基于卡尔曼类滤波方法的利率期限结构模型估计研究
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摘要
利率期限结构的理论和模型是金融研究中最具挑战性的课题之一,也是目前金融工程领域的一项十分重要的基础性研究工作。而利率期限结构的模型估计又是利率理论研究和实证工作的基础和关键环节。卡尔曼类滤波估计理论是经典最优滤波理论的组成部分,由于其实时、快速、精确以及稳定和易操作等性质和特点而广泛应用于信号处理、通讯和控制等领域,取得了很好的效果。本论文的目的就在于通过回顾利率期限结构模型和卡尔曼类滤波估计理论和方法的发展历程,系统的将卡尔曼类滤波估计理论和方法引入到利率期限结构的模型估计上来,为利率期限结构模型理论和实证研究提供模型估计方法和应用基础。
本论文首先将利率期限结构模型划分为均衡模型和无套利模型两大类,分别具体介绍了两大类模型中具体模型理论的提出、构建以及模型特点。接着,系统介绍了卡尔曼类滤波估计理论和方法,包括卡尔曼滤波估计(Kalman Filter)、扩展卡尔曼滤波估计(Extended Kalman Filter)、无损卡尔曼滤波估计(Unscented Kalman Filter)的理论和方法,以及在利率期限结构模型估计上的具体应用。
最后,本论文在Matlab 7.0环境下实现了扩展卡尔曼滤波估计(EKF,下同)和无损卡尔曼滤波估计(UKF)对Vasicek模型的参数估计,并通过对两种滤波方法的运算速度、估计效果等方面进行对比,探讨了EKF和UKF的特点、适用范围以及性能优劣。
本论文的研究内容受国家自然科学基金项目“固定收益证券利率风险动态定价与对冲方法研究”(项目编号:70471051)资助,是其部分研究成果。
The theories and models on term structure of interest rates are one of the most challenging works in finance research and an important fundamental branch in financial engineering field. And the model estimation of term structure of interest rates is the foundation and key link for the theoretical and empirical research on interest rates. The family of Kalman filters is part of the classical theory of optimal filtering. They have got advantages such as real time corresponding, speed computing, precise estimating, steady running and easy operating, which makes them widely used in various fields such as signal processing, communication, control and so on. In light of the excellent estimation effects these filters have obtained in the fields above, this dissertation, by reviewing the evolving process of both theories on modeling term structure of interest rates and algorithms of the family of Kalman filters, pursues a systemic application of the family of Kalman filters to the model estimation of term structure of interest rates, which supplies an estimation method and application foundation for the theoretical and empirical research on modeling the term structure of interest rates.
The dissertation firstly divided the models into two families: equilibrium models and no-arbitrage models and makes a detailed discussion on models in each family about their origin, construction and model characters. Then, the dissertation systematically introduce the theories and algorithms of the family of Kalman filters, including Kalman filter (KF), extended Kalman filter (EKF) and unscented Kalman filter (UKF), as well as their applications to the model estimation of the term structure of interest rates.
Finally, the dissertation carries out parameter estimation of Vasicek model, in Matlab 7.0, using EKF and UKF respectively. And then a discussion is given on the two filters in the aspects of characters, applicability scopes and superiority, by contrasting their estimation results in estimation effects, computing speed and so on.
As one of its research achievements, the dissertation is financed by the National Natural Science Fund project‘Research on Approach to Dynamically Pricing and Hedging of Interest Rate Risk of Fixed Income Securities’(No. 70471051).
本论文首先将利率期限结构模型划分为均衡模型和无套利模型两大类,分别具体介绍了两大类模型中具体模型理论的提出、构建以及模型特点。接着,系统介绍了卡尔曼类滤波估计理论和方法,包括卡尔曼滤波估计(Kalman Filter)、扩展卡尔曼滤波估计(Extended Kalman Filter)、无损卡尔曼滤波估计(Unscented Kalman Filter)的理论和方法,以及在利率期限结构模型估计上的具体应用。
最后,本论文在Matlab 7.0环境下实现了扩展卡尔曼滤波估计(EKF,下同)和无损卡尔曼滤波估计(UKF)对Vasicek模型的参数估计,并通过对两种滤波方法的运算速度、估计效果等方面进行对比,探讨了EKF和UKF的特点、适用范围以及性能优劣。
本论文的研究内容受国家自然科学基金项目“固定收益证券利率风险动态定价与对冲方法研究”(项目编号:70471051)资助,是其部分研究成果。
The theories and models on term structure of interest rates are one of the most challenging works in finance research and an important fundamental branch in financial engineering field. And the model estimation of term structure of interest rates is the foundation and key link for the theoretical and empirical research on interest rates. The family of Kalman filters is part of the classical theory of optimal filtering. They have got advantages such as real time corresponding, speed computing, precise estimating, steady running and easy operating, which makes them widely used in various fields such as signal processing, communication, control and so on. In light of the excellent estimation effects these filters have obtained in the fields above, this dissertation, by reviewing the evolving process of both theories on modeling term structure of interest rates and algorithms of the family of Kalman filters, pursues a systemic application of the family of Kalman filters to the model estimation of term structure of interest rates, which supplies an estimation method and application foundation for the theoretical and empirical research on modeling the term structure of interest rates.
