连续时间金融模型的时变性检验
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摘要
在金融数学中,已经有大量的文献研究连续时间模型估计问题,但有关连续时间模型设定方面的工作却相对较少。模型的错误设定往往对推断和检验造成误导,进一步,错误拟合的模型可能会导致定价,套期保值以及风险管理上大的错误。因此,模型设定检验是非常必要的。
本文首先回顾了连续时间模型估计问题在理论和实证分析中的一些成果,然后针对连续时间模型的非参数检验做了如下工作：
第二章主要介绍了一种非参数设定检验方法,该方法通过转移密度对连续时间模型做非参数设定检验。此方法需要求得模型的广义残差,再求联合密度的边界修正核估计,最终构造检验统计量Q(j)。
第三章在第二章的理论基础上,选取几何Brown运动作为原假设,CKLS模型和CIR模型作为备择假设。通过对计算得出的检验统计量Q(j)进行分析,说明本文所用方的可行性。同时讨论了分段几何Brown运动时变性检验问题。
第四章运用前面提到的非参数检验方法对上证指数日数据进行分析,得出上证指数服从分段几何Brown运动,即上证指数具有时变性。接下来,选择一个具有持续增长趋势的个股的收盘价进行分析,检验得出用几何Brown运动可以描述该股票的收盘价。
In mathematical finance, there has been a large number of literature on the estimation of the continuous-time models, however, there is relatively little effort on specification analysis for continuous-times models. Model misspecificaton generally leads to misleading conclusions in inference and hypothesis testing, more, misspecified model can yield large errors in pricing, hedging, and risk management. It is necessary to develop reliable specification tests for continuous-times models.
In this paper, we first reviewed the fruit of continuous-time models estimation in theory and applications, and then discussed the nonparametric specification test problem in continuous-time models as follows:
In chapter 2 we introduced the nonparametric specification test for continuous-time models using the transition density, which need to compute the model generalized residuals, then compute the boundary-modified kernel joint density estimator, and finally compute the test statistic Q(j).
In chapter 3, on the theoretical basis of chapter 2, we chose geometric Brown motion as the null hypothesis, CKLS model and CIR model as the alternative hypothesis. Through analyzing the test statistic, we concluded that the method in this paper can obtain the good result. Furthermore, we discussed the time-dependent features of segmented geometric Brown motion.
In chapter 4, we tested our method by analyzing Shanghai A Share Index and concluded that Shanghai A Share Index is the time-dependent segmented geometric Brown motion. We analyzed a sustainable growth stock, and concluded it can be described by the geometric Brown motion.
本文首先回顾了连续时间模型估计问题在理论和实证分析中的一些成果,然后针对连续时间模型的非参数检验做了如下工作：
第二章主要介绍了一种非参数设定检验方法,该方法通过转移密度对连续时间模型做非参数设定检验。此方法需要求得模型的广义残差,再求联合密度的边界修正核估计,最终构造检验统计量Q(j)。
第三章在第二章的理论基础上,选取几何Brown运动作为原假设,CKLS模型和CIR模型作为备择假设。通过对计算得出的检验统计量Q(j)进行分析,说明本文所用方的可行性。同时讨论了分段几何Brown运动时变性检验问题。
第四章运用前面提到的非参数检验方法对上证指数日数据进行分析,得出上证指数服从分段几何Brown运动,即上证指数具有时变性。接下来,选择一个具有持续增长趋势的个股的收盘价进行分析,检验得出用几何Brown运动可以描述该股票的收盘价。
In mathematical finance, there has been a large number of literature on the estimation of the continuous-time models, however, there is relatively little effort on specification analysis for continuous-times models. Model misspecificaton generally leads to misleading conclusions in inference and hypothesis testing, more, misspecified model can yield large errors in pricing, hedging, and risk management. It is necessary to develop reliable specification tests for continuous-times models.
In this paper, we first reviewed the fruit of continuous-time models estimation in theory and applications, and then discussed the nonparametric specification test problem in continuous-time models as follows:
In chapter 2 we introduced the nonparametric specification test for continuous-time models using the transition density, which need to compute the model generalized residuals, then compute the boundary-modified kernel joint density estimator, and finally compute the test statistic Q(j).
In chapter 3, on the theoretical basis of chapter 2, we chose geometric Brown motion as the null hypothesis, CKLS model and CIR model as the alternative hypothesis. Through analyzing the test statistic, we concluded that the method in this paper can obtain the good result. Furthermore, we discussed the time-dependent features of segmented geometric Brown motion.
