住房抵押贷款强度定价模型研究
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摘要
住房抵押贷款作为银行的一项重要资产,其定价的准确与否直接影响银行的资产管理水平和进一步证券化的可行性。住房抵押贷款强度定价模型是简约化模型与结构化模型的有效整合。虽已有一些国外学者在强度模型研究上取得一些初步成果,但这些成果大都发表于次贷危机爆发之前,已不能适应住房抵押贷款市场的新形势。并且国外学者建立的强度模型难以对我国住房抵押贷款市场数据进行较准确地拟合。我国的住房抵押贷款的违约强度和提前还款强度具有明显的尖峰厚尾特征。现在国内外学者对于这一特征的刻画主要采用跳扩散过程和Copula函数这两种方法。受制于庞大的计算量,无法同时使用跳扩散过程和Copula函数来刻画违约强度和提前还款强度。本文分别采用这两种方法来度量违约强度和提前还款强度,构建考虑跳扩散过程的住房抵押贷款强度定价模型和基于Clayton Copula住房抵押贷款强度定价模型,并对模型中主要参数变量一贴现率、违约强度和提前还款强度提出了相应的参数估计方法,相关研究成果简述如下:
(1)本文构建可综合考虑违约、提前还款、抵押房产和利率四种风险因素的住房抵押贷款定价模型—考虑跳扩散过程的住房抵押贷款强度定价模型。
该模型从中国个人住房抵押贷款市场的实际出发,改进了KKS模型的四个关键因素:折现方法、追偿值计算、违约强度(提前还款强度)模型和贴现率模型;对每月的还款金额采用不同的折现率,用抵押房产价格和未偿付本金的最小值取代KKS模型中的固定追偿值,采用跳扩散过程中的CIR模型刻画违约强度和提前还款强度,采用跳扩散过程中的带跳的Vasicek模型刻画贴现率。该模型的稳定性测试结果表明,考虑跳扩散过程的住房抵押贷款强度模型稳定性较好,可满足商业银行住房抵押贷款定价的要求。最后,考察了首付比例、贷款利率、贷款期限的变化对住房抵押贷款价值的影响。研究结论对我国银行制定合理的住房抵押贷款政策具有重要的借鉴价值。
(2)以三月期Shibor作为住房抵押贷款的贴现率,遴选出带跳的Vasicek模型作为刻画Shibor的最佳单因素利率模型,并提出了基于粒子滤波和随机逼近的参数和状态联合估计方法来估计该模型的参数。
统计特征检验的结果表明Shibor的收益率数据有均值回复性和厚尾性,故可选择带跳的Vasicek模型或带跳的指数Vasicek模型来刻画。但这两个模型都比较复杂,常用的最小二乘法、最大似然估计和卡尔曼滤波都不能很好地对这两个模型进行参数估计。而粒子滤波方法摆脱了求解非线性滤波问题时随机量必须满足高斯分布的制约,适于本文的研究。所以,本文提出基于粒子滤波和随机逼近的参数和状态联合估计方法,通过对两模型的拟合优度和预测精度的比较,最终遴选出带跳Vasicek单因素利率模型作为描述Shibor的适用模型。
(3)解决了考虑跳扩散过程的住房抵押贷款强度定价模型中的两个参数变量一违约强度和提前还款强度的小样本参数估计问题,提出了两阶段MCMC参数估计方法。
采用我国第一支个人住房抵押贷款支持证券“建元2005—1”的违约和提前还款数据作为违约强度和提前还款强度的实证样本。但该样本的规模较小,致使小样本偏差成为违约强度模型和提前还款模型参数估计的一道障碍。违约强度模型和提前还款模型都属于跳扩散模型,许多学者的研究成果已证明MCMC方法是跳扩散模型参数估计的有效工具。但现有的MCMC方法对小样本下违约强度模型和提前还款强度模型的参数估计显得无能为力。有鉴于此,本文提出两阶段MCMC参数估计方法:第一阶段用Lee和Mykland的跳辨识方法估计跳跃项参数;第二阶段用MCMC方法估计扩散和漂移项参数。误差分析和稳定性分析的结果表明两阶段MCMC方法小样本下违约模型参数估计的效果要明显好于单纯的MCMC方法。
(4)除了违约、提前还款、抵押房产和利率等风险因素外,特别考察了相关性因素,进而构建了基于Clayton Copula的住房抵押贷款强度定价模型。
在对住房抵押贷款违约情况深入分析的基础上,重点利用Clayton Copula模型考察违约相关性和提前还款相关性对住房抵押贷款价值的影响,进而构建一个基于ClaytonCopula的住房抵押贷款强度定价模型。违约相关性和提前还款相关性对住房抵押贷款价值影响的分析表明,Clayton Copula模型可以很好地描述违约和提前还款强度的尖峰厚尾现象。主要参数的敏感度分析结果表明,本文对违约强度和提前还款强度模型的改进是必要和有效的。压力测试结果说明:如果近年内房价出现大幅下跌,将给大量发行低首付比例住房抵押贷款的银行带来巨大损失:并且相较于考虑跳扩散过程的住房抵押贷款强度定价模型,基于Clayton Copula的住房抵押贷款强度定价模型对房价暴跌的极端情况反应更为敏感,如果银行采用这一模型来度量住房抵押贷款价值,可以更早地预警风险。
综上所述,本文在国外先进的住房抵押贷款定价模型的基础上,结合中国市场的实际和次贷危机的经验教训,提出两个能够更加全面、准确地度量违约风险、提前还款风险、抵押房产风险、贴现率风险和相关性风险的住房抵押贷款强度定价模型;为了克服了数据缺乏的困难,充分利用各种参数估计和数值分析的手段,对这两个模型进行了深入地模拟分析和实证分析。从理论和实践两方面推进了住房抵押贷款定价研究的进展。相应的研究成果可为银行提供一种更准确的住房抵押贷款估值方法,有利于提高我国银行信贷资产的管理效率和经营的安全性。
Mortgages are important assets of commercial banks. The pricing of mortgages has direct impact on bank asset management and the feasibility of securitization. Mortgage intensity pricing model integrates reduced-form model and structure model effectively. Although there are some results on intensity model study, most of them were published before the sub-prime crisis, thus they are unable to meet the new situation. Furthermore, it is unreasonable to apply existing results of intensity model based on other markets directly to China's housing mortgage loan market. Default intensity and prepayment intensity of Chinese housing mortgage have the