The dissertation firstly divided the models into two families: equilibrium models and no-arbitrage models and makes a detailed discussion on models in each family about their origin, construction and model characters. Then, the dissertation systematically introduce the theories and algorithms of the family of Kalman filters, including Kalman filter (KF), extended Kalman filter (EKF) and unscented Kalman filter (UKF), as well as their applications to the model estimation of the term structure of interest rates.
Finally, the dissertation carries out parameter estimation of Vasicek model, in Matlab 7.0, using EKF and UKF respectively. And then a discussion is given on the two filters in the aspects of characters, applicability scopes and superiority, by contrasting their estimation results in estimation effects, computing speed and so on.
As one of its research achievements, the dissertation is financed by the National Natural Science Fund project‘Research on Approach to Dynamically Pricing and Hedging of Interest Rate Risk of Fixed Income Securities’(No. 70471051).
引文
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[2] Nowman, B., Gaussian Estimation of Single-Factor Continuous Time Models of the Term Structure of Interest Rates, Journal of Finance, Vol. 52, 1997, pp. 1695-1706.
[3] Kalman, R. E. 1960.“A New Approach to Linear Filtering and Prediction Problems,”Transaction of the ASME—Journal of Basic Engineering, pp. 35-45 (March 1960).
[4] N. Wiener. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. New York; The Technology Press Wiley, 1950.
[5] Greg Welch and Gary Bishop, an introduction to Kalman Filter, Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3175.
[6] Bucy R. S., Renne K. D. Digital Synthesis of Nonlinear Filters [J]. Automatica, 1971, 7(3) :287～289.
[7] Sunahara, Y., Ohsumi, A. Performance evaluation of nonlinear filter dynamics for the stochastic optimal control. Symposium on Nonlinear Estimation Theory and Its Applications, 2nd, San Diego, Calif , United States. 13-15 Sept. 1971. pp. 329-332. 1972.
[8] Maybeck P. S. Stochastic models estimation and control [M]. New York: Academic, 1982.
[9] Jazwinski A. H. Stochastic processes and filtering theory [M], New York: Academic, 1970.
[10] Caballero-Gil P., Fuster-Sabaer A., A wide family of nonlinear filter functions with a large linear span [J], Information Science, 2003, 164(1-4): 197-207.
[11] Gordon N. J., Salmond D. J., Smith A. F. M. Novel approach to nonlinear/non-Gaussian Bayesian state estimation [J]. IEEE Proc on Radar and Signal Processing, 1993, 140(2): 107-113
[12] Doucet A, De Freitas A, Gordon N. Sequential Monte Carlo methods in practice [M]. New York: Springer-Verlag, 2001.
[13] Arulampalam S, Maskell S, Gordon N, et al. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking [J]. IEEE Trans on Signal Processing, 2002, 50(2): 174-188.
[14] Julier S. J., Uhlmann J. K., Durrant-Whyten H. F. A new approach for filtering nonlinear system[A]. Proc of the American Control Conf [C]. Washington: Seattle, 1995: 1628-1632.
[15] Julier S. J., Uhlmann J. K. A general method for approximating nonlinear transformations of probability distributions [EB/OL]. http://www.robots.ox.ac.uk/～siju/work/ publications/ Unscented.zip, 1997-09-27.
[16] Julier S. J., Uhlmann J. K. A consistent, debiased method for converting between polar and Cartesian coordinate systems [A]. The Proc of Aerospace: The 11th Int Symposium on Aerospace/Defense sensing, Simulation and Controls [C]. Orlando, 1997: 110-121.
[17] Julier S. J.,Uhlmann J. K. A new extension of the Kalman filter to nonlinear system [A]. The Proc of Aerospace: The 11th Int Symposium on Aerospace/Defense sensing, Simulation and Controls [C]. Orlando, 1997: 54-65.
[18] Julier S. J. A skewed approach to filtering [A]. The Proc of Aerospace: The 12th Int Symposium on Aerospace/Defense sensing, Simulation and Controls [C]. Orlando, 1998: 271-282
[19] Julier S. J. The spherical simplex unscented transformation [A]. American Control Conf [C]. Denver, 2003: 2430-2434.
[20] Julier S. J., Uhlmann J. K., Durrant-Whyten H. F. A new approach for the nonlinear transformation of means and covariances in filters and estimators [J]. IEEE Trans on Automatic Control, 2000, 45(3): 477-482.
