分数布朗运动下的回望期权定价研究
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摘要
期权是金融数学领域内研究最广泛的一类金融工具,而期权定价是其核心工作。
回望期权是为了满足金融市场及不同投资者的需求,在标准期权合同的基础上,运用期权理论和分析方法,设计创造出一种路径依赖型奇异期权,它在到期日的收益依赖整个期权有效期内标的资产价格经历的最大值或最小值。由于具有路径的强依赖性,所以使得回望期权定价比标准期权定价要复杂。
本文是用分数布朗运动来刻画股票价格的变化,用Poisson跳跃过程来刻画当有重大消息到达时股票价格的较大波动。主要应用随机过程等数学工具,讨论了分数布朗运动模型下和分数-跳扩散模型下并且在股票预期收益率、波动率和无风险利率均为时间函数的情况下的回望期权定价问题,建立了回望期权定价模型,结合Wick积分理论,推导出回望期权价格所满足的显示表达式。
Options are financial mathematics area of study of the most extensive one type of financial, while the option pricing is the core work.
In order to meet the financial markets and the needs of different investors in the standard option contract based on the use of option theory and analysis methods, Look-back on options,which designed to create a path-dependent exotic options, it returns the due date dependent upon the options validity of the underlying asset price experiences the maximum or minimum value. Because of the strong path dependence, it makes the look-back option pricing is more complex than the standard option pricing.
This article is a fractional Brownian motion to characterize the changes in stock prices, with Poisson jump process to characterize when there is big news when they arrive in greater volatility in the stock price. Main application of stochastic processes such as mathematical tools to discuss the fractional Brownian motion model and fractional- jump-diffusion model and the expected rate of return in the stock, volatility and risk-free interest rates are a function of time look-back on the context of option pricing problem the establishment of a look-back option pricing model,combined with Wick product theory is derived Looking back on option prices to meet the display expressions.
回望期权是为了满足金融市场及不同投资者的需求,在标准期权合同的基础上,运用期权理论和分析方法,设计创造出一种路径依赖型奇异期权,它在到期日的收益依赖整个期权有效期内标的资产价格经历的最大值或最小值。由于具有路径的强依赖性,所以使得回望期权定价比标准期权定价要复杂。
本文是用分数布朗运动来刻画股票价格的变化,用Poisson跳跃过程来刻画当有重大消息到达时股票价格的较大波动。主要应用随机过程等数学工具,讨论了分数布朗运动模型下和分数-跳扩散模型下并且在股票预期收益率、波动率和无风险利率均为时间函数的情况下的回望期权定价问题,建立了回望期权定价模型,结合Wick积分理论,推导出回望期权价格所满足的显示表达式。
Options are financial mathematics area of study of the most extensive one type of financial, while the option pricing is the core work.
In order to meet the financial markets and the needs of different investors in the standard option contract based on the use of option theory and analysis methods, Look-back on options,which designed to create a path-dependent exotic options, it returns the due date dependent upon the options validity of the underlying asset price experiences the maximum or minimum value. Because of the strong path dependence, it makes the look-back option pricing is more complex than the standard option pricing.
This article is a fractional Brownian motion to characterize the changes in stock prices, with Poisson jump process to characterize when there is big news when they arrive in greater volatility in the stock price. Main application of stochastic processes such as mathematical tools to discuss the fractional Brownian motion model and fractional- jump-diffusion model and the expected rate of return in the stock, volatility and risk-free interest rates are a function of time look-back on the context of option pricing problem the establishment of a look-back option pricing model,combined with Wick product theory is derived Looking back on option prices to meet the display expressions.
引文
[1] Black F.Scholes M.The pricing of options and corporate liabilities[J].Journal of Political Economy,1973.81(3):637—654.
[2] Merton R C.Option pricing when underling stock returns are discontinuous[J]. Journal of Economy,1976.3:125-144.
[3] B. B. Mandelbrot and J. W. Van Ness. Fractional Brownian motions, fractional noises and applications[J]. SIAM Review,1968.10(4),422-437.
[4] Lin, S. J. Stochastic analysis of fractional Brownian motion, fractional noises and application[J].SIAM Review.1995.10:422-437.
[5] L. C. G. Rogers, Arbitrage with fractional Brownian motion[J], Math. Finance. 1997.7:95-105.
[6] T. E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion, I. Theory[J], SIAM J. Control Optim. 2000.38: 582-612.
[7] HU Y . KENDAL B . Fractional white noise calculus and applications to finance[J].Infinite Dim Anal Quantum Prob Related Topics,2003.6:1-32.
[8]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.
[9]刘韶跃,杨向群.分数布朗运动环境中欧式未定权益定价[J].应用概率统计. 2004.20(4):429-434.
[10]王旭,薛红.分数布朗运动环境下的美式看涨期权的定价方法[J].价值工程.2007.11.159-161.