In chapter 4, we tested our method by analyzing Shanghai A Share Index and concluded that Shanghai A Share Index is the time-dependent segmented geometric Brown motion. We analyzed a sustainable growth stock, and concluded it can be described by the geometric Brown motion.
引文
[1]Wong, E. Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York,1971
[2]Duffie, D. Dynamic Asset Pricing Theory.2nd edition. Princeton University Press, Princeton, New Jersey,1996
[3]Merton, R. C. Continuous-Time Finance. Blackwell, Cambridge, Massachusetts,1992
[4]Vasicek, O. An equilibrium characterization of the term structure. J. Finan. Econom., 1977,5:177-188
[5]Cox, J. C., Ingersoll, J. E., Ross,S. A. An analysis of variable rate loan contracts. J. Finance,1980,35:389-403
[6]Cox, J. C., Ingersoll, J. E., Ross, S. A. A theory of the term structure of interest rates. Econometrica,1985,53:385-467
[7]Chan, K. C., Karolyi, A. G., Longstaff, F. A., Sanders, A. B. An empirical comparison of alternative models of the short-term interest rate. J.,1992
[8]Ho, T. S. Y., Lee, S. B. Term structure movements and pricing interest rate contingent claims. J. Finance,1986,41:1011-1029
[9]Black, F., Derman, E., Toy, W. A one-factor model of interest rates and its application to treasury bond options. Finan. Analysts'J.1990,46:33-39
[10]Black, F., Karasinski, P. Bond and option pricing when short rates are lognormal. Finan. Analysts'J,1991,47:52-59
[11]Donoho, D.L, Johnstone.I.M, Kerkyacharian.G,Picard.D. Density estimation by wavelet thresholding. Ann.Statist,1996,24(2):508-539
[12]GenonCatalot,V.,Laredo,C.,Picard,D. Nonparametric estimation of the diffusion coefficient by wavelet methods. Scand.J.Statist,1992,19:317-335
[13]肖庆宪,郑祖康.股票价格过程方差函数的统计推断.应用概率统计,2000,16(2)：182—190
[14]J.Fan,J.Jiang,C.zhang,Z.Zhou, Time-dependent diffusion models for term structure dynamics. Statistics Sinica,2003,13:965-992. Finance,47:1209-1227
[15]Chen Ping, Wang Jinde. Wavelet estimation in time dependent diffusion models.2006
[16]Ait-Sahalia, Y. Transition densities for interest rate and other nonlinear diffusions. Journal of Finance,1999,54:1361-1395.
[17]Stanton, R. A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance,1997,52:1973-2002
[18]Fan, J., Zhang, C. Are-examination of diffusion estimators with applications to financial model validation. Journal of the American Statistical Association,20003,98:118-134
[19]Hong, Y., Li, H. Nonparametric specification testing for continuous-time models with applications to interest rate term structures. The Review of Financial Studies,2005,18: 37-84
[20]Ait-Sahalia, Y. Testing Continuous-Time Models of the Spot Interest Rate. Review of Financial Studies,1996b,9:385-426
[21]Bliss, R.R., Smith, D. The elasticity of interest rate volatility:Chan, Karolyi, Longstaff, and Sanders revisited. Journal of Risk,1998,1:21-46
[22]Lo, A.W. Maximum likelihood estimation of generalized Ito processes with discretely sampled data. Econometric Theory,1988,4:231-247
[23]Duffie, D., K.J. Singleton. Simulated moments estimation of Markov models of asset prices. Econometrica,1993,61:929-952
[24]Gourieroux, C., A. Monfort, E. Renault. Indirect inference. Journal of Applied Econometrics,1993,8:85-118
[25]Hansen, L.P., J.A. Scheinkman. Back to the future:Generating moment implications for continuous time Markov processes, Econometrica,1995,63:767-804
[26]Gallant, A.R., G. Tauchen. Which moments to match? Econometric Theory, 1996,12:657-681
[27]Jacquier, E., N.G Polson, P. Rossi. Bayesian analysis of stochastic volatility models. Journal of Business and Economic Statistics,1994,12:371-389
[28]Eraker, B. Markov chain Monte Carlo analysis of diffusion models with application to finance. HAE Thesis. Norwegian School of Economics and Business Administration,1998