characteristic of higher peak and fat tail. Now Domestic and foreign scholars use jump-diffusion process and Copula function to describe this characteristic. Subject to the large amount of computation, we can not simultaneously use jump-diffusion process and Copula function to measure default intensity and prepayment intensity. Therefore this paper respectively uses those two methods and builds a jump diffusion process-based intensity housing mortgage pricing model and a Clayton Copula-based intensity housing mortgage pricing model; also I propose two new methods to estimate parameters such as discount rate, default intensity and prepayment intensity. The main results are listed as follows:
(1) The jump diffusion process-based intensity housing mortgage pricing model built in this paper considers default, prepayment, building property and interest rate risk factors. The model improves four key factors of KKS model: discounting method, calculation approach of recovery value and interest rate stochastic model. Payment per month is discounted by different rate. Fixed recovery value in KKS model is replaced by minimum of mortgaged house price and remained mortgage principle. In addition, two-factor CIR interest model is substituted by Vasicek model with jumps. The default intensity and prepayment intensity are measured by the CIR model which belongs to jump-diffusion process. The discount rate is measured by Vasicek model with jumps which belong to jump-diffusion process. Stability analysis of the model shows that the jump diffusion process-based intensity housing mortgage pricing model can meet commercial bank's requirement. Finally, this paper examines the impact of housing price trend, the proportion of initial payment to principal, loan interest, scale and term to mortgage value.
(2) This paper uses Vasicek model with jumps as the one-factor interest rate model to estimate the 3-month Shibor. I propose an adaptive estimation algorithm based on combination of particle filter and simultaneous perturbation stochastic approximation method to estimate parameters of this model.
Statistical tests indicate that three-month Shibor has the characteristics of mean reversion and fat tails. Therefore I use Vasicek model with jumps or exponential Vasicek model with jumps as alternatives since these two are nonlinear and non-Gaussian models. Common methods such as least squares, maximum likelihood estimation and Kalman filtering can not estimate parameters of these two models very well. The particle filter algorithm is suitable for nonlinear and non-Gaussian model. Therefore, this paper proposes adaptive estimation algorithm based on combination of particle filter and simultaneous perturbation stochastic approximation method. At last, we compare goodness-of-fit and forecast effectiveness between the two models, and the result shows that Vasicek model with jumps does better than exponential Vasicek model with jumps in fitting Shibor.
