分数布朗运动下的欧式外汇期权定价及实证分析
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摘要
浮动汇率制度的普遍执行,导致金融市场货币汇率风险日益暴露.作为重要避险工具的外汇期权地位更加突显.
B-S公式很好地解决了期权的定价问题,但是它却依赖于标的资产服从正态分布.而分数布朗运动特有的长期性和自相似性与金融资产价格变化特性相吻合,使得用分数布朗运动来描述金融资产的价格变化更逼真.
本文基于分数布朗运动建立了汇率满足的随机微分方程,利用分形积分理论,在风险中性定价原理下得到了新模型下的外汇期权定价公式.本文建立的新模型中假设本国和外国的无风险利率均是时间的函数,这使得新模型更贴近现实.另外本文还利用修正的R/S分析方法对我国外汇市场做了研究,结果表明我国外汇市场存在很明显的分形结构.这一结论为本文建立的模型提供了现实依据.最后本文通过实证对新模型和传统的外汇期权模型做了比较,表明分数布朗运动下的外汇期权定价更优越.
Floating exchange rate system is generally carried out, which leads to exchange rate risk has become increasingly exposed in financial market. Foreign exchange option has become more prominent as a hedging tool.
B-S formula solves the problem of option pricing well, but it depends on the underlying assets subject to the normal distribution. However, the fraction Brow-nian motion inosculates well with the change of financial assets price by its unique long-term and the like character, so it describes the change of financial assets price more realistic.
This article established a stochastic differential equation of rate based on frac-tion Brownian motion, it used the theory of fractal integral and risk-neutral pricing to get a new model of foreign exchange rate option pricing. In the new model, we assume that the risk-free interest rate of domestic and foreign are functions of time, which makes the new model more close to reality. In addition, the modified R/S analysis is applied to study China's foreign exchange market, the conclusion indi-cates that China's foreign exchange market has a clear fractal structure. The result provides a realistic basis for the new model. At last, this article compares the new model and the traditional model by empirical analysis and gets that the new model is superior.
B-S公式很好地解决了期权的定价问题,但是它却依赖于标的资产服从正态分布.而分数布朗运动特有的长期性和自相似性与金融资产价格变化特性相吻合,使得用分数布朗运动来描述金融资产的价格变化更逼真.
本文基于分数布朗运动建立了汇率满足的随机微分方程,利用分形积分理论,在风险中性定价原理下得到了新模型下的外汇期权定价公式.本文建立的新模型中假设本国和外国的无风险利率均是时间的函数,这使得新模型更贴近现实.另外本文还利用修正的R/S分析方法对我国外汇市场做了研究,结果表明我国外汇市场存在很明显的分形结构.这一结论为本文建立的模型提供了现实依据.最后本文通过实证对新模型和传统的外汇期权模型做了比较,表明分数布朗运动下的外汇期权定价更优越.
Floating exchange rate system is generally carried out, which leads to exchange rate risk has become increasingly exposed in financial market. Foreign exchange option has become more prominent as a hedging tool.
B-S formula solves the problem of option pricing well, but it depends on the underlying assets subject to the normal distribution. However, the fraction Brow-nian motion inosculates well with the change of financial assets price by its unique long-term and the like character, so it describes the change of financial assets price more realistic.
This article established a stochastic differential equation of rate based on frac-tion Brownian motion, it used the theory of fractal integral and risk-neutral pricing to get a new model of foreign exchange rate option pricing. In the new model, we assume that the risk-free interest rate of domestic and foreign are functions of time, which makes the new model more close to reality. In addition, the modified R/S analysis is applied to study China's foreign exchange market, the conclusion indi-cates that China's foreign exchange market has a clear fractal structure. The result provides a realistic basis for the new model. At last, this article compares the new model and the traditional model by empirical analysis and gets that the new model is superior.
引文
[1]Black F.Scholes M.The pricing of options and corporate liabilities[J]. Journal of Po-litical Economy,1973,81:637-655.
[2]John Hull,Alan White.The pricing of options on Assets with Stochastic volatili-ties[J].The Journal of Finance,1987,42:281-300.
[3]孔庆雨.具有随机波动率的期权定价的研究进展[J].经济师,2006,12.
[4]陈俊霞.随机波动率情形下期权定价的解析解[J].统计与决策,2007,8：21-22.
