期权定价公式的分形几何证明
摘要
二十世纪八十年代初,由Mandelbrot所创立的分形几何理论在诸多领域都显示出了广泛的应用性,其中金融领域就是分形理论应用的一个典型领域.Mandelbrot曾经从统计学的观点发现了关于股票价格变动两条的规律,但这只是统计学意义上的规律,我们无法利用这些法则来预测股票的未来价格.
为了减少股票价格变动的不确定性所带来的损失以提高收益的稳定性,人们创造了期权这一金融衍生品.随之而来的问题是如何为期权确定一个合理的价格,以使交易的双方都能接受.布莱克和舒尔斯两人经过不懈的努力,找出了为期权定价的公式,也就是现在所熟知的布莱克-舒尔斯公式,这一公式已成为金融学领域的一大经典.本文探讨了分形几何理论在期权定价中的应用,主要利用分数布朗运动证明了期权定价的分数次布莱克-舒尔斯公式,并把所得的结果与建立在标准布朗运动之上的经典结论进行比较,发现经典的结果只是分数次布莱克-舒尔斯的一种特殊形式.从这点可以进一步看出分形理论在金融分析中的重要作用.
In 1980s, the fractal geometry theory founded by Mandelbrot had been greatly used in many fields, among which financial field is one of the typical. Mandelbrot has found two rules about the change of stock price from statistical analysis. But it’s just the statistical analysis, we can’t use the results to estimate the future price of stock.
In order to increase the stability of yield, people create the option which is a derivative financial instrument. Then the coming problem is how to make a reasonable price for an option. With great efforts, Black and Scholes got a wonderful formula, famous for Black-Scholes formula. This paper is mainly about the applications of fractal geometry theory in the aspects of option pricing . I mainly use fractional Brownian motion to prove a fractional Black-Scholes formula for the price of an option and compare the results with the classical ones which based on the standard Brownian motion. I concludes that the classical results is just one special case of the fractional Black-Scholes formula. From this point we can see the wide usage of fractal geometry in financial fields.
为了减少股票价格变动的不确定性所带来的损失以提高收益的稳定性,人们创造了期权这一金融衍生品.随之而来的问题是如何为期权确定一个合理的价格,以使交易的双方都能接受.布莱克和舒尔斯两人经过不懈的努力,找出了为期权定价的公式,也就是现在所熟知的布莱克-舒尔斯公式,这一公式已成为金融学领域的一大经典.本文探讨了分形几何理论在期权定价中的应用,主要利用分数布朗运动证明了期权定价的分数次布莱克-舒尔斯公式,并把所得的结果与建立在标准布朗运动之上的经典结论进行比较,发现经典的结果只是分数次布莱克-舒尔斯的一种特殊形式.从这点可以进一步看出分形理论在金融分析中的重要作用.
In 1980s, the fractal geometry theory founded by Mandelbrot had been greatly used in many fields, among which financial field is one of the typical. Mandelbrot has found two rules about the change of stock price from statistical analysis. But it’s just the statistical analysis, we can’t use the results to estimate the future price of stock.
In order to increase the stability of yield, people create the option which is a derivative financial instrument. Then the coming problem is how to make a reasonable price for an option. With great efforts, Black and Scholes got a wonderful formula, famous for Black-Scholes formula. This paper is mainly about the applications of fractal geometry theory in the aspects of option pricing . I mainly use fractional Brownian motion to prove a fractional Black-Scholes formula for the price of an option and compare the results with the classical ones which based on the standard Brownian motion. I concludes that the classical results is just one special case of the fractional Black-Scholes formula. From this point we can see the wide usage of fractal geometry in financial fields.
引文
[1]文志英.分形几何的数学基础[M].上海:上海科技教育出版社,1998.
[2]法尔科内.分形几何—数学基础及其应用(第二版)[M].北京:人民邮电出版社,2007.
[3]张济忠.分形[M].北京:清华大学出版社,1997.
[4]陈松男.金融工程学[M].上海:复旦大学出版社,2002.