[11]刘韶跃,方秋莲,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.
[12] Knut K,Aase.Contingent claims valuation when the security price is combination of an Itóprocess and a random point process[J].Stochastic processes and their Applications。1998.28(2):185-220.
[13] Martin Schweizer.Option heading for semi-martingales[J].Stochastic processes and their Applications,1991.37(3):1339-360.
[14] Chan T.Pricing contingent claims on stocks driven by Levy processes[J].Annals of Appl Prob,1999.9(2):1504-528.
[15]宁丽娟.刘新平.股票价格服从跳-扩散过程的期权定价模型[J].陕西师范大学学报(自然科学版).2003.31(4):16-19.
[16]梅正阳,杨玉孔.基于鞅方法的分数Brown运动模型的期权定价[J].应用数学.2008,21(4):727-730.
[17] Ciprian Necula. Option Pricing in a Fractional Brownian Motion Environment[J],Preprint, Academy of Economic Studies Bucharest. Romania, www.dofin.ase.ro/
[18]张学莲,薛红.股票价格遵循分数一跳扩散过程的期权定价[J].纺织高校基础科学学报.2008.21(4):446-450.
[19] Ducan T E, Hu Y, Pasik-Ducan B. Stochastic calculus for fractional Brownian motion [J]. Theroy.SIAM J. Control Option. 2000.38:582-612.
[20] Yue-kuen kwok. Mathematical models of Financial Derivatives [M]. Springer, 1999.267-282.
[21] Bender C, An Ito formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter [J]. Stochastic Processes and their Applications 2003.104:81-106.
[22] K. Aase, B.? ksendal, N. Privault and J. Ub?e, White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance[J], Finance Stochastics 2002. 4:465-496.
[23] F. E. Benth, Integrals in the Hida distribution space( S )? , in Stochastic Analysis and Related Topics, eds. T. Lindstr?m, B.? ksendal and A. S.üstünel (Gordon and Breach, 1993), pp. 327-351.
[24] N. J. Cutland, P. E. Kopp and W. Willinger, Stock price returns and the Joseph effect: A fractional version of the Black-Scholes model[J], Seminar on Stochastic Analysis, Random Fields and Applications (Ascona,1993), Prog. Probab., Vol. 36 (Birkh?user, 1995), pp. 327-351.
[25] W. Dai and C. C. Heyde, Itóformula with respect to fractional Brownian motion and its application[J], J. Appl. Math. Stoch. Anal. 9 (1996) 439-448.
[26] A. Dasgupta, Fractional Brownian motion: Its properties and applications to stochastic integration[D], Ph. D. Thesis, Department of Statistics, University of North Carolina at Chapel Hill, 1997.
[27] L. Decreusefond and A. S.üstünel, Stochastic analysis of the fractional Brownian motion[J], Potential Anal. 10 (1999) 177-214.
[28] Y. Hu, Itó-Wiener chaos expansion with exact residual and correlation, variance inequalities[J], Theor. Probab. 10 (1997) 835-848.
[29] Y. Hu and B. ?ksendal, Wick approximation of quasilinear stochastic differential equations[J], in Stochastic Analysis and Related Topics, Vol. 5, eds. H. K?rezlioglu, B. ?ksendal and A. S.üstünel (Birkh?user, 1996), pp. 203-231.
[30] Y. Hu, B. ?ksendal and A. Sulem, Optimal portfolio in a fractional Black-Scholes market[J], in Mathematical Physics and Stochastic Analysis, eds. S. Albeverio et al. (World Scientific, 2000), pp. 267-279.
[31] Y. Hu, B. ?ksendal and A. Sulem, Optimal consumption and portfolio in aBlack-Scholes market driven by fractional Brownian motion[J], Inf. Dim. Anal. Quantum Probab. Related Topics, to appear .
[32] G. Kallianpur and R. L. Karandikar, Introduction to Option Pricing Theory[J](Birkh?user, 2000).
[33] H. Holden, B.?ksenda, J.Ub?e and T. Zhang, Stochastic Partial Differential Equations[M] (Birkh?user, 1996).
[34] H.-H. Kuo, White Noise Distribution Theory[M] (CRC Press,1996).
[35] S. J. Lin, Stochastic analysis of fractional Brownian motion[J], Stochastics Stochastic Rep. 55 (1995) 121-140.
[36] T. Lindstr_m, B.?ksendal and J. Ub?e, Wick multiplication and Itó-Skorohod stochastic differential equations, in Ideas and Methods in Mathematical Analysis[J], Stochastics and Applications, eds. S. Albeverio, J. E. Fenstad, H. Holden and T. Lindstr?m (Cambridge Univ. Press, 1992), pp. 183-206.
[37] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications[J], SIAM Rev. 10 (1968) 422-437.
[38] B. B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk[M], (Springer-Verlag,1997).