[5]朱利芝.随机波动率下股价服从O-U过程的期权定价[J].经济数学,2010.27.
[6]Merton,Robert.Option pricing when underlying stock returns are discontinu-ous[J].Journal of Financial Economics may,1976,3:125-144.
[7]Philippe Jorion.On jump process in the foreign exchange and stock markets[J].The Review of Financial Studies,1989,4:427-445.
[8]Scott,Louis.Pricing stock options in a jump-diffusion model with Stochastic Volatil-ity and Interest Rate:Applications of Fourier Tnversion Methords[J]. Journal of Mathematical Finance.1997,4:413-426.
[9]闫海峰.带有Poisson跳的股票价格模型的期权定价[J].工程数学学报,2003,2.
[10]姜礼尚.跳扩散模型下永久美式看跌期权定价[J].系统工程理论与实践.2008,2.
[11]袁国军.跳-扩散价格过程下有交易成本的期权定价研究[J].数学的实践与认识.2007,21.
[12]Merton RC.On the pricing of corporate debt:The risk Structure of intere-strate[J].Journal of Finance,1974,29:449-470.
[13]刘文平.利率服从马尔科夫过程时的期权定价[J].华中师范大学学报(自然科学版),2004,2.
[14]姚落根,王雄,杨向群Vasicek利率模型下几何亚式期权定价[J].湘潭大学自然科学报,2004,3.
[15]薛红.随机利率情形下的多维Black-Scholes模型[J].工程数学学报.2005,4.
[16]Eberlein.E,U.Keller.Hyperbolic distributions in finance[J].Bernoulli,1995,1:281-299.
[17]Chan T.Pricing Contigent Claims on Stock Driven by Levy Process[J].Annals of Appl Prob.1999,9.
[18]Kallsen Jan.Optimal Portfolios for Exponential Levy Process [J]. Math Meth oper Res,2005,51:357-377.
[19]杨芝艳Levy模型下复合期权的定价[J].西南师范大学学报(自然科学版),2010,35：103-106.
[20]T.E.Duncan,Y.Hu.Pasik-Duncan.Stochastic calculus for fractional Brownian mo-tion[J].Theory SIAMJ control optim.2000,38:582-612.
[21]E.Alos,O.Mazet,D.Nualart.Stochastic calculus with respect to fractional Brown mo-tion with Hurst parameter lesser than 1/2[J].Stochastic Process Appl,2000,86:121-139.
[22]刘韶跃.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,2002.19：35-39.
[23]周颖.分型统计在投资项目期权价值评估中的应用[J].国际金融研究,2003,2：26-31.
[24]张超.分数布朗运动下随机利率情形的欧式期权定价公式[J].上海师范大学学报(自然科学版),2010,39：558-562.
[25]Garman.Kohlhagen.Foreign currency option values [J]. Journal of Internation Money and Finance,1983,23:1 7.
[26]T.Harikumar,Maria E.Evaluation of Black-Scholes and GARCH model Us-ing Currency Cash options Date[J].Review of Quantitative Finance and Accounting,2004,23:299-312.
[27]吴文吴.外汇期权定价方法的论述及比较[J].东方企业文化,2011,8.
[28]徐静.波动率不确定模型在外汇期权定价中的应用[J].统计与决策,2011,13：169-171.
[29]Lieu Derming.Pricing foreign currency options:a comparison of the modified Black Scholes model and a modified Merton model [J].Journal of Financial Studies,1994.2:75-104.
[30]Dofou A,Hilliard JE.Testing a jump-diffusion stochastic interest rates model in currency options market [R].University of Texas,University of Geor-gia, Workongpaper,1998.
[31]贾品壹,陈燕武.基于跳跃-扩散过程的外汇期权定价模型实证分析[J].湖北财经高等专科学校学报,2010,22：11-13.
[32]张运良,苗芳,刘新平.带跳扩散过程的外汇期权定价[J].陕西师范大学学报(自然科学版),2009.37：15-18.
[33]邓国和.跳跃扩散型汇率过程的外汇期权定价[J].经济数学,2003,20：13-18.
[34]余星.跳跃扩散模型下外汇期权的模糊定价[J].湖南人文科技学院学报,2010,2：5-6.
[35]Grabbe J.The pricing of call and put on foreign exchange[J]. Journal of International Money and Finance,1983,2:239-253.