[5] (英)理查德A.布雷利,(美)斯图尔特C.迈尔斯,(美)弗兰克林?艾伦.公司财务原理[M].北京:机械工业出版社,2008.
[6]宋逢明.金融工程原理-无套利均衡分析[M].北京:清华大学出版社,2004.
[7]约翰?马歇尔,维普尔?班赛尔.金融工程[M].北京:清华大学出版社,1998.
[8]张波,张景肖.应用随机过程[M].北京:清华大学出版社,2007.
[9]吴建民,庄菁.分形分布在股票市场中的应用研究[J].中国流通经济, 2007, 3: 59-61.
[10]邹志勇.金融数据的分形实证分析[J].电脑与信息技术,2001,6:62-64.
[11] Hu,Y., B.Oksendal. Stochastic calculus and application to Finance[J]. Preprint, University of Oslo, 2000.
[12] Hu Y, Oksendal B, Sulem A. Optimal consumption and portfolio in a Black-Scholes market driven by fractional brownian motion [J]. Preprint 23/2000, University of Oslo, 2000.
[13] Necula C. Stochastic Calculus of the Fractional Brownian motion and Applications in Mathematical Finance, MSc Dissertation Paper, Faculty of Mathematics, University of Bucharest, 2003.
[14] Oksendal. B., Fractional Brownian motion in Finance, Preprint, University of Oslo, 2003, 28.
[15] Duncan T E, Hu Y, Pasik B Duncan. Stochastic calculus for fractional Brownian motion Theory [J]. SIAMJ. Control Optim.,2000,38:582-612.
[16] Bender C. The Fractional Ito Integral, Change of Measure and Absence ofArbitrage,Preprint,2002
[17] Peters E E. Fractal Structure in the Capital Markets [J]. Financial Analysis Journal, 1989, 4:45.
[18] Devroye, Luc. Non-Uniform Random Variate Generation[M]. New York: Springer- Verlag Inc,1986.
[19] Kanter, Stable M. Densities under change of Scale and Total Variation Inequalities [J].Annals of Probability, 1975, 3:697-707.
[20] Tudor C. On the Wiener integral with respect to the fractional Brownian motion [J].Bol.Mex.Mat.Soc,2002, 8: 97-106.
[21] Aase A, ?ksendal B, Privault N, Uboe J. White noise generalizations of the Clark Haussmann Ocone theorem with application to mathematical finance [J]. Finance & Stochastics 2000, 4: 465-496.
[22] Ciprian Necula. Option Pricing in a Fractional Brownian Motion Environment [J].Academy of Economic Studies Bucharest, 2002, 12:1-17.
[23] Benth F E. Integrals in the Hida distribution space .In T. Lindstrom, B. ?ksendal and A.S.Ustunel: Stochastic Analysis and Related Topics. Gordon and Breach 1993, 89-99.
[24] Cutland N J, Kopp P E, Willinger W. Stock price returns and the Joseph effect : A fractional version of the Black-Scholes model[M]. Seminar on Stochastic Analysis, Random Fields and Applications 1993, 327-351.
[25] Dai W, Heyde C C. It? formula with respect to fractional Brownian motion and its application [J]. J. Appl. Math. Stoch. Anal. 9 (1996), 439-448.
[26] Dasgupta A. Fractional Brownian motion: Its properties and applications to stochastic integration [D]. Ph. D Thesis, Dept of Statistics, University of North Carolina at Chapel Hill, 1997.
[27] Dasgupta A, Kallianpur G. Arbitrage opportunities for a class of Gladyshev processes [J]. Appl. Math. Optim., 2000, 41: 377-385.
[28] Dasgupta A, Kallianpur G. Chaos decomposition of multiple fractional integrals and applications [J]. Probab. Theory Related Fields,1999, 115: 527-548.
[29] Duncan T E, Hu Y, Pasik Duncan B. Stochastic calculus for fractional Brownianmotion [J]. I. Theory. SIAM J. Control Optim. 2000, 38: 582-612
[30] Chambers J M, Mallows C L, Stuck B W. A Method for Simulating Stable Random Variables [J] .Journal of the American Statistical Association, 1976, 71: 340-342.