[21] Lefebvre T, Bruyninckx H, De Schutter J. Comment on“A new method for the nonlinear transformation of means and covariances in filters and estimators”[J]. IEEE Trans on Automatic Control, 2002, 47(8): 1406-1408.
[22] Julier S. J., Uhlmann J. K. Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations [A]. Proc of the American Control Conf [C]. Jefferson City, 2002: 887-892..
[23] Julier S. J. The scaled unscented transformation [A]. Proc of the American Control Conf [C]. Jefferson City, 2002: 4555-4559.
[24] Wan E. A., Van de Merwe R. The unscented Kalman filter for nonlinear estimation, Proceedings of Symposiums 2000 on Adaptive Systems for Signal Processing. Communication and Control ( AS-SPCC), IEEE, Lake Louise, Alberta, Canada. 2000.
[25] Wan E. A., Van de Merwe R. The unscented Kalman filter, in Kalman filtering and neural networks [DB/OL]. http://www.cse.ogi.edu/PacSoft/projects/sec/ wan01b.ps, 2004-03-10.
[26] Van de Merwe R., Arnaud Doucet, Nando de Freitas, Eric Wan, The Unscented Particle Filter, Technical Report CUED/F-INFENG/TR 380, Cambridge University, 2000.
[27] Julier S. J., Uhlmann J. K. Unscented filtering and nonlinear estimation [J]. Proc of the IEEE Aerospace and Electronic Systems, 2004, 92(3): 401-422.
[28] J. C. Jimenez, T. Ozaki, Linear Estimation of Continuous-Discrete Linear State Space Models with Multiplicative Noise [J], Systems and Control Letters 47, 91-101. 2003, Vol.49, 161.
[29] J. C. Jimenez, T. Ozaki, Local linearization filters for nonlinear continuous-discrete state space models with multiplicative noise [J], International Journal of Control., 2003, Vol.76, No.12, 1159-1170.
[30] Pennacchi, George G, Identifying the Dynamics of Real Interest Rates and Inflation: Evidence Using Survey Data [J]. Review of Financial Studies, Volume: 4 , Issue: 1 , Pages: 53-86, 1991.
[31] Lund, J., ``Econometric Analysis of Continuous-Time Arbitrage-Free Models of the Term Structure of Interest Rates,'' Unpublished manuscript, Aarhus School of Business, 1994.
[32] Chen, R. and L. Scott, ``Pricing Interest Rate Options in a Two-Factor Cox-Ingersoll-Ross Model of the Term Structure.'' Review of Financial Studies 5, 613-636, (1992).
[33] Chen, R. and L. Scott, ``Maximum Likelihood Estimation for a Multi-Factor Equilibrium Model of the Term Structure of Interest Rates.'' Journal of Fixed Income 4, 14-31, (1993a).
[34] Chen, R. and L. Scott, ``Multi-Factor Cox- Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a State-Space Model Using a Kalman Filter, Unpublished manuscript, University of Georgia.'' (1993b).
[35] Duan, J.C. and J.G. Simonato (1995), Estimating and Testing Exponential-Affine Term Structure Models by Kalman Filter," Manuscript, McGill University.
[36] Narasimhan Jegadeesh , George G. Pennacchi , Arturo Estrella. The Behavior of Interest Rates Implied by the Term Structure of Eurodollar Futures [J]. Journal of Money, Credit & Banking, Vol. 28, 1996.
[37] Ball, C.A. and W.N. Torous (1996), \Unit Roots and the Estimation of Interest Rate Dynamics," Journal of Empirical Finance, 3, 215-238.
[38] Santa-Clara, P. (1995),“Markovian Arbitrage-Free Models of the Term-Structure of Interest Rates," Manuscript, Anderson Graduate School of Management, University of California, Los Angeles, 1995.
[39] de Jong, F. (1996),“Time-Series and Cross-Section Information in Term Structure Models," Manuscript, Tilburg University. 1996.
[40] Zheng, C.C. (1993),“Testing Term Structure Models with Unspecied Dynamic Factors," Manuscript, First National Bank of Chicago, Chicago, IL. 1993.
[41] Gong, F.F. and E.M. Remolona (1996a),“Two Factors Along the Yield Curve," Manuscript, Capital Markets Function, Federal Reserve Bank of New York, forthcoming in Manchester School. 1996.
[42] Gong, F.F. and E.M. Remolona (1996b),“A Three-Factor Econometric Model of the U.S. Term Structure," Manuscript, Capital Markets Function, Federal Reserve Bank of New York. 1996.
[43] Geyer, A.L.J. and S. Pichler (1996),“A State-Space Approach to Estimate the Term Structure of Interest Rates: Some Empirical Evidence," Manuscript, University of Economics and Business Administration, Vienna. 1996.
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