[39] T. Sottinen, Fractional Brownian motion, random walks and binary market models[J], preprint, University of Helsinki, June 1999.
[40] E. Valkeila, On some properties of geometric fractional Brownian motion[J], preprint, University of Helsinki, May 1999.
[41] Merton Robert C. Continuous-Time Finance (Revised Edition),Blackwell Publishers Inc., (1997).中文版.R.C.默顿,连续时间金融(修订版) [M],北京:中国人民大学出版社,(2005).
[42]谢振羽,蒋国中.数据仓库技术及其应用[J].河海大学常州分校学报,2003.17(2):63-67.
[43] Claudia I, Nicholas G, Geiger J G.数据仓库设计[M].于戈,鲍玉斌,王大玲译.北京:机械工业出版社,2004.10-11.
[44]高红兵,潘谨,陈宏民.我国证券波动的Hurst指数[J].东华大学学报, 2001.27(4):22-25.
[45]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,2003.295-304.
[46]张艳秋,杜雪樵.随机利率下的回望期权的定价[J].合肥工业大学学报(自然科学版),2007.30(4):515-517.
[47]吴奕东.跳跃-扩散过程中一些新型期权的定价[D].湖南师范大学硕士论文.
[48]彭勃.一类更新跳扩散模型下的期权定价研究[D].合肥工业大学硕士论文.
[49] A.N.Kolnogorov, Wienersche Spiralen und einige andere interessante Kurvenim Hilbertschen Raum. C.R.(Doklady)Acad. URSS(N.S).1940.26:115-118.
[50] A.M.Yaglom, Correlation theory of processes with random stationary n th increments[J]. AMS Transl. 1958.2(8):87-141.
[51] B.Francesca, HU Y, B.KENDAL and Zhang T, Stpchastic calculus for Fractional Brownian Motions and applications[M].Springer,2008.5-97.
[2] Merton R C.Option pricing when underling stock returns are discontinuous[J]. Journal of Economy,1976.3:125-144.
[3] B. B. Mandelbrot and J. W. Van Ness. Fractional Brownian motions, fractional noises and applications[J]. SIAM Review,1968.10(4),422-437.
[4] Lin, S. J. Stochastic analysis of fractional Brownian motion, fractional noises and application[J].SIAM Review.1995.10:422-437.
[5] L. C. G. Rogers, Arbitrage with fractional Brownian motion[J], Math. Finance. 1997.7:95-105.
[6] T. E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion, I. Theory[J], SIAM J. Control Optim. 2000.38: 582-612.
[7] HU Y . KENDAL B . Fractional white noise calculus and applications to finance[J].Infinite Dim Anal Quantum Prob Related Topics,2003.6:1-32.
[8]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.
[9]刘韶跃,杨向群.分数布朗运动环境中欧式未定权益定价[J].应用概率统计. 2004.20(4):429-434.
[10]王旭,薛红.分数布朗运动环境下的美式看涨期权的定价方法[J].价值工程.2007.11.159-161.
[11]刘韶跃,方秋莲,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.
[12] Knut K,Aase.Contingent claims valuation when the security price is combination of an Itóprocess and a random point process[J].Stochastic processes and their Applications。1998.28(2):185-220.
[13] Martin Schweizer.Option heading for semi-martingales[J].Stochastic processes and their Applications,1991.37(3):1339-360.
[14] Chan T.Pricing contingent claims on stocks driven by Levy processes[J].Annals of Appl Prob,1999.9(2):1504-528.
[15]宁丽娟.刘新平.股票价格服从跳-扩散过程的期权定价模型[J].陕西师范大学学报(自然科学版).2003.31(4):16-19.
[16]梅正阳,杨玉孔.基于鞅方法的分数Brown运动模型的期权定价[J].应用数学.2008,21(4):727-730.
[17] Ciprian Necula. Option Pricing in a Fractional Brownian Motion Environment[J],Preprint, Academy of Economic Studies Bucharest. Romania, www.dofin.ase.ro/
[18]张学莲,薛红.股票价格遵循分数一跳扩散过程的期权定价[J].纺织高校基础科学学报.2008.21(4):446-450.
[19] Ducan T E, Hu Y, Pasik-Ducan B. Stochastic calculus for fractional Brownian motion [J]. Theroy.SIAM J. Control Option. 2000.38:582-612.
[20] Yue-kuen kwok. Mathematical models of Financial Derivatives [M]. Springer, 1999.267-282.
[21] Bender C, An Ito formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter [J]. Stochastic Processes and their Applications 2003.104:81-106.
[22] K. Aase, B.? ksendal, N. Privault and J. Ub?e, White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance[J], Finance Stochastics 2002. 4:465-496.
[23] F. E. Benth, Integrals in the Hida distribution space( S )? , in Stochastic Analysis and Related Topics, eds. T. Lindstr?m, B.? ksendal and A. S.üstünel (Gordon and Breach, 1993), pp. 327-351.
