利率期限结构理论、模型及应用研究
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摘要
利率期限结构反映了时间因素对利率的影响,也即在当前条件下市场对未来利率的预期,是整个金融体系的基准和参照系,故利率期限结构是资产定价和风险管理的基础,也是理解货币政策效应及其传导机制的关键。因此,利率期限结构研究是目前金融工程领域的一项十分重要的基础性研究工作。
本论文分别从利率期限结构形成机制理论、静态利率期限结构模型理论、动态利率期限结构建模的一般均衡方法、动态利率期限结构建模的无套利方法、HJM框架下违约利率期限结构研究以及利率期限结构模型估计理论与方法六个方面对利率期限结构理论、模型及其估计和应用进行了系统而深入的研究。
首先,在介绍利率期限结构形成机制理论的基础上,基于上海银行间同业拆放利率数据利用单位根和协整检验对预期理论对于收益率曲线不同部分的适用性差异进行研究,发现预期理论对于整个SHIBOR收益率曲线是不适用的,但对于其短端和长端则分别适用。在此基础上,利用向量误差修正模型对货币政策对于SHIBOR收益率曲线的传导效应进行分析,发现货币政策的效应沿SHIBOR收益率曲线衰减,因此SHIBOR市场的货币政策传导机制有待进一步完善。此外,通过主成分分析发现,SHIBOR市场利率期限结构的动态特性可主要由水平因子、斜率因子以及扭曲因子三个影响因子来刻画。
其次,在对静态利率期限结构估计理论进行简要介绍基础上,通过引入遗传算法对扩展Nelson-Siegel模型进行估计以改进其拟合精度,并基于上证所国债数据与基于三次样条插值的息票剥离法以及基于非线性回归的扩展Nelson-Siegel模型的估计效果进行对比研究,最后利用基于遗传算法的扩展Nelson-Siegel模型对所选取的三个样本交易日的收益率曲线进行估计,并分析了金融危机和货币政策等宏观因素对国债收益率曲线的影响。
再次,在对动态利率期限结构建模的一般均衡框架进行系统介绍基础上,分别从模型估计方法、市场适用性以及制度转换等角度对利率期限结构的均衡模型进行了研究和实证分析。首先,在CKLS广义模型框架下,系统地引入了基于扩展卡尔曼滤波和无损卡尔曼滤波的利率期限结构均衡模型的估计方法,并对以上两种方法的模型估计效果进行实证对比分析。进一步,基于无损卡尔曼滤波估计方法对Vasicek模型和CIR模型对于国外和国内市场的适用性进行了对比研究。最后,在带制度转换的CIR模型下利用基于Kim滤波的极大似然估计法对SHIBOR市场利率期限结构及其风险溢价的动态特性进行了研究,对其制度转换特征及相应的市场意义进行了重点刻画和分析,并基于SHIBOR数据就RSCIR模型与CIR模型对风险溢价动态特性的刻画能力进行了对比研究。
然后,在对无套利模型的HJM模型框架进行了深入研究基础上,基于一类特定的波动率结构设定对HJM模型框架进行有限维马尔可夫仿射实现。在此基础上,分别从两个角度引入HJM模型的直接估计法和基于无损卡尔曼滤波的极大似然估计法。最后,分别利用以上引入的两种估计方法对SHIBOR市场利率期限结构进行实证研究。结果表明,三因子HJM模型可以很好的刻画SHIBOR期限结构的动态特性和波动率结构;而三个因子中,水平因子和斜率因子是驱动SHIBOR利率系统的主要因素。
此外,基于HJM模型框架,将远期利率波动率设定为服从广义均值回归平方根过程的随机变量,以刻画隐性随机波动因子的动态特性;并通过将漂移项限制条件推广至波动因子之间,以及利率波动率的变化与利率变动之间存在相关性情形,建立了一个具有相关波动因子的广义随机波动HJM模型,进而在对模型进行有限维马尔科夫仿射实现基础上推导出零息债券的准解析定价公式。
最后,将随机波动引入可违约HJM模型框架,建立了可违约随机波动HJM模型,并通过对波动率进行适当设定,对模型进行有限维马尔科夫仿射实现,推导出了可违约债券价格的显性解析式。在此基础上,基于AAA级企业债券数据利用三因子可违约随机波动HJM模型对我国违约利率期限结构的动态特性进行刻画,发现无风险短期利率、短期信用利差以及随机波动过程之间存在显著的相关关系。此外,还发现在违约利率上行期间,无风险短期利率波动因子对于违约短期利率波动率的贡献较大;而在违约利率下行期间,短期信用利差波动因子对于违约短期利率波动率的贡献则占优。
本论文的研究内容受国家自然科学基金项目(No. 70771075)及教育部博士点基金项目(No. 200800560032)资助,是其部分研究成果。
The term structure of interest rates reflects the influence of the time factor on interest rates, that is the market's expectations of future interest rates given the current market conditions, which makes it the benchmark for the entire financial system and the basis for asset pricing and hedging, and thus the key to interpret the effect of the monetary policy and its transmission mechanism. So research on the term structure of interest rates is the fundamental work of critical importance in the financial engineering field.