[29]Jones, C.S. Bayesian estimation of continuous-time finance models.Working Paper, Simon School of Business.University of Rochester,1998
[30]Jiang, G.J., J.L. Knight. Estimation of continuous time processes via the empirical characteristic function. Journal of Business and Economic Statistics,2002,20:198-212
[31]Singleton, K.J. Estimation of affine asset pricing models using the empirical characteristic function. Journal of Econometrics,2001,102:111-141
[32]Ait-Sahalia, Y. Maximum likelihood estimation of discretely sampled diffusions:A closed-form approach. Econometrica,2002a,70:223-262
[33]Egorov, A., H. Li, Y. Xu. Maximum likelihood estimation of time-inhomogeneous diffusions, Journal of Econometrics,2003,114:107-139
[34]Ait-Sahalia, Y. Closed-form likelihood expansions for multivariate diffusion, Annals of Statistics,2008,36:906-937
[35]Ait-Sahalia, Y., Kimmel, R. Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics,2007,83:413-452
[36]Pritsker, M. Nonparametric density estimation and tests of continuous time interest rate models, Review of Financial Studies,1998,11:449-487
[37]Rosenblatt, M. Remarks on a multivariate transformation, The Annals of Mathe-matical Statistics,1952,23:470-472
[38]Diebold, F.X., T. Gunther, Tay, A. Evaluating density forecasts with applications to financial risk management, International Economic Review,1998,39:863-883
[39]Hong, Y., Lee, T.H. Diagnostic checking for nonlinear time series models,Econometric Theory,2003b,19:1065-1121
[40]Chapman, D., Pearson, N. Is the short rate drift actually nonlinear? Journal of Finance, 2000,55,355-388
[41]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
[42]王军,王娟.随机过程及其在金融领域中的应用.清华大学出版社,2007.
[43]林元烈.应用随机过程.清华大学出版社,2006
[44]陈桂明,戚红雨.MATLAB数理统计.科学出版社,2009
[45]龚纯,王正林.MATLAB语言常用算法程序集.电子工业出版社,2008
[46]叶中行,王蓉华.概率论与数理统计.北京大学出版社,2009
[47]周品,赵新芬.MATLAB数理统计分析.国防工业出版社,2009
[48]徐士良.计算方法.人民邮电出版社,2009
[49]同济大学计算数学研究室.现代数值计算.人民邮电出版社,2009
[50]朱长青.数值计算方法与应用.科学出版社,2006
[2]Duffie, D. Dynamic Asset Pricing Theory.2nd edition. Princeton University Press, Princeton, New Jersey,1996
[3]Merton, R. C. Continuous-Time Finance. Blackwell, Cambridge, Massachusetts,1992
[4]Vasicek, O. An equilibrium characterization of the term structure. J. Finan. Econom., 1977,5:177-188
[5]Cox, J. C., Ingersoll, J. E., Ross,S. A. An analysis of variable rate loan contracts. J. Finance,1980,35:389-403
[6]Cox, J. C., Ingersoll, J. E., Ross, S. A. A theory of the term structure of interest rates. Econometrica,1985,53:385-467
[7]Chan, K. C., Karolyi, A. G., Longstaff, F. A., Sanders, A. B. An empirical comparison of alternative models of the short-term interest rate. J.,1992
[8]Ho, T. S. Y., Lee, S. B. Term structure movements and pricing interest rate contingent claims. J. Finance,1986,41:1011-1029
[9]Black, F., Derman, E., Toy, W. A one-factor model of interest rates and its application to treasury bond options. Finan. Analysts'J.1990,46:33-39
[10]Black, F., Karasinski, P. Bond and option pricing when short rates are lognormal. Finan. Analysts'J,1991,47:52-59
[11]Donoho, D.L, Johnstone.I.M, Kerkyacharian.G,Picard.D. Density estimation by wavelet thresholding. Ann.Statist,1996,24(2):508-539
[12]GenonCatalot,V.,Laredo,C.,Picard,D. Nonparametric estimation of the diffusion coefficient by wavelet methods. Scand.J.Statist,1992,19:317-335
[13]肖庆宪,郑祖康.股票价格过程方差函数的统计推断.应用概率统计,2000,16(2)：182—190
[14]J.Fan,J.Jiang,C.zhang,Z.Zhou, Time-dependent diffusion models for term structure dynamics. Statistics Sinica,2003,13:965-992. Finance,47:1209-1227
[15]Chen Ping, Wang Jinde. Wavelet estimation in time dependent diffusion models.2006
[16]Ait-Sahalia, Y. Transition densities for interest rate and other nonlinear diffusions. Journal of Finance,1999,54:1361-1395.