[36]Hilliard J.Currency option pricing with stochastic domestic and foreign interest rates [J]. Journal of Financial and Quantitative Analysis,1991,26:139-151.
[37]Amin KI.Discrete-time Valuation of American options with stochastic interest r ate [J]. Re view of Financial Studies,1995,8:193-234.
[38]陈迪红.一个外汇期权定价模型[J].财经理论与实践,201.22：59-60.
[39]徐根新Vasicek利率模型下欧式看涨外汇期权定价分析[J].同济大学学报(自然科学版),2006.34.
[40]黄丁伟.外汇期权定价：VG模型与BS模型谁更适用[J].金融发展研究,2009,3：24-25.
[41]陈荣达.基于汇率回报厚尾性的外汇期权定价模型[J].运筹与管理,2006,5：137-140.
[42]Hu Y.0 Ksendal B.Fractional white noise calculus and applications to fi-nance[J].Infinite Dim Anal Quantum Probab Related Topics.2003,6:1-32.
[43]Necula C.Option pricing in a fractional Brownian motion environment[R].Workong paper of the Academy of Economic studies,Bucharest,2002,27.
[44]Rostek S,Schobel R.Risk preference based option pricing in a fractional Brownian Market [J]. Applied Probability Trust.2007,30:1-29.
[45]张敏,李旭,何穗.几何分形Brown运动下的欧式外汇期权定价[J].湖北工业大学学报,2006,6：75-78.
[46]刘目楼.何春雄.分数布朗运动下的欧式外汇期权定价[J].科学技术与工程,2007,7.
[47]张卫国.分数布朗运动下偶是汇率期权定价[J].系统工程理论与实践,2009,6：68-77.
[48]黄礼文.基于分数布朗运动模型的金融衍生产品定价.浙江大学：博士学位论文.2011,4.
[49]叶永刚,黄志坚.外汇期权[M].武汉：武汉大学出版社,2004,7.
[50]Stampli,J著.蔡明超译.金融数学[M].北京：机械工业出版社,2004,3.
[2]John Hull,Alan White.The pricing of options on Assets with Stochastic volatili-ties[J].The Journal of Finance,1987,42:281-300.
[3]孔庆雨.具有随机波动率的期权定价的研究进展[J].经济师,2006,12.
[4]陈俊霞.随机波动率情形下期权定价的解析解[J].统计与决策,2007,8：21-22.
[5]朱利芝.随机波动率下股价服从O-U过程的期权定价[J].经济数学,2010.27.
[6]Merton,Robert.Option pricing when underlying stock returns are discontinu-ous[J].Journal of Financial Economics may,1976,3:125-144.
[7]Philippe Jorion.On jump process in the foreign exchange and stock markets[J].The Review of Financial Studies,1989,4:427-445.
[8]Scott,Louis.Pricing stock options in a jump-diffusion model with Stochastic Volatil-ity and Interest Rate:Applications of Fourier Tnversion Methords[J]. Journal of Mathematical Finance.1997,4:413-426.
[9]闫海峰.带有Poisson跳的股票价格模型的期权定价[J].工程数学学报,2003,2.
[10]姜礼尚.跳扩散模型下永久美式看跌期权定价[J].系统工程理论与实践.2008,2.
[11]袁国军.跳-扩散价格过程下有交易成本的期权定价研究[J].数学的实践与认识.2007,21.
[12]Merton RC.On the pricing of corporate debt:The risk Structure of intere-strate[J].Journal of Finance,1974,29:449-470.
[13]刘文平.利率服从马尔科夫过程时的期权定价[J].华中师范大学学报(自然科学版),2004,2.
[14]姚落根,王雄,杨向群Vasicek利率模型下几何亚式期权定价[J].湘潭大学自然科学报,2004,3.
[15]薛红.随机利率情形下的多维Black-Scholes模型[J].工程数学学报.2005,4.
[16]Eberlein.E,U.Keller.Hyperbolic distributions in finance[J].Bernoulli,1995,1:281-299.
[17]Chan T.Pricing Contigent Claims on Stock Driven by Levy Process[J].Annals of Appl Prob.1999,9.
[18]Kallsen Jan.Optimal Portfolios for Exponential Levy Process [J]. Math Meth oper Res,2005,51:357-377.
[19]杨芝艳Levy模型下复合期权的定价[J].西南师范大学学报(自然科学版),2010,35：103-106.