In this dissertation, a systematic and intensive study is made on theories, models and applications of the term structure of interest rates from six aspects below respectively: formation mechanism theories of term structure of interest rates, static term structure models of interest rates, generalized equilibrium models of term structure of interest rates, no-arbitrage models of term structure of interest rates, HJM framework of defaultable term structure models and estimation theories for term structure models.
First, based on a concise introduction to formation mechanism theories of term structure of interest rates, an empirical research is made on the applicability of the expectation hypothesis (EH) to different parts of the term structure of SHIBOR using the unit root test and cointegration method, which results in the find that EH is not applicable to the whole yield curve of SHIBOR. However, EH works on the short and long parts of the SHIBOR term structure respectively. Thus, vector error correction models are established to investigate the effect of the monetary policy on the yield curve and it is found that the effect tends to decline along the term structure of SHIBOR, which means that monetary policy transmission in SHIBOR market need further improvements. Furthermore, the result of principal component analysis indicates that the dynamics of the term structure of SHIBOR can be captured by three main factors, that is the level factor, the slope factor and the curvature factor.
Second, based on a brief introduction to static term structure models of interest rates, an estimation approach based on the genetic algorithm (hereafter GA) is introduced for the extended Nelson-Siegel model to improve its fitness. Then an empirical comparison is made between the bootstrap method based on cubic spline interpolation, the extended Nelson-Siegel model based on nonlinear regression and the model introduced above. Then yield curves are estimated based on the method proposed for the three sample dates and the influence of the financial crisis and monetary policy on the curve is analyzed.
Third, based on a systematic introduction to the generalized equilibrium framework of term structure models of interest rates, intensive research is made on equilibrium models from the aspects of estimation methods, applicability and regime switching respectively. Firstly, in the CKLS general framework, the dissertation introduces approaches of estimation for equilibrium models of term structure of interest rates based on EKF and UKF, and an empirical contrast is made between the estimation performances of the EKF-based and UKF-based algorithms. Furthermore, the fitness of Vasicek model and CIR model is compared with both the foreign and domestic data using the UKF-based algorithm. Finally, under a RSCIR model, an empirical research is carried out on the dynamics of the term structure and risk premium of SHIBOR using the Kim filter based maximum likelihood estimator, with especial attention given to the regime-switching characteristic of the SHIBOR evolution and the corresponding market implications, and a comparison is also made between RSCIR model and CIR model based on the SHIBOR data above.
Fourth, based on an in-depth study on the HJM framework of no-arbitrage models, the dissertation performs finite-dimensional Markovian affine realization for HJM models under a certain volatility specification, based on which a direct estimation method and a maximum likelihood estimator based on the unscented Kalman filter are introduced for HJM models in two different ways. Finally empirical research is made on the SHIBOR term structure based on the two estimation methods above respectively, which comes to a conclusion that the dynamics and volatility structure of SHIBOR are both well captured by a three-factor HJM model, and the level and slope factors explain the majority of the variation of the yield curve.
Fifth, Heath-Jarrow-Morton model is generalized by extending the no-arbitrage drift restriction with nonzero instantaneous correlations between volatility factors and setting forward rate volatilities subject to generalized mean-reverting square-root processes and correlated with innovations to forward rates. Then a quasi-analytical formula for zero coupon bond prices is derived based on the finite-dimensional Markovian affine realization of the model framework above.
Finally, a stochastic volatility HJM framework of defaultable term structure models is proposed by introducing stochastic volatility process into the defaultable HJM framework. Then finite-dimensional Markovian affine realization is performed for the model framework established above under a certain volatility specification, based on which a explicit analytical pricing formula is derived for defaultable bonds. On the basis of the above, an empirical study is made on the dynamics of the term structure of defaultable bonds in China in a three-factor defaultable HJM framework with stochastic volatility, using AAA corporate bond data, which demonstrates that there are significant correlations among the default-free short rate, the instantaneous short-term credit spread and the stochastic volatility. It is also found that the default-free short rate makes more contribution to the volatility of the defaultable short rate when the defaultable rate goes up, while the short-term credit spread makes more contribution when the defaultable rate goes down.