(3) To estimate parameters of default intensity and prepayment intensity of jump diffusion process-based intensity housing mortgage pricing model with small-size samples, this paper proposes a two-stage MCMC (Markov chain Monte Carlo) approach.
I use "Jianyuan 2005-1", the first MBS (Mortgage Backed Securities) in China to construct my sample. However, the sample size is limited so that it is difficult to estimate parameters of default intensity and prepayment intensity properly. Existing literature suggests that MCMC can effectively estimate parameters of the jump-diffusion process, but only for large samples. Therefore, this paper proposes a two-stage MCMC (Markov chain Monte Carlo) approach. In the first stage, I adopt non-parametric estimation method developed by Lee and Mykland to estimate the parameters of jumps. In the second stage, I use MCMC to estimate parameters of diffusion and drift. Then we compare the estimation error of the two-stage MCMC approach and MCMC approach, and analyze their stability. All of the above show that the former is less than the latter in error.
(4) This paper establishes a Clayton Copula-based intensity housing mortgage pricing model which considers the default, prepayment, building property, interest rate and correlation.
To analyse mortgage default, we use Copula models to measure the influence of default correlation and prepayment correlation on the mortgage value, i.e. establishing an intensity housing mortgage pricing model based on Clayton Copula. The analysis about the influence of default correlation and prepayment correlation on the mortgage value shows that Clayton Copula model describes fat-tailedness better. The sensitivity analysis results of main parameters indicate that this model is necessary and effective. Stress test results show that if house prices fall sharply in recent years, the bank that issued many housing mortgages with low proportion of initial payment to principal is exposed to huge losses. Clayton Copula-based intensity housing mortgage pricing model is more sensitive to the risk of housing price falling quickly than jump diffusion process-based intensity housing mortgage pricing model. If the bank uses this model to measure the value of housing mortgages, the risk will be warned earlier.
In summary, this paper builds two housing mortgage pricing models and adapts the two models to China's housing mortgage market. These two models improve existing mortgage pricing models in measuring dafault risk, prepayment risk, building property risk and interest risk. For overcoming the difficulties of data limitation, this paper does simulation analysis and empirical analysis to the two pricing model by use of various methods of parameters estimation and numerical analysis. Research on housing mortgage pricing model is promoted in the theoretical and practical aspects. Thus it is in favor of improving the asset management efficiency and operational safety of Chinese banks.
(1)本文构建可综合考虑违约、提前还款、抵押房产和利率四种风险因素的住房抵押贷款定价模型—考虑跳扩散过程的住房抵押贷款强度定价模型。
该模型从中国个人住房抵押贷款市场的实际出发,改进了KKS模型的四个关键因素:折现方法、追偿值计算、违约强度(提前还款强度)模型和贴现率模型;对每月的还款金额采用不同的折现率,用抵押房产价格和未偿付本金的最小值取代KKS模型中的固定追偿值,采用跳扩散过程中的CIR模型刻画违约强度和提前还款强度,采用跳扩散过程中的带跳的Vasicek模型刻画贴现率。该模型的稳定性测试结果表明,考虑跳扩散过程的住房抵押贷款强度模型稳定性较好,可满足商业银行住房抵押贷款定价的要求。最后,考察了首付比例、贷款利率、贷款期限的变化对住房抵押贷款价值的影响。研究结论对我国银行制定合理的住房抵押贷款政策具有重要的借鉴价值。
(2)以三月期Shibor作为住房抵押贷款的贴现率,遴选出带跳的Vasicek模型作为刻画Shibor的最佳单因素利率模型,并提出了基于粒子滤波和随机逼近的参数和状态联合估计方法来估计该模型的参数。
统计特征检验的结果表明Shibor的收益率数据有均值回复性和厚尾性,故可选择带跳的Vasicek模型或带跳的指数Vasicek模型来刻画。但这两个模型都比较复杂,常用的最小二乘法、最大似然估计和卡尔曼滤波都不能很好地对这两个模型进行参数估计。而粒子滤波方法摆脱了求解非线性滤波问题时随机量必须满足高斯分布的制约,适于本文的研究。所以,本文提出基于粒子滤波和随机逼近的参数和状态联合估计方法,通过对两模型的拟合优度和预测精度的比较,最终遴选出带跳Vasicek单因素利率模型作为描述Shibor的适用模型。
