目标区域下的汇率扩散模型及其期权定价
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摘要
汇率目标区域是20世纪90年代以来,国际金融界最受重视与争议的汇率课题之一。本文将汇率目标区域下的扩散模型应用到中国,探讨人民币对美元的汇率满足的扩散方程。
首先,选择两种目标区域下的汇率扩散模型,并给出这两种模型的统计特征,为后面的参数估计和实证分析作准备。
其次,对上述提到的两种模型分别用鞅估计方法和基于条件矩的GMM方法对模型进行参数估计。通过数值模拟得出,基于条件矩的GMM方法要优于鞅估计方法的结果。然后选用人民币对美元的汇率数据在两种模型下进行参数估计和拟合。由于第二种模型估计出的人民币和美元的相关系数失真,从而第一种模型更适合我国的短期汇率市场。
最后,我们研究两种模型下的外汇期权定价。对于期权的定价一般采用鞅方法,本文采用套期保值的方法给出了两种模型的欧式不支付红利、欧式支付红利以及强路径依赖的亚式外汇期权满足的偏微分方程。并初步研究了波动率服从CIR模型的目标区域下的汇率扩散模型的欧式外汇期权定价。
Since the 90's in the 20th century, the exchange rate in a target zone was one of the most important and controversial issue about the exchange rate in the international financial community .In this paper, diffusion models for exchange rate in a target zone are applied to China, in order to explore which diffusion model the exchange rate of RMB against U.S. dollar to satisfy.
First of all, we select two diffusion models for exchange rate in a target zone,and study the statistical characteristics of them to make preparation for the parameter estimation and empirical analysis.
Secondly, the above-mentioned two models were estimated using martingale estimation method and the GMM based on the conditional moments. Through numerical simulation, we educe that the GMM based on the conditional moments is better than the martingale estimation method. Then we choose the exchange rate data of RMB against the U.S. dollar to estimate parameters and draw the fitting pictures. As a result of the correlation coefficient estimated in the second model is out of distortion, so the first model is more suitable for the Chinese short-term exchange rate market.
Finally, we study the foreign currency option pricing about the two above-mentioned models. Generally, the option is priced using the martingale method . In this paper, non-payment of dividend of European options、European option to pay dividend and strong path dependence of the Asian foreign exchange options are priced using the method of hedging. In the end, we derive the partial differential equation to satisfy about them. And we preliminary study the pricing of the European foreign exchange option in the diffusion model for exchange rate in a target zone which the volatility meets the CIR model.
首先,选择两种目标区域下的汇率扩散模型,并给出这两种模型的统计特征,为后面的参数估计和实证分析作准备。
其次,对上述提到的两种模型分别用鞅估计方法和基于条件矩的GMM方法对模型进行参数估计。通过数值模拟得出,基于条件矩的GMM方法要优于鞅估计方法的结果。然后选用人民币对美元的汇率数据在两种模型下进行参数估计和拟合。由于第二种模型估计出的人民币和美元的相关系数失真,从而第一种模型更适合我国的短期汇率市场。
最后,我们研究两种模型下的外汇期权定价。对于期权的定价一般采用鞅方法,本文采用套期保值的方法给出了两种模型的欧式不支付红利、欧式支付红利以及强路径依赖的亚式外汇期权满足的偏微分方程。并初步研究了波动率服从CIR模型的目标区域下的汇率扩散模型的欧式外汇期权定价。
Since the 90's in the 20th century, the exchange rate in a target zone was one of the most important and controversial issue about the exchange rate in the international financial community .In this paper, diffusion models for exchange rate in a target zone are applied to China, in order to explore which diffusion model the exchange rate of RMB against U.S. dollar to satisfy.
First of all, we select two diffusion models for exchange rate in a target zone,and study the statistical characteristics of them to make preparation for the parameter estimation and empirical analysis.
Secondly, the above-mentioned two models were estimated using martingale estimation method and the GMM based on the conditional moments. Through numerical simulation, we educe that the GMM based on the conditional moments is better than the martingale estimation method. Then we choose the exchange rate data of RMB against the U.S. dollar to estimate parameters and draw the fitting pictures. As a result of the correlation coefficient estimated in the second model is out of distortion, so the first model is more suitable for the Chinese short-term exchange rate market.