The research is sponsored by National Natural Science Foundation of China (Grant No. 70771075) and Doctoral Fund of Ministry of Education of China (Grant No. 200800560032).
本论文分别从利率期限结构形成机制理论、静态利率期限结构模型理论、动态利率期限结构建模的一般均衡方法、动态利率期限结构建模的无套利方法、HJM框架下违约利率期限结构研究以及利率期限结构模型估计理论与方法六个方面对利率期限结构理论、模型及其估计和应用进行了系统而深入的研究。
首先,在介绍利率期限结构形成机制理论的基础上,基于上海银行间同业拆放利率数据利用单位根和协整检验对预期理论对于收益率曲线不同部分的适用性差异进行研究,发现预期理论对于整个SHIBOR收益率曲线是不适用的,但对于其短端和长端则分别适用。在此基础上,利用向量误差修正模型对货币政策对于SHIBOR收益率曲线的传导效应进行分析,发现货币政策的效应沿SHIBOR收益率曲线衰减,因此SHIBOR市场的货币政策传导机制有待进一步完善。此外,通过主成分分析发现,SHIBOR市场利率期限结构的动态特性可主要由水平因子、斜率因子以及扭曲因子三个影响因子来刻画。
其次,在对静态利率期限结构估计理论进行简要介绍基础上,通过引入遗传算法对扩展Nelson-Siegel模型进行估计以改进其拟合精度,并基于上证所国债数据与基于三次样条插值的息票剥离法以及基于非线性回归的扩展Nelson-Siegel模型的估计效果进行对比研究,最后利用基于遗传算法的扩展Nelson-Siegel模型对所选取的三个样本交易日的收益率曲线进行估计,并分析了金融危机和货币政策等宏观因素对国债收益率曲线的影响。
再次,在对动态利率期限结构建模的一般均衡框架进行系统介绍基础上,分别从模型估计方法、市场适用性以及制度转换等角度对利率期限结构的均衡模型进行了研究和实证分析。首先,在CKLS广义模型框架下,系统地引入了基于扩展卡尔曼滤波和无损卡尔曼滤波的利率期限结构均衡模型的估计方法,并对以上两种方法的模型估计效果进行实证对比分析。进一步,基于无损卡尔曼滤波估计方法对Vasicek模型和CIR模型对于国外和国内市场的适用性进行了对比研究。最后,在带制度转换的CIR模型下利用基于Kim滤波的极大似然估计法对SHIBOR市场利率期限结构及其风险溢价的动态特性进行了研究,对其制度转换特征及相应的市场意义进行了重点刻画和分析,并基于SHIBOR数据就RSCIR模型与CIR模型对风险溢价动态特性的刻画能力进行了对比研究。
然后,在对无套利模型的HJM模型框架进行了深入研究基础上,基于一类特定的波动率结构设定对HJM模型框架进行有限维马尔可夫仿射实现。在此基础上,分别从两个角度引入HJM模型的直接估计法和基于无损卡尔曼滤波的极大似然估计法。最后,分别利用以上引入的两种估计方法对SHIBOR市场利率期限结构进行实证研究。结果表明,三因子HJM模型可以很好的刻画SHIBOR期限结构的动态特性和波动率结构;而三个因子中,水平因子和斜率因子是驱动SHIBOR利率系统的主要因素。
此外,基于HJM模型框架,将远期利率波动率设定为服从广义均值回归平方根过程的随机变量,以刻画隐性随机波动因子的动态特性;并通过将漂移项限制条件推广至波动因子之间,以及利率波动率的变化与利率变动之间存在相关性情形,建立了一个具有相关波动因子的广义随机波动HJM模型,进而在对模型进行有限维马尔科夫仿射实现基础上推导出零息债券的准解析定价公式。
最后,将随机波动引入可违约HJM模型框架,建立了可违约随机波动HJM模型,并通过对波动率进行适当设定,对模型进行有限维马尔科夫仿射实现,推导出了可违约债券价格的显性解析式。在此基础上,基于AAA级企业债券数据利用三因子可违约随机波动HJM模型对我国违约利率期限结构的动态特性进行刻画,发现无风险短期利率、短期信用利差以及随机波动过程之间存在显著的相关关系。此外,还发现在违约利率上行期间,无风险短期利率波动因子对于违约短期利率波动率的贡献较大;而在违约利率下行期间,短期信用利差波动因子对于违约短期利率波动率的贡献则占优。
本论文的研究内容受国家自然科学基金项目(No. 70771075)及教育部博士点基金项目(No. 200800560032)资助,是其部分研究成果。
The term structure of interest rates reflects the influence of the time factor on interest rates, that is the market's expectations of future interest rates given the current market conditions, which makes it the benchmark for the entire financial system and the basis for asset pricing and hedging, and thus the key to interpret the effect of the monetary policy and its transmission mechanism. So research on the term structure of interest rates is the fundamental work of critical importance in the financial engineering field.
