,借用数学的逻辑思维对美国次贷危机发生的原因和启示进行了深入思考。
Against the backdrop of economic globalization, Finance being the core of the China's economy became marked. Financial issues will be the primary strategic problem of economic and social development, reform, stability. It must be handled with careful. Options are derivatives of financial market.Its theoretical study are important in theory and practice.It has an extremely important application in the development of financial markets and preventing financial risks.
First, The paper give a detailed introduction for the main results and current research focus of the option pricing theory, as well as financial risks.And expound the practical significance of the subject in depth. Then, The paper introduce the concept and nature of options. From the advantages and disadvantages of the efficient market theory(EMH),The fractal market hypothesis(FMH) which based on nonlinear market structure be introduced and be compare with the EMH. Then introduce the definition and nature of the Brownian motion and fractional Brownian motion.
Next, The paper study the general form and basic properties of the subject of fractional stochastic differential equations, whose underlying assets subject to fractional Brownian motion.And study the fractional Black-Scholes formula, fractional Black-Scholes differential equations and their deduction. The paper focus on the study of the barrier option and warrant pricing formula under fractional Brownian motion.
Again,The paper gives an overview of the basic principles, calculation methods an application of VaR and R/S,according to the idea of financial risks and the option pricing theory of fractal market.
As we known,The international financial crisis,that caused by the US subprime mortgage crisis,is not yet bottomed out and still spreading. Finally, according to the current situation particularly,The paper consider the cause and revelation of international financial crisis in depth by mathematical logic and the excellent idea of many scholars.
首先,文章对期权定价理论、金融风险的主要成果以及目前的研究热点进行了详尽的介绍,深入阐述了课题的现实意义。接着,介绍了期权的有关概念及相关性质,从有效市场理论优缺点出发,引出了基于非线性市场结构的分形市场理论,并与有效市场理论进行比较,介绍了布朗运动与分数布朗运动的定义及相关性质。
其次,文章研究了标的资产服从分数布朗运动的分数随机微分方程解的一般形式及其基本性质、分数Black-Scholes公式、分数Black-Scholes微分方程以及其对应的推广形式,重点研究了基于分数布朗运动下的障碍期权定价和权证定价公式。
再次,文章基于风险思想和分形市场的期权定价理论,综述性的介绍了风险价值(VaR)法和重标极差法(R/S)的基本原理、计算方法以及有关应用。
最后,文章特别地结合当前形势,针对由美国次贷危机引发的尚未见底、仍在蔓延的国际金融危机。在汲取众多学者的优秀思想的基础上,借用数学的逻辑思维对美国次贷危机发生的原因和启示进行了深入思考。
Against the backdrop of economic globalization, Finance being the core of the China's economy became marked. Financial issues will be the primary strategic problem of economic and social development, reform, stability. It must be handled with careful. Options are derivatives of financial market.Its theoretical study are important in theory and practice.It has an extremely important application in the development of financial markets and preventing financial risks.
First, The paper give a detailed introduction for the main results and current research focus of the option pricing theory, as well as financial risks.And expound the practical significance of the subject in depth. Then, The paper introduce the concept and nature of options. From the advantages and disadvantages of the efficient market theory(EMH),The fractal market hypothesis(FMH) which based on nonlinear market structure be introduced and be compare with the EMH. Then introduce the definition and nature of the Brownian motion and fractional Brownian motion.
Next, The paper study the general form and basic properties of the subject of fractional stochastic differential equations, whose underlying assets subject to fractional Brownian motion.And study the fractional Black-Scholes formula, fractional Black-Scholes differential equations and their deduction. The paper focus on the study of the barrier option and warrant pricing formula under fractional Brownian motion.
Again,The paper gives an overview of the basic principles, calculation methods an application of VaR and R/S,according to the idea of financial risks and the option pricing theory of fractal market.
As we known,The international financial crisis,that caused by the US subprime mortgage crisis,is not yet bottomed out and still spreading. Finally, according to the current situation particularly,The paper consider the cause and revelation of international financial crisis in depth by mathematical logic and the excellent idea of many scholars.
引文
[1]张虹.分数期权定价及其在公司价值评估中的应用[D].湖南:湖南大学,2006.
[2]王伟.谈期权理论的现实应用.吉林省经济管理干部学院学报[J],2005,19(1):27-29.
[3]吴拥政.金融风险测度统计研究[D].湖南:湖南大学,2004.
[4] Hull J.C.Option,Futures,and Other Derivaties.Fourth Edition[M].北京:清华大学出版社,2001:237-240.
[5]彭实戈,倒向随机微分方程及应用[J],数学进展,1997,26(2):97-112.
[6] DuffieD. , GlynnP. , Eficient Monte Carlo estimation of security prices [J].Ann.Appl.Porbab.,1996,(5):897-905.