[17]Stanton, R. A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance,1997,52:1973-2002
[18]Fan, J., Zhang, C. Are-examination of diffusion estimators with applications to financial model validation. Journal of the American Statistical Association,20003,98:118-134
[19]Hong, Y., Li, H. Nonparametric specification testing for continuous-time models with applications to interest rate term structures. The Review of Financial Studies,2005,18: 37-84
[20]Ait-Sahalia, Y. Testing Continuous-Time Models of the Spot Interest Rate. Review of Financial Studies,1996b,9:385-426
[21]Bliss, R.R., Smith, D. The elasticity of interest rate volatility:Chan, Karolyi, Longstaff, and Sanders revisited. Journal of Risk,1998,1:21-46
[22]Lo, A.W. Maximum likelihood estimation of generalized Ito processes with discretely sampled data. Econometric Theory,1988,4:231-247
[23]Duffie, D., K.J. Singleton. Simulated moments estimation of Markov models of asset prices. Econometrica,1993,61:929-952
[24]Gourieroux, C., A. Monfort, E. Renault. Indirect inference. Journal of Applied Econometrics,1993,8:85-118
[25]Hansen, L.P., J.A. Scheinkman. Back to the future:Generating moment implications for continuous time Markov processes, Econometrica,1995,63:767-804
[26]Gallant, A.R., G. Tauchen. Which moments to match? Econometric Theory, 1996,12:657-681
[27]Jacquier, E., N.G Polson, P. Rossi. Bayesian analysis of stochastic volatility models. Journal of Business and Economic Statistics,1994,12:371-389
[28]Eraker, B. Markov chain Monte Carlo analysis of diffusion models with application to finance. HAE Thesis. Norwegian School of Economics and Business Administration,1998
[29]Jones, C.S. Bayesian estimation of continuous-time finance models.Working Paper, Simon School of Business.University of Rochester,1998
[30]Jiang, G.J., J.L. Knight. Estimation of continuous time processes via the empirical characteristic function. Journal of Business and Economic Statistics,2002,20:198-212
[31]Singleton, K.J. Estimation of affine asset pricing models using the empirical characteristic function. Journal of Econometrics,2001,102:111-141
[32]Ait-Sahalia, Y. Maximum likelihood estimation of discretely sampled diffusions:A closed-form approach. Econometrica,2002a,70:223-262
[33]Egorov, A., H. Li, Y. Xu. Maximum likelihood estimation of time-inhomogeneous diffusions, Journal of Econometrics,2003,114:107-139
[34]Ait-Sahalia, Y. Closed-form likelihood expansions for multivariate diffusion, Annals of Statistics,2008,36:906-937
[35]Ait-Sahalia, Y., Kimmel, R. Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics,2007,83:413-452
[36]Pritsker, M. Nonparametric density estimation and tests of continuous time interest rate models, Review of Financial Studies,1998,11:449-487
[37]Rosenblatt, M. Remarks on a multivariate transformation, The Annals of Mathe-matical Statistics,1952,23:470-472
[38]Diebold, F.X., T. Gunther, Tay, A. Evaluating density forecasts with applications to financial risk management, International Economic Review,1998,39:863-883
[39]Hong, Y., Lee, T.H. Diagnostic checking for nonlinear time series models,Econometric Theory,2003b,19:1065-1121
[40]Chapman, D., Pearson, N. Is the short rate drift actually nonlinear? Journal of Finance, 2000,55,355-388
[41]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
[42]王军,王娟.随机过程及其在金融领域中的应用.清华大学出版社,2007.
[43]林元烈.应用随机过程.清华大学出版社,2006
[44]陈桂明,戚红雨.MATLAB数理统计.科学出版社,2009
[45]龚纯,王正林.MATLAB语言常用算法程序集.电子工业出版社,2008
[46]叶中行,王蓉华.概率论与数理统计.北京大学出版社,2009
[47]周品,赵新芬.MATLAB数理统计分析.国防工业出版社,2009
[48]徐士良.计算方法.人民邮电出版社,2009
[49]同济大学计算数学研究室.现代数值计算.人民邮电出版社,2009
[50]朱长青.数值计算方法与应用.科学出版社,2006