Finally, we study the foreign currency option pricing about the two above-mentioned models. Generally, the option is priced using the martingale method . In this paper, non-payment of dividend of European options、European option to pay dividend and strong path dependence of the Asian foreign exchange options are priced using the method of hedging. In the end, we derive the partial differential equation to satisfy about them. And we preliminary study the pricing of the European foreign exchange option in the diffusion model for exchange rate in a target zone which the volatility meets the CIR model.
引文
[1]Ball,C.and A.Roma.Target zone modeling and estimation for European Monetary System exchange rates.Journal of Empirical Finance.1994(1):385-420
[2]Bekaert,G.,and S.F.Gray.Target zones and exchange rates:An empirical investigation,working paper Stanford University.1996(45):1-35
[3]Bibby,B.M.and Sorensen M.Martingale estimation functions for Discretely observed diffusion peocesses.Bernoulli,1995(1):17-39
[4]Bibby.B.M,I.M.Skovgaard and M.Sorensen.Diffusion-Type Models with Given Marginals Distribution and Autocorrelation Function.Bernoulli.2005(11):191-220
[5]Bibby M.Martin Jacobsen and Michael Sorensen.Estimation Function for Discretely Sampled Diffusion-Type Models.http://citeseer.ist.psu.edu/613760.html,2004
[6]Cheung.M.T,Yeung D.A.Non random walk theory of exchange rate dynamic with applicatio(?)s to option pricing.Stochastic Analysis and Applications.1994(12):141-157
[7]Cox,J.,J.Ingersoll and S.Ross.A theory of the term structure of interest rates.Econometrical.1985(53):385-407
[8]De Jong F.C.Drost and B.J.M.Werker.A Jump-Diffusion Model for Exchange Rates in a Target Zone.Statistica Neerlandica.2001(55):270-300
[9]Dumas B.,L.P.Jennergren and B.Naslund.Realignment risk and currency option pricing in target zones,European Economic Review.1995(39):1523-1544
[10]Frankel Jeffrey.Eduardo Fajnzylber.Sergio Schmukler and Luis Serven.Verifying exchange rate regimes:The World Bank Development Research Group Macroeconomics and Growth.Policy research working paper.http://catalogue.nla.gov.au/Record/827862,2000
[11]Garman M.and S.Kohlhagen.Foreign Currency Option Values.Journal International Money and Finance.1983(2):231-237
[12]Goldstein.Morris and Turner P.Banking Crisis in Economic:Origins and Policy Option.BIS Economic Paper.1990(46)
[131 Goldstein.Morris and Nicholas Lardy.Two-Stage Currency Reform For China.Asian Wall Street Journal.2003(12)
[14]Joh(?) Maynard Keynes.The General Theory of Employment,Interest and Money.Macmillan Cambridge University Press.1935
[15]Harrison M.and S.Pliska.Martingales and stochastic integrals in the theory of continuous trading.Stochastic Processes and Their applications.1981(20):381-408.
[16] Hyungsik Roger Moon. Peter C. B. Phillips. GMM Estimation of Autoregressive Roots near Unity with Panel Data, Econometrical. 2004(72):467-522.
[17] Kristian Stegenborg Larsen and Michael Sorensen. Diffusion models for exchange rate in a target zone.Mathematical Finance. 2007(4):285-306
[18] Paul R. Krugman. Target Zones and Exchange Rate Dynamics. The Quarterly Journal of Economics. 1991(106).669-682
[19] Lansen,K. S., andM.Sorensen. Diffusion Model for Exchange Rates in a Target Zone,University of Copenhagen. 2003
[20] Marine Carrasco; Jean-Pierre Florens. Generalization of GMM to a Continuum of Moment Conditions. Econometric Theory, 2000(16):797-834.
[21] Meddahi, N.and E. Renault. Linear statistical inference for ARCH-type Processes.working paper University Toulouse. 1996(11):258—289
[22] Merton, R.C. Option pricing when the underlying stock returns are discontinuous.Journal of Financial Economics. 1976(3): 125-144.
[23] Michael Sorensen .Discretely observed diffusion: Approximation of the continuous-time score function.Scand.J.Statist.2001(11) :113-121
[24] Michael Sorensen.M .Parametric Inference for Discretely Sampled Stochastic Differential Equations. Department of Applied Mathematics and Statistics. University of Copenhagen. 2008(35):438-465
[25] Musiela M. Rutkowski M.. Martingale Methods in Financial Modeling. Berlin:Springer-Verlag.l997(36):256-280
[26] Rangvid, J., and C. Sorensen. Determinants of the Implied Shadow Exchange Rates from a Target Zone. Eur. Econ. Rev. 2001(45):1665-1696.