[31] Decreusefond L,ǖstǖnel A S. Stochastic analysis of the fractional Brownian motion [J]. Potential Analysis, 1999, 10: 177-214.
[32]孙霞,吴自勤,黄畇.分形原理及其应用[M].合肥:中国科学技术大学出版社,2003.10.
[33] Gripenberg G, Norros I. On the prediction of fractional Brownian motion[J]. J. Appl. Prob. ,1996, 33: 400-410.
[34] Hida T, Kuo H H, Pottho J, Streit L. White Noise Analysis[M]. Kluwer, 1993.
[35] Y.Hu: It?-Wiener chaos expansion with exact residual and correlation, variance in equalities [J]. Journal of Theoretical Probability, 1997, 10: 835-848
[36] Y. Hu, B.?ksendal. Wick approximation of quasi-linear stochastic differential equations. In H.K?rezlioglu, B. ?ksendal and A.S. Ustunel: Stochastic Analysis and Related Topics, Birkh?user 1996, 5: 203-231
[37] Hu Y, Ksendal B, Sulem A. Optimal portfolio in a fractional Black-Scholes market. In S. Albeverio et al.: Mathematical Physics and Stochastic Analysis. World Scienti 2000
[38] Y. Hu, B.?ksendal, A. Sulem. Optimal consumption and portfolio in a Black-Scholes market driven by fractional Brownian motion. Inf. Dim. Analysis, Quantum Probab. Rel. Topics
[39] H. Holden, B. ?ksendal, J. Ube and T. Zhang. Stochastic Partial Di Equations. Birkh?user 1996.
[40] T. Sottinen: Fractional Brownian motion, random walks and binary market models. Preprint, University of Helsinki, June 1999.
[2]法尔科内.分形几何—数学基础及其应用(第二版)[M].北京:人民邮电出版社,2007.
[3]张济忠.分形[M].北京:清华大学出版社,1997.
[4]陈松男.金融工程学[M].上海:复旦大学出版社,2002.
[5] (英)理查德A.布雷利,(美)斯图尔特C.迈尔斯,(美)弗兰克林?艾伦.公司财务原理[M].北京:机械工业出版社,2008.
[6]宋逢明.金融工程原理-无套利均衡分析[M].北京:清华大学出版社,2004.
[7]约翰?马歇尔,维普尔?班赛尔.金融工程[M].北京:清华大学出版社,1998.
[8]张波,张景肖.应用随机过程[M].北京:清华大学出版社,2007.
[9]吴建民,庄菁.分形分布在股票市场中的应用研究[J].中国流通经济, 2007, 3: 59-61.
[10]邹志勇.金融数据的分形实证分析[J].电脑与信息技术,2001,6:62-64.
[11] Hu,Y., B.Oksendal. Stochastic calculus and application to Finance[J]. Preprint, University of Oslo, 2000.
[12] Hu Y, Oksendal B, Sulem A. Optimal consumption and portfolio in a Black-Scholes market driven by fractional brownian motion [J]. Preprint 23/2000, University of Oslo, 2000.
[13] Necula C. Stochastic Calculus of the Fractional Brownian motion and Applications in Mathematical Finance, MSc Dissertation Paper, Faculty of Mathematics, University of Bucharest, 2003.
[14] Oksendal. B., Fractional Brownian motion in Finance, Preprint, University of Oslo, 2003, 28.
[15] Duncan T E, Hu Y, Pasik B Duncan. Stochastic calculus for fractional Brownian motion Theory [J]. SIAMJ. Control Optim.,2000,38:582-612.
[16] Bender C. The Fractional Ito Integral, Change of Measure and Absence ofArbitrage,Preprint,2002
[17] Peters E E. Fractal Structure in the Capital Markets [J]. Financial Analysis Journal, 1989, 4:45.