[7] Courtadon G.A.,A more acurate finite differences approxation for the valuation of options[J].J.Fiacnial Quant Anal.,1982,(17):679-703.
[8] CoxJ,C.,RossS.A.,RubinsteinM.,Option Pricing:asimplified approch[J].J.Finance Econ.,1979,(7):229-263.
[9] Amin K. , JarrowR. , Pricing foreign currency options Under stochastic interest rates[J].J.Internt.Money Fian ce,1991,(10):310-329.
[10] Hull J. , White A. , The pricing of options on assets with stochastic volatilities[J].J.finance,1987,(42):281-300.
[11] Ball C.A.,Torous W.N.,On jumps incomon stock prices and their impact on call option pricing[J].J.Finance,1985,(40):155-173.
[12]薛红,聂赞坎.借贷利率不同情形下具有随机寿命的未定权益定价[J].系统工程理论与实践,2001,12(2):44-46.
[13]周少甫.期权定价理论和倒向随机微分方程[J].科学进步与对策,2000,17(10):100-102.
[14] Bender C. An It? fomula for generalized functionals of a fractional brownian motion with arbitrary hurst parameter[J].stochastic proceses applications,2003,104:81-106.
[15] Bernt ?ksendal. Stochastic differential equations:An introduction with applications [M].Fourth Edition.New York:Springer Verlag,1995:18-69.
[16] Guy Jumarie. On the solution of the stochastic differential equation of exponentialgrowth driven by fractional Brownian motion[J].Applied Mathematics letters.2005,(18):817-826.
[17]刘韶跃,杨向群.分数布朗运动环境中标的资产有红利支付的欧式期权定价[J].经济数学,2002,19(4):35-39.
[18]刘韶跃,杨向群,王剑君.多个分数次布朗运动影响时的混合期权定价[J].系统工程,2005,23(6):110-114.
[19]刘韶跃,杨向群.分数布朗运动环境中欧式未定权益定价[J].应用统计概率,2004,20(4):429-434.
[20]朱淑珍.金融创新与金融风险[M].第一版.上海:复旦大学出版社,2002.
[21] Stephen Ross.The Arbitrage Theory of Capital Asset Pricing[J]. Journal of Economic Theory,1976(9):341-360.
[22] PyleD.H.and S.J. Turnovsky. Safety First and Expected Utility Maximizition in Mean–Standard Deviation Portfolio Analysis[J].Review of Economics and Statistics,1970,( 52 ):75-81.
[23] Atwood J.Demonstration of the Use of Lower Partial Moments to Improve Safet y-First Probability Limits[J].American Jounral of Agricultural Economics,1985,( 67 ):787-793.
[24]王春峰.金融市场风险管理[M].第一版.天津:天津大学出版社,2001.
[25] Jorion P..Value at Risk:The New Benchmark for Controlling Market Risk[M].New York:McGraw-Hill Companies,lnc,1997.
[26]埃德加·E·彼德斯.资本市场的混沌与秩序[M].第二版.王小东译.北京:经济科学出版社,1999.
[27]埃德加·E·彼德斯.分形市场分析[M].第一版.储海林,殷勤译.北京:经济科学出版社,2002.
[28]戴国强,徐龙炳,陆蓉.经济系统的非线性:混沌与分形[J].财经研究,1999,(9):29-34.
[29]魏宇.基于多标度分形理论的金融风险管理方法研究[D].四川:西南交通大学, 2004.
[30]黄永强,郑术专.分形与混沌理论及在现代金融体系中的应用[J].台声新视角,2006,(1):119-120.
[31]尤??金融期权与实物期权——比较与应用[M].上海:上海财经出版社,2003:32-36.
[32]任彪,齐二石,马军海.从有效市场假说到分形市场假说[J].河北大学学报(哲学社会科学版),2004,29(4):82-85.
[33]邱晓坚,孟卫东,王榆.信息机制在有效市场与分形市场的比较[J].重庆交通学院学报,2006,25(1):134-137.
[34]庄后响,吴治民.分形市场假说对有效市场假说的挑战[J].市场周刊·财经论坛,2005,(5):55-56.
[35]樊智,张世英.分形市场理论[J].预测,1999,18(7):117-119.
[36]林勇,郭林军.有效市场假设与分形市场假设[J].预测,2002,21(2):34-38.
[37]史美景,邱长溶.分形金融市场特征及应用[J].财经论坛,2005,(4):103-104.
[38]韩旺红.资本市场理论:有效、协同与分形[J].经济研究参考,2002,(32):20-22.
[39]樊智,张世英.金融市场的效率与分形市场理论[J].系统工程理论与实践,2002,3(3):13-19.