[27] G.Cassel .Gustav Cassel's early purchasing power parity theory .History of Political Economy. 1982(14):242-245
[28] Williamson John.Bergsten. Target Zone and Policy Coordination. Washington D.C Institute for International Economics.l985(11):54-57
[29] Wefelmeyer W. Quasi-likelihood Models and Optimal Inference, The Annals of Statistics.1996(24):405-422
[30] Godambe, V.P.. The Foundation of Finite Sample Estimation in Stochastic Processes.Biometrica. 1985(72):419-428
[31] Newey W.. Semiparametric Efficiency Bounds. Journal of Applied Econometrics.1990(5):99-135
[32] Lucas Robert Jr. Interest rates and currency prices in a two country world. Journal of Monetary Economics.1982(10):335-359
[33]陈萍.随机波动率模型的统计推断及其衍生证券的定价.南京理工大学博士论文.2004
[34]陈萍,杨孝平.CIR模型的统计推断,应用概率统计.2005
[35]段梅丽.人民币汇率目标区的可行性分析.东北财经大学学报.2003(5):43-45
[36]姜凌,韩璐.汇率目标区理论与人民币汇率机制的改革思路.经济评论.2003(2):110-114
[37]马德功.汇率目标区理论回顾与展望.生产力研究.2003(6):287-289
[38]王国青.人民币汇率目标区域模型初探[J].南开经济研究.1996
[39]肖建军.汇率目标区制度在中国实用性研究.复旦大学硕士论文.2003
[40]姜礼尚.期权定价的数学模型和方法.第1版.高等教育出版社.2003
[41]刘广应.随机波动率模型的参数估计与期权定价.南京理工大学硕士论文.2004
[42]王树禾,侯定丕.经济与管理科学中的数学模型[M].合肥:中国科学技术出版.2004
[43]魏巍贤.人民币汇率的稳定机制及其动态过程—目标区域模型[J].系统工程理论与实践.1999
[44]魏巍贤.人民币汇率决定模型的实证分析,系统工程理论与实践,第3期.2000
[45]雷志卫.欧洲货币联盟的理论基础与运作机制.系统工程理论与实践.1999
[2]Bekaert,G.,and S.F.Gray.Target zones and exchange rates:An empirical investigation,working paper Stanford University.1996(45):1-35
[3]Bibby,B.M.and Sorensen M.Martingale estimation functions for Discretely observed diffusion peocesses.Bernoulli,1995(1):17-39
[4]Bibby.B.M,I.M.Skovgaard and M.Sorensen.Diffusion-Type Models with Given Marginals Distribution and Autocorrelation Function.Bernoulli.2005(11):191-220
[5]Bibby M.Martin Jacobsen and Michael Sorensen.Estimation Function for Discretely Sampled Diffusion-Type Models.http://citeseer.ist.psu.edu/613760.html,2004
[6]Cheung.M.T,Yeung D.A.Non random walk theory of exchange rate dynamic with applicatio(?)s to option pricing.Stochastic Analysis and Applications.1994(12):141-157
[7]Cox,J.,J.Ingersoll and S.Ross.A theory of the term structure of interest rates.Econometrical.1985(53):385-407
[8]De Jong F.C.Drost and B.J.M.Werker.A Jump-Diffusion Model for Exchange Rates in a Target Zone.Statistica Neerlandica.2001(55):270-300
[9]Dumas B.,L.P.Jennergren and B.Naslund.Realignment risk and currency option pricing in target zones,European Economic Review.1995(39):1523-1544
[10]Frankel Jeffrey.Eduardo Fajnzylber.Sergio Schmukler and Luis Serven.Verifying exchange rate regimes:The World Bank Development Research Group Macroeconomics and Growth.Policy research working paper.http://catalogue.nla.gov.au/Record/827862,2000
[11]Garman M.and S.Kohlhagen.Foreign Currency Option Values.Journal International Money and Finance.1983(2):231-237
[12]Goldstein.Morris and Turner P.Banking Crisis in Economic:Origins and Policy Option.BIS Economic Paper.1990(46)
[131 Goldstein.Morris and Nicholas Lardy.Two-Stage Currency Reform For China.Asian Wall Street Journal.2003(12)
[14]Joh(?) Maynard Keynes.The General Theory of Employment,Interest and Money.Macmillan Cambridge University Press.1935
[15]Harrison M.and S.Pliska.Martingales and stochastic integrals in the theory of continuous trading.Stochastic Processes and Their applications.1981(20):381-408.