[24] N. J. Cutland, P. E. Kopp and W. Willinger, Stock price returns and the Joseph effect: A fractional version of the Black-Scholes model[J], Seminar on Stochastic Analysis, Random Fields and Applications (Ascona,1993), Prog. Probab., Vol. 36 (Birkh?user, 1995), pp. 327-351.
[25] W. Dai and C. C. Heyde, Itóformula with respect to fractional Brownian motion and its application[J], J. Appl. Math. Stoch. Anal. 9 (1996) 439-448.
[26] A. Dasgupta, Fractional Brownian motion: Its properties and applications to stochastic integration[D], Ph. D. Thesis, Department of Statistics, University of North Carolina at Chapel Hill, 1997.
[27] L. Decreusefond and A. S.üstünel, Stochastic analysis of the fractional Brownian motion[J], Potential Anal. 10 (1999) 177-214.
[28] Y. Hu, Itó-Wiener chaos expansion with exact residual and correlation, variance inequalities[J], Theor. Probab. 10 (1997) 835-848.
[29] Y. Hu and B. ?ksendal, Wick approximation of quasilinear stochastic differential equations[J], in Stochastic Analysis and Related Topics, Vol. 5, eds. H. K?rezlioglu, B. ?ksendal and A. S.üstünel (Birkh?user, 1996), pp. 203-231.
[30] Y. Hu, B. ?ksendal and A. Sulem, Optimal portfolio in a fractional Black-Scholes market[J], in Mathematical Physics and Stochastic Analysis, eds. S. Albeverio et al. (World Scientific, 2000), pp. 267-279.
[31] Y. Hu, B. ?ksendal and A. Sulem, Optimal consumption and portfolio in aBlack-Scholes market driven by fractional Brownian motion[J], Inf. Dim. Anal. Quantum Probab. Related Topics, to appear .
[32] G. Kallianpur and R. L. Karandikar, Introduction to Option Pricing Theory[J](Birkh?user, 2000).
[33] H. Holden, B.?ksenda, J.Ub?e and T. Zhang, Stochastic Partial Differential Equations[M] (Birkh?user, 1996).
[34] H.-H. Kuo, White Noise Distribution Theory[M] (CRC Press,1996).
[35] S. J. Lin, Stochastic analysis of fractional Brownian motion[J], Stochastics Stochastic Rep. 55 (1995) 121-140.
[36] T. Lindstr_m, B.?ksendal and J. Ub?e, Wick multiplication and Itó-Skorohod stochastic differential equations, in Ideas and Methods in Mathematical Analysis[J], Stochastics and Applications, eds. S. Albeverio, J. E. Fenstad, H. Holden and T. Lindstr?m (Cambridge Univ. Press, 1992), pp. 183-206.
[37] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications[J], SIAM Rev. 10 (1968) 422-437.
[38] B. B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk[M], (Springer-Verlag,1997).
[39] T. Sottinen, Fractional Brownian motion, random walks and binary market models[J], preprint, University of Helsinki, June 1999.
[40] E. Valkeila, On some properties of geometric fractional Brownian motion[J], preprint, University of Helsinki, May 1999.
[41] Merton Robert C. Continuous-Time Finance (Revised Edition),Blackwell Publishers Inc., (1997).中文版.R.C.默顿,连续时间金融(修订版) [M],北京:中国人民大学出版社,(2005).
[42]谢振羽,蒋国中.数据仓库技术及其应用[J].河海大学常州分校学报,2003.17(2):63-67.
[43] Claudia I, Nicholas G, Geiger J G.数据仓库设计[M].于戈,鲍玉斌,王大玲译.北京:机械工业出版社,2004.10-11.
[44]高红兵,潘谨,陈宏民.我国证券波动的Hurst指数[J].东华大学学报, 2001.27(4):22-25.
[45]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,2003.295-304.
[46]张艳秋,杜雪樵.随机利率下的回望期权的定价[J].合肥工业大学学报(自然科学版),2007.30(4):515-517.
[47]吴奕东.跳跃-扩散过程中一些新型期权的定价[D].湖南师范大学硕士论文.
[48]彭勃.一类更新跳扩散模型下的期权定价研究[D].合肥工业大学硕士论文.
[49] A.N.Kolnogorov, Wienersche Spiralen und einige andere interessante Kurvenim Hilbertschen Raum. C.R.(Doklady)Acad. URSS(N.S).1940.26:115-118.
[50] A.M.Yaglom, Correlation theory of processes with random stationary n th increments[J]. AMS Transl. 1958.2(8):87-141.
[51] B.Francesca, HU Y, B.KENDAL and Zhang T, Stpchastic calculus for Fractional Brownian Motions and applications[M].Springer,2008.5-97.