(3)解决了考虑跳扩散过程的住房抵押贷款强度定价模型中的两个参数变量一违约强度和提前还款强度的小样本参数估计问题,提出了两阶段MCMC参数估计方法。
采用我国第一支个人住房抵押贷款支持证券“建元2005—1”的违约和提前还款数据作为违约强度和提前还款强度的实证样本。但该样本的规模较小,致使小样本偏差成为违约强度模型和提前还款模型参数估计的一道障碍。违约强度模型和提前还款模型都属于跳扩散模型,许多学者的研究成果已证明MCMC方法是跳扩散模型参数估计的有效工具。但现有的MCMC方法对小样本下违约强度模型和提前还款强度模型的参数估计显得无能为力。有鉴于此,本文提出两阶段MCMC参数估计方法:第一阶段用Lee和Mykland的跳辨识方法估计跳跃项参数;第二阶段用MCMC方法估计扩散和漂移项参数。误差分析和稳定性分析的结果表明两阶段MCMC方法小样本下违约模型参数估计的效果要明显好于单纯的MCMC方法。
(4)除了违约、提前还款、抵押房产和利率等风险因素外,特别考察了相关性因素,进而构建了基于Clayton Copula的住房抵押贷款强度定价模型。
在对住房抵押贷款违约情况深入分析的基础上,重点利用Clayton Copula模型考察违约相关性和提前还款相关性对住房抵押贷款价值的影响,进而构建一个基于ClaytonCopula的住房抵押贷款强度定价模型。违约相关性和提前还款相关性对住房抵押贷款价值影响的分析表明,Clayton Copula模型可以很好地描述违约和提前还款强度的尖峰厚尾现象。主要参数的敏感度分析结果表明,本文对违约强度和提前还款强度模型的改进是必要和有效的。压力测试结果说明:如果近年内房价出现大幅下跌,将给大量发行低首付比例住房抵押贷款的银行带来巨大损失:并且相较于考虑跳扩散过程的住房抵押贷款强度定价模型,基于Clayton Copula的住房抵押贷款强度定价模型对房价暴跌的极端情况反应更为敏感,如果银行采用这一模型来度量住房抵押贷款价值,可以更早地预警风险。
综上所述,本文在国外先进的住房抵押贷款定价模型的基础上,结合中国市场的实际和次贷危机的经验教训,提出两个能够更加全面、准确地度量违约风险、提前还款风险、抵押房产风险、贴现率风险和相关性风险的住房抵押贷款强度定价模型;为了克服了数据缺乏的困难,充分利用各种参数估计和数值分析的手段,对这两个模型进行了深入地模拟分析和实证分析。从理论和实践两方面推进了住房抵押贷款定价研究的进展。相应的研究成果可为银行提供一种更准确的住房抵押贷款估值方法,有利于提高我国银行信贷资产的管理效率和经营的安全性。
Mortgages are important assets of commercial banks. The pricing of mortgages has direct impact on bank asset management and the feasibility of securitization. Mortgage intensity pricing model integrates reduced-form model and structure model effectively. Although there are some results on intensity model study, most of them were published before the sub-prime crisis, thus they are unable to meet the new situation. Furthermore, it is unreasonable to apply existing results of intensity model based on other markets directly to China's housing mortgage loan market. Default intensity and prepayment intensity of Chinese housing mortgage have the characteristic of higher peak and fat tail. Now Domestic and foreign scholars use jump-diffusion process and Copula function to describe this characteristic. Subject to the large amount of computation, we can not simultaneously use jump-diffusion process and Copula function to measure default intensity and prepayment intensity. Therefore this paper respectively uses those two methods and builds a jump diffusion process-based intensity housing mortgage pricing model and a Clayton Copula-based intensity housing mortgage pricing model; also I propose two new methods to estimate parameters such as discount rate, default intensity and prepayment intensity. The main results are listed as follows:
(1) The jump diffusion process-based intensity housing mortgage pricing model built in this paper considers default, prepayment, building property and interest rate risk factors. The model improves four key factors of KKS model: discounting method, calculation approach of recovery value and interest rate stochastic model. Payment per month is discounted by different rate. Fixed recovery value in KKS model is replaced by minimum of mortgaged house price and remained mortgage principle. In addition, two-factor CIR interest model is substituted by Vasicek model with jumps. The default intensity and prepayment intensity are measured by the CIR model which belongs to jump-diffusion process. The discount rate is measured by Vasicek model with jumps which belong to jump-diffusion process. Stability analysis of the model shows that the jump diffusion process-based intensity housing mortgage pricing model can meet commercial bank's requirement. Finally, this paper examines the impact of housing price trend, the proportion of initial payment to principal, loan interest, scale and term to mortgage value.