[16] Hyungsik Roger Moon. Peter C. B. Phillips. GMM Estimation of Autoregressive Roots near Unity with Panel Data, Econometrical. 2004(72):467-522.
[17] Kristian Stegenborg Larsen and Michael Sorensen. Diffusion models for exchange rate in a target zone.Mathematical Finance. 2007(4):285-306
[18] Paul R. Krugman. Target Zones and Exchange Rate Dynamics. The Quarterly Journal of Economics. 1991(106).669-682
[19] Lansen,K. S., andM.Sorensen. Diffusion Model for Exchange Rates in a Target Zone,University of Copenhagen. 2003
[20] Marine Carrasco; Jean-Pierre Florens. Generalization of GMM to a Continuum of Moment Conditions. Econometric Theory, 2000(16):797-834.
[21] Meddahi, N.and E. Renault. Linear statistical inference for ARCH-type Processes.working paper University Toulouse. 1996(11):258—289
[22] Merton, R.C. Option pricing when the underlying stock returns are discontinuous.Journal of Financial Economics. 1976(3): 125-144.
[23] Michael Sorensen .Discretely observed diffusion: Approximation of the continuous-time score function.Scand.J.Statist.2001(11) :113-121
[24] Michael Sorensen.M .Parametric Inference for Discretely Sampled Stochastic Differential Equations. Department of Applied Mathematics and Statistics. University of Copenhagen. 2008(35):438-465
[25] Musiela M. Rutkowski M.. Martingale Methods in Financial Modeling. Berlin:Springer-Verlag.l997(36):256-280
[26] Rangvid, J., and C. Sorensen. Determinants of the Implied Shadow Exchange Rates from a Target Zone. Eur. Econ. Rev. 2001(45):1665-1696.
[27] G.Cassel .Gustav Cassel's early purchasing power parity theory .History of Political Economy. 1982(14):242-245
[28] Williamson John.Bergsten. Target Zone and Policy Coordination. Washington D.C Institute for International Economics.l985(11):54-57
[29] Wefelmeyer W. Quasi-likelihood Models and Optimal Inference, The Annals of Statistics.1996(24):405-422
[30] Godambe, V.P.. The Foundation of Finite Sample Estimation in Stochastic Processes.Biometrica. 1985(72):419-428
[31] Newey W.. Semiparametric Efficiency Bounds. Journal of Applied Econometrics.1990(5):99-135
[32] Lucas Robert Jr. Interest rates and currency prices in a two country world. Journal of Monetary Economics.1982(10):335-359
[33]陈萍.随机波动率模型的统计推断及其衍生证券的定价.南京理工大学博士论文.2004
[34]陈萍,杨孝平.CIR模型的统计推断,应用概率统计.2005
[35]段梅丽.人民币汇率目标区的可行性分析.东北财经大学学报.2003(5):43-45
[36]姜凌,韩璐.汇率目标区理论与人民币汇率机制的改革思路.经济评论.2003(2):110-114
[37]马德功.汇率目标区理论回顾与展望.生产力研究.2003(6):287-289
[38]王国青.人民币汇率目标区域模型初探[J].南开经济研究.1996
[39]肖建军.汇率目标区制度在中国实用性研究.复旦大学硕士论文.2003
[40]姜礼尚.期权定价的数学模型和方法.第1版.高等教育出版社.2003
[41]刘广应.随机波动率模型的参数估计与期权定价.南京理工大学硕士论文.2004
[42]王树禾,侯定丕.经济与管理科学中的数学模型[M].合肥:中国科学技术出版.2004
[43]魏巍贤.人民币汇率的稳定机制及其动态过程—目标区域模型[J].系统工程理论与实践.1999
[44]魏巍贤.人民币汇率决定模型的实证分析,系统工程理论与实践,第3期.2000
[45]雷志卫.欧洲货币联盟的理论基础与运作机制.系统工程理论与实践.1999