In this dissertation, a systematic and intensive study is made on theories, models and applications of the term structure of interest rates from six aspects below respectively: formation mechanism theories of term structure of interest rates, static term structure models of interest rates, generalized equilibrium models of term structure of interest rates, no-arbitrage models of term structure of interest rates, HJM framework of defaultable term structure models and estimation theories for term structure models.
First, based on a concise introduction to formation mechanism theories of term structure of interest rates, an empirical research is made on the applicability of the expectation hypothesis (EH) to different parts of the term structure of SHIBOR using the unit root test and cointegration method, which results in the find that EH is not applicable to the whole yield curve of SHIBOR. However, EH works on the short and long parts of the SHIBOR term structure respectively. Thus, vector error correction models are established to investigate the effect of the monetary policy on the yield curve and it is found that the effect tends to decline along the term structure of SHIBOR, which means that monetary policy transmission in SHIBOR market need further improvements. Furthermore, the result of principal component analysis indicates that the dynamics of the term structure of SHIBOR can be captured by three main factors, that is the level factor, the slope factor and the curvature factor.
Second, based on a brief introduction to static term structure models of interest rates, an estimation approach based on the genetic algorithm (hereafter GA) is introduced for the extended Nelson-Siegel model to improve its fitness. Then an empirical comparison is made between the bootstrap method based on cubic spline interpolation, the extended Nelson-Siegel model based on nonlinear regression and the model introduced above. Then yield curves are estimated based on the method proposed for the three sample dates and the influence of the financial crisis and monetary policy on the curve is analyzed.
Third, based on a systematic introduction to the generalized equilibrium framework of term structure models of interest rates, intensive research is made on equilibrium models from the aspects of estimation methods, applicability and regime switching respectively. Firstly, in the CKLS general framework, the dissertation introduces approaches of estimation for equilibrium models of term structure of interest rates based on EKF and UKF, and an empirical contrast is made between the estimation performances of the EKF-based and UKF-based algorithms. Furthermore, the fitness of Vasicek model and CIR model is compared with both the foreign and domestic data using the UKF-based algorithm. Finally, under a RSCIR model, an empirical research is carried out on the dynamics of the term structure and risk premium of SHIBOR using the Kim filter based maximum likelihood estimator, with especial attention given to the regime-switching characteristic of the SHIBOR evolution and the corresponding market implications, and a comparison is also made between RSCIR model and CIR model based on the SHIBOR data above.
Fourth, based on an in-depth study on the HJM framework of no-arbitrage models, the dissertation performs finite-dimensional Markovian affine realization for HJM models under a certain volatility specification, based on which a direct estimation method and a maximum likelihood estimator based on the unscented Kalman filter are introduced for HJM models in two different ways. Finally empirical research is made on the SHIBOR term structure based on the two estimation methods above respectively, which comes to a conclusion that the dynamics and volatility structure of SHIBOR are both well captured by a three-factor HJM model, and the level and slope factors explain the majority of the variation of the yield curve.
Fifth, Heath-Jarrow-Morton model is generalized by extending the no-arbitrage drift restriction with nonzero instantaneous correlations between volatility factors and setting forward rate volatilities subject to generalized mean-reverting square-root processes and correlated with innovations to forward rates. Then a quasi-analytical formula for zero coupon bond prices is derived based on the finite-dimensional Markovian affine realization of the model framework above.
Finally, a stochastic volatility HJM framework of defaultable term structure models is proposed by introducing stochastic volatility process into the defaultable HJM framework. Then finite-dimensional Markovian affine realization is performed for the model framework established above under a certain volatility specification, based on which a explicit analytical pricing formula is derived for defaultable bonds. On the basis of the above, an empirical study is made on the dynamics of the term structure of defaultable bonds in China in a three-factor defaultable HJM framework with stochastic volatility, using AAA corporate bond data, which demonstrates that there are significant correlations among the default-free short rate, the instantaneous short-term credit spread and the stochastic volatility. It is also found that the default-free short rate makes more contribution to the volatility of the defaultable short rate when the defaultable rate goes up, while the short-term credit spread makes more contribution when the defaultable rate goes down.
The research is sponsored by National Natural Science Foundation of China (Grant No. 70771075) and Doctoral Fund of Ministry of Education of China (Grant No. 200800560032).
引文
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