(2) This paper uses Vasicek model with jumps as the one-factor interest rate model to estimate the 3-month Shibor. I propose an adaptive estimation algorithm based on combination of particle filter and simultaneous perturbation stochastic approximation method to estimate parameters of this model.
Statistical tests indicate that three-month Shibor has the characteristics of mean reversion and fat tails. Therefore I use Vasicek model with jumps or exponential Vasicek model with jumps as alternatives since these two are nonlinear and non-Gaussian models. Common methods such as least squares, maximum likelihood estimation and Kalman filtering can not estimate parameters of these two models very well. The particle filter algorithm is suitable for nonlinear and non-Gaussian model. Therefore, this paper proposes adaptive estimation algorithm based on combination of particle filter and simultaneous perturbation stochastic approximation method. At last, we compare goodness-of-fit and forecast effectiveness between the two models, and the result shows that Vasicek model with jumps does better than exponential Vasicek model with jumps in fitting Shibor.
(3) To estimate parameters of default intensity and prepayment intensity of jump diffusion process-based intensity housing mortgage pricing model with small-size samples, this paper proposes a two-stage MCMC (Markov chain Monte Carlo) approach.
I use "Jianyuan 2005-1", the first MBS (Mortgage Backed Securities) in China to construct my sample. However, the sample size is limited so that it is difficult to estimate parameters of default intensity and prepayment intensity properly. Existing literature suggests that MCMC can effectively estimate parameters of the jump-diffusion process, but only for large samples. Therefore, this paper proposes a two-stage MCMC (Markov chain Monte Carlo) approach. In the first stage, I adopt non-parametric estimation method developed by Lee and Mykland to estimate the parameters of jumps. In the second stage, I use MCMC to estimate parameters of diffusion and drift. Then we compare the estimation error of the two-stage MCMC approach and MCMC approach, and analyze their stability. All of the above show that the former is less than the latter in error.
(4) This paper establishes a Clayton Copula-based intensity housing mortgage pricing model which considers the default, prepayment, building property, interest rate and correlation.
To analyse mortgage default, we use Copula models to measure the influence of default correlation and prepayment correlation on the mortgage value, i.e. establishing an intensity housing mortgage pricing model based on Clayton Copula. The analysis about the influence of default correlation and prepayment correlation on the mortgage value shows that Clayton Copula model describes fat-tailedness better. The sensitivity analysis results of main parameters indicate that this model is necessary and effective. Stress test results show that if house prices fall sharply in recent years, the bank that issued many housing mortgages with low proportion of initial payment to principal is exposed to huge losses. Clayton Copula-based intensity housing mortgage pricing model is more sensitive to the risk of housing price falling quickly than jump diffusion process-based intensity housing mortgage pricing model. If the bank uses this model to measure the value of housing mortgages, the risk will be warned earlier.
In summary, this paper builds two housing mortgage pricing models and adapts the two models to China's housing mortgage market. These two models improve existing mortgage pricing models in measuring dafault risk, prepayment risk, building property risk and interest risk. For overcoming the difficulties of data limitation, this paper does simulation analysis and empirical analysis to the two pricing model by use of various methods of parameters estimation and numerical analysis. Research on housing mortgage pricing model is promoted in the theoretical and practical aspects. Thus it is in favor of improving the asset management efficiency and operational safety of Chinese banks.
引文
[1]陈永华,毕玉升.抵押贷款支持证券定价的蒙特卡罗方法研究[J].上海经济研究,2008,(1):78-84.
[2]陈天宇.当前我国个人住房抵押贷款风险分析及对策研究.青岛大学硕士论文,2008.
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