[20]T.E.Duncan,Y.Hu.Pasik-Duncan.Stochastic calculus for fractional Brownian mo-tion[J].Theory SIAMJ control optim.2000,38:582-612.
[21]E.Alos,O.Mazet,D.Nualart.Stochastic calculus with respect to fractional Brown mo-tion with Hurst parameter lesser than 1/2[J].Stochastic Process Appl,2000,86:121-139.
[22]刘韶跃.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,2002.19：35-39.
[23]周颖.分型统计在投资项目期权价值评估中的应用[J].国际金融研究,2003,2：26-31.
[24]张超.分数布朗运动下随机利率情形的欧式期权定价公式[J].上海师范大学学报(自然科学版),2010,39：558-562.
[25]Garman.Kohlhagen.Foreign currency option values [J]. Journal of Internation Money and Finance,1983,23:1 7.
[26]T.Harikumar,Maria E.Evaluation of Black-Scholes and GARCH model Us-ing Currency Cash options Date[J].Review of Quantitative Finance and Accounting,2004,23:299-312.
[27]吴文吴.外汇期权定价方法的论述及比较[J].东方企业文化,2011,8.
[28]徐静.波动率不确定模型在外汇期权定价中的应用[J].统计与决策,2011,13：169-171.
[29]Lieu Derming.Pricing foreign currency options:a comparison of the modified Black Scholes model and a modified Merton model [J].Journal of Financial Studies,1994.2:75-104.
[30]Dofou A,Hilliard JE.Testing a jump-diffusion stochastic interest rates model in currency options market [R].University of Texas,University of Geor-gia, Workongpaper,1998.
[31]贾品壹,陈燕武.基于跳跃-扩散过程的外汇期权定价模型实证分析[J].湖北财经高等专科学校学报,2010,22：11-13.
[32]张运良,苗芳,刘新平.带跳扩散过程的外汇期权定价[J].陕西师范大学学报(自然科学版),2009.37：15-18.
[33]邓国和.跳跃扩散型汇率过程的外汇期权定价[J].经济数学,2003,20：13-18.
[34]余星.跳跃扩散模型下外汇期权的模糊定价[J].湖南人文科技学院学报,2010,2：5-6.
[35]Grabbe J.The pricing of call and put on foreign exchange[J]. Journal of International Money and Finance,1983,2:239-253.
[36]Hilliard J.Currency option pricing with stochastic domestic and foreign interest rates [J]. Journal of Financial and Quantitative Analysis,1991,26:139-151.
[37]Amin KI.Discrete-time Valuation of American options with stochastic interest r ate [J]. Re view of Financial Studies,1995,8:193-234.
[38]陈迪红.一个外汇期权定价模型[J].财经理论与实践,201.22：59-60.
[39]徐根新Vasicek利率模型下欧式看涨外汇期权定价分析[J].同济大学学报(自然科学版),2006.34.
[40]黄丁伟.外汇期权定价：VG模型与BS模型谁更适用[J].金融发展研究,2009,3：24-25.
[41]陈荣达.基于汇率回报厚尾性的外汇期权定价模型[J].运筹与管理,2006,5：137-140.
[42]Hu Y.0 Ksendal B.Fractional white noise calculus and applications to fi-nance[J].Infinite Dim Anal Quantum Probab Related Topics.2003,6:1-32.
[43]Necula C.Option pricing in a fractional Brownian motion environment[R].Workong paper of the Academy of Economic studies,Bucharest,2002,27.
[44]Rostek S,Schobel R.Risk preference based option pricing in a fractional Brownian Market [J]. Applied Probability Trust.2007,30:1-29.
[45]张敏,李旭,何穗.几何分形Brown运动下的欧式外汇期权定价[J].湖北工业大学学报,2006,6：75-78.
[46]刘目楼.何春雄.分数布朗运动下的欧式外汇期权定价[J].科学技术与工程,2007,7.
[47]张卫国.分数布朗运动下偶是汇率期权定价[J].系统工程理论与实践,2009,6：68-77.
[48]黄礼文.基于分数布朗运动模型的金融衍生产品定价.浙江大学：博士学位论文.2011,4.
[49]叶永刚,黄志坚.外汇期权[M].武汉：武汉大学出版社,2004,7.
[50]Stampli,J著.蔡明超译.金融数学[M].北京：机械工业出版社,2004,3.