[18] Devroye, Luc. Non-Uniform Random Variate Generation[M]. New York: Springer- Verlag Inc,1986.
[19] Kanter, Stable M. Densities under change of Scale and Total Variation Inequalities [J].Annals of Probability, 1975, 3:697-707.
[20] Tudor C. On the Wiener integral with respect to the fractional Brownian motion [J].Bol.Mex.Mat.Soc,2002, 8: 97-106.
[21] Aase A, ?ksendal B, Privault N, Uboe J. White noise generalizations of the Clark Haussmann Ocone theorem with application to mathematical finance [J]. Finance & Stochastics 2000, 4: 465-496.
[22] Ciprian Necula. Option Pricing in a Fractional Brownian Motion Environment [J].Academy of Economic Studies Bucharest, 2002, 12:1-17.
[23] Benth F E. Integrals in the Hida distribution space .In T. Lindstrom, B. ?ksendal and A.S.Ustunel: Stochastic Analysis and Related Topics. Gordon and Breach 1993, 89-99.
[24] Cutland N J, Kopp P E, Willinger W. Stock price returns and the Joseph effect : A fractional version of the Black-Scholes model[M]. Seminar on Stochastic Analysis, Random Fields and Applications 1993, 327-351.
[25] Dai W, Heyde C C. It? formula with respect to fractional Brownian motion and its application [J]. J. Appl. Math. Stoch. Anal. 9 (1996), 439-448.
[26] Dasgupta A. Fractional Brownian motion: Its properties and applications to stochastic integration [D]. Ph. D Thesis, Dept of Statistics, University of North Carolina at Chapel Hill, 1997.
[27] Dasgupta A, Kallianpur G. Arbitrage opportunities for a class of Gladyshev processes [J]. Appl. Math. Optim., 2000, 41: 377-385.
[28] Dasgupta A, Kallianpur G. Chaos decomposition of multiple fractional integrals and applications [J]. Probab. Theory Related Fields,1999, 115: 527-548.
[29] Duncan T E, Hu Y, Pasik Duncan B. Stochastic calculus for fractional Brownianmotion [J]. I. Theory. SIAM J. Control Optim. 2000, 38: 582-612
[30] Chambers J M, Mallows C L, Stuck B W. A Method for Simulating Stable Random Variables [J] .Journal of the American Statistical Association, 1976, 71: 340-342.
[31] Decreusefond L,ǖstǖnel A S. Stochastic analysis of the fractional Brownian motion [J]. Potential Analysis, 1999, 10: 177-214.
[32]孙霞,吴自勤,黄畇.分形原理及其应用[M].合肥:中国科学技术大学出版社,2003.10.
[33] Gripenberg G, Norros I. On the prediction of fractional Brownian motion[J]. J. Appl. Prob. ,1996, 33: 400-410.
[34] Hida T, Kuo H H, Pottho J, Streit L. White Noise Analysis[M]. Kluwer, 1993.
[35] Y.Hu: It?-Wiener chaos expansion with exact residual and correlation, variance in equalities [J]. Journal of Theoretical Probability, 1997, 10: 835-848
[36] Y. Hu, B.?ksendal. Wick approximation of quasi-linear stochastic differential equations. In H.K?rezlioglu, B. ?ksendal and A.S. Ustunel: Stochastic Analysis and Related Topics, Birkh?user 1996, 5: 203-231
[37] Hu Y, Ksendal B, Sulem A. Optimal portfolio in a fractional Black-Scholes market. In S. Albeverio et al.: Mathematical Physics and Stochastic Analysis. World Scienti 2000
[38] Y. Hu, B.?ksendal, A. Sulem. Optimal consumption and portfolio in a Black-Scholes market driven by fractional Brownian motion. Inf. Dim. Analysis, Quantum Probab. Rel. Topics
[39] H. Holden, B. ?ksendal, J. Ube and T. Zhang. Stochastic Partial Di Equations. Birkh?user 1996.
[40] T. Sottinen: Fractional Brownian motion, random walks and binary market models. Preprint, University of Helsinki, June 1999.