[40]林元烈.应用随机过程[M].北京:清华大学出版社,2002:168-170.
[41]曹宏泽,韩文秀,李旲.投资机会决策中分数布朗运动理论[J].系统工程学报,2001,16(1):45-49.
[42] Nathana?l,Enriquez.A simple construction of the fractional Brownian motion [J].stochastic proceses and their applications,2004,(109):203-223.
[43] T.E.Duncan.Prediction for some processes related to a fractional Brownian motion [J].statistics probability letters,2006,(76):128-134.
[44] Bohdan Maslowski,Dacid Nualart.Evolution equations driven by a fractional Brownian motion[J].journal of functional analysis,2003,(202):277-305.
[45] Guy Jumarie.On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion[J].Applied Mathematics letters.2005,(18):817-826.
[46] John C.Hull.期权,期货和衍生证券[M].张陶伟译.北京:华夏出版社,1997:237-245,161.
[47] Z.-M.YIN.New Method for Simulation of Fractional Brownian Motion [J].Journal of computational physics,1996,127(1):66-72.
[48] Joseph Stampfli,Victor Goodman.金融数学.蔡明超译.北京:机械出版社,2004:108-109.
[49]李霞,金治命.障碍期权的定价问题[J].经济数学,2004,21(3):200-208.
[50] Necula C. . Option pricing in a fractional Brownian motion environment [Z].Preprint Academy of Economic Studies,Burcharest (Romania) ,www. dofin. ase. ro,2002.
[51]王寿仁.概率论基础和随机过程[M].北京:科学出版社,1986.
[52] Wei-Guo Zhang,Wei-Lin Xiao,Chun-Xiong He.Equity warrants pricing model under Fractional Brownian motion and an empirical study[J]. Expert Systems with Applications,36 ,(2009):3056–3065.
[53]张辉.金融创新条件下的金融风险管理[J].重庆社会科学,2007,(5):89-109.
[54]郁俊莉.我国资本市场分形特性研究[J].中南财经政法大学学报,2005,(3):89-93.
[55]黄丹华.美国金融危机对中国金融市场的启示[J].时事资料手册,2008,(6):5-6.
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[59]刘英奎.美国次贷危机的成因及其对中国经济发展的启示[J].北京邮电大学学报(社会科学版),2008,10(3):43-83.
[60]郑良芳.美国次贷危机的影响、成因剖析和十点警示[J].经济研究参考,2007,(66):4-21.
[2]王伟.谈期权理论的现实应用.吉林省经济管理干部学院学报[J],2005,19(1):27-29.
[3]吴拥政.金融风险测度统计研究[D].湖南:湖南大学,2004.
[4] Hull J.C.Option,Futures,and Other Derivaties.Fourth Edition[M].北京:清华大学出版社,2001:237-240.
[5]彭实戈,倒向随机微分方程及应用[J],数学进展,1997,26(2):97-112.
[6] DuffieD. , GlynnP. , Eficient Monte Carlo estimation of security prices [J].Ann.Appl.Porbab.,1996,(5):897-905.
[7] Courtadon G.A.,A more acurate finite differences approxation for the valuation of options[J].J.Fiacnial Quant Anal.,1982,(17):679-703.
[8] CoxJ,C.,RossS.A.,RubinsteinM.,Option Pricing:asimplified approch[J].J.Finance Econ.,1979,(7):229-263.
[9] Amin K. , JarrowR. , Pricing foreign currency options Under stochastic interest rates[J].J.Internt.Money Fian ce,1991,(10):310-329.
[10] Hull J. , White A. , The pricing of options on assets with stochastic volatilities[J].J.finance,1987,(42):281-300.
[11] Ball C.A.,Torous W.N.,On jumps incomon stock prices and their impact on call option pricing[J].J.Finance,1985,(40):155-173.
[12]薛红,聂赞坎.借贷利率不同情形下具有随机寿命的未定权益定价[J].系统工程理论与实践,2001,12(2):44-46.
[13]周少甫.期权定价理论和倒向随机微分方程[J].科学进步与对策,2000,17(10):100-102.
[14] Bender C. An It? fomula for generalized functionals of a fractional brownian motion with arbitrary hurst parameter[J].stochastic proceses applications,2003,104:81-106.
[15] Bernt ?ksendal. Stochastic differential equations:An introduction with applications [M].Fourth Edition.New York:Springer Verlag,1995:18-69.
[16] Guy Jumarie. On the solution of the stochastic differential equation of exponentialgrowth driven by fractional Brownian motion[J].Applied Mathematics letters.2005,(18):817-826.
[17]刘韶跃,杨向群.分数布朗运动环境中标的资产有红利支付的欧式期权定价[J].经济数学,2002,19(4):35-39.
[18]刘韶跃,杨向群,王剑君.多个分数次布朗运动影响时的混合期权定价[J].系统工程,2005,23(6):110-114.
[19]刘韶跃,杨向群.分数布朗运动环境中欧式未定权益定价[J].应用统计概率,2004,20(4):429-434.
[20]朱淑珍.金融创新与金融风险[M].第一版.上海:复旦大学出版社,2002.
[21] Stephen Ross.The Arbitrage Theory of Capital Asset Pricing[J]. Journal of Economic Theory,1976(9):341-360.
[22] PyleD.H.and S.J. Turnovsky. Safety First and Expected Utility Maximizition in Mean–Standard Deviation Portfolio Analysis[J].Review of Economics and Statistics,1970,( 52 ):75-81.
[23] Atwood J.Demonstration of the Use of Lower Partial Moments to Improve Safet y-First Probability Limits[J].American Jounral of Agricultural Economics,1985,( 67 ):787-793.
[24]王春峰.金融市场风险管理[M].第一版.天津:天津大学出版社,2001.
[25] Jorion P..Value at Risk:The New Benchmark for Controlling Market Risk[M].New York:McGraw-Hill Companies,lnc,1997.
[26]埃德加·E·彼德斯.资本市场的混沌与秩序[M].第二版.王小东译.北京:经济科学出版社,1999.
[27]埃德加·E·彼德斯.分形市场分析[M].第一版.储海林,殷勤译.北京:经济科学出版社,2002.
[28]戴国强,徐龙炳,陆蓉.经济系统的非线性:混沌与分形[J].财经研究,1999,(9):29-34.
[29]魏宇.基于多标度分形理论的金融风险管理方法研究[D].四川:西南交通大学, 2004.
[30]黄永强,郑术专.分形与混沌理论及在现代金融体系中的应用[J].台声新视角,2006,(1):119-120.
[31]尤??金融期权与实物期权——比较与应用[M].上海:上海财经出版社,2003:32-36.
[32]任彪,齐二石,马军海.从有效市场假说到分形市场假说[J].河北大学学报(哲学社会科学版),2004,29(4):82-85.
[33]邱晓坚,孟卫东,王榆.信息机制在有效市场与分形市场的比较[J].重庆交通学院学报,2006,25(1):134-137.
[34]庄后响,吴治民.分形市场假说对有效市场假说的挑战[J].市场周刊·财经论坛,2005,(5):55-56.
[35]樊智,张世英.分形市场理论[J].预测,1999,18(7):117-119.
[36]林勇,郭林军.有效市场假设与分形市场假设[J].预测,2002,21(2):34-38.
[37]史美景,邱长溶.分形金融市场特征及应用[J].财经论坛,2005,(4):103-104.
[38]韩旺红.资本市场理论:有效、协同与分形[J].经济研究参考,2002,(32):20-22.
[39]樊智,张世英.金融市场的效率与分形市场理论[J].系统工程理论与实践,2002,3(3):13-19.
[40]林元烈.应用随机过程[M].北京:清华大学出版社,2002:168-170.
[41]曹宏泽,韩文秀,李旲.投资机会决策中分数布朗运动理论[J].系统工程学报,2001,16(1):45-49.
[42] Nathana?l,Enriquez.A simple construction of the fractional Brownian motion [J].stochastic proceses and their applications,2004,(109):203-223.
[43] T.E.Duncan.Prediction for some processes related to a fractional Brownian motion [J].statistics probability letters,2006,(76):128-134.
[44] Bohdan Maslowski,Dacid Nualart.Evolution equations driven by a fractional Brownian motion[J].journal of functional analysis,2003,(202):277-305.
[45] Guy Jumarie.On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion[J].Applied Mathematics letters.2005,(18):817-826.
[46] John C.Hull.期权,期货和衍生证券[M].张陶伟译.北京:华夏出版社,1997:237-245,161.
[47] Z.-M.YIN.New Method for Simulation of Fractional Brownian Motion [J].Journal of computational physics,1996,127(1):66-72.
[48] Joseph Stampfli,Victor Goodman.金融数学.蔡明超译.北京:机械出版社,2004:108-109.
[49]李霞,金治命.障碍期权的定价问题[J].经济数学,2004,21(3):200-208.
[50] Necula C. . Option pricing in a fractional Brownian motion environment [Z].Preprint Academy of Economic Studies,Burcharest (Romania) ,www. dofin. ase. ro,2002.
[51]王寿仁.概率论基础和随机过程[M].北京:科学出版社,1986.
[52] Wei-Guo Zhang,Wei-Lin Xiao,Chun-Xiong He.Equity warrants pricing model under Fractional Brownian motion and an empirical study[J]. Expert Systems with Applications,36 ,(2009):3056–3065.
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