利率期限结构模型及其在融资决策问题中的应用
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摘要
金融数学是一门新兴的边缘科学，是数学与金融学的交叉。其核心问题是不确定环境下的最优投资策略的选择理论和资产定价理论。而短期利率对金融资产定价和金融风险管理有着决定性的影响，所以利率期限结构模型以及利率的行为特征一直以来就是金融学研究的重点。本文在回顾利率期限结构理论近二十年发展历程的基础上，对这一领域的主要研究成果进行了简要的总结和探讨。主要做了以下几方面的工作：
第一，介绍利率期限结构的有关概念、定义和相关的金融背景。
第二，传统的利率期限结构模型主要集中于研究收益率曲线的形状及其形成的原因。本文介绍了传统利率期限结构的几种理论，并对各种理论的特点做了比较。现代利率期限结构的研究与利率衍生证券的定价是密不可分的。因此，本文对常用离散时间和连续时间的利率期限结构模型分别进行了分析，并结合债券的定价问题，得出了各模型的优缺点。
第三，新型的利率期限结构模型的目的是将依赖于状态变量数的短期利率动态化所造成的不确定的多源因素包涵到模型中，用现代数学的方法对利率行为加以研究。本文在Chan，Karolyi，Longstaff and Sanders等人提出的连续时间模型统一框架的基础上，讨论了一系列水平模型的特点，这些模型显示了利率变动的条件均值和条件方差与利率水平之间的关系。利率的结构转换模型是对Vasicek模型和CIR模型的推广。本文对这三种模型进行了分析和比较，利用中国金融市场上银行间同业拆借利率的数据，对加入结构转换变量的GARCH利率期限结构模型做了实证分析，发现利率结构转换模型更适合中国金融市场上利率行为的特点。随机跳跃过程模型考虑了金融市场上稀有事件对股票价格、期权定价等问题的影响，将利率的动态过程分为连续部分和跳跃部分，用Brown运动来描述金融资产的价格过程是连续的，而用跳跃部分来描述不可预测的随机事件对这种连续性的破坏。本文从理论上探讨了用Poisson过程和Levy过程描述跳跃部分的利率动态模型，并和扩散过程下的模型作了比较，得到随机跳跃过程模型只是多了一个跳跃项，且这一模型是由一个具有均值回复特性的Ornstein-Uhlenbek过程和Poisson过程混合而成的。
第四，鞅方法的基本思想是先选取适当的随机贴现因子，使利率衍生产品
的价格除以贴现因子后得到相对价格，而这一相对价格过程在风险中性测度尸’
下是一个鞍过程.本文从零息债券的鞍表示法出发，探讨了在确定短期利率的模
型后，偏微分方程法(尸Z)E法)和教方法结果的一致性，但鞍方法更直观，更
容易求解，也更便于应用.从而得到了一个更一般化的结论，它是对Hull一White
模型的推广.并利用其结论从理论上讨论了如何将利率期限结构模型用于企业的
融资决策问题.
Finance mathematics is a new bi-disciplinary science, and is a intersection between mathematics and finance. Their key problems are the choice theory of optimal strategy for investment and the pricing theory of assets. And short interest rate has a determinative importance to the pricing of financial assets and risk managements. So term structure models of interest rate and characteristics of the behavior of interest rate are always a focal point to financial researches. In this paper, recalling the development process about the term structure of interest rate in twenty years, the author wants to get a clear sequence of ideas in all kinds of complicate mathematics models, and gives a brief summary to the major researches achievements in this field. Mainly there are the following contents:
First, introduce some conceptions, definitions and relational backgrounds in finance.
Second, traditional term structure models of interest rate focus on what the yield curve's shape is and how it forms. The article introduce four theories and compares their characteristics. The research on the modern term structure of interest rate is relative to the pricing of derivatives. Therefore, the paper analyses these models which are often used discrete time and continuous time, combining the problem about the pricing of bond, and gets their advantages and shortcomings in theory.
Third, new term structure models of interest rate include some uncertain multi-factors, which the dynamic state of short interest rate depends on the variable about state, and researching the behavior of interest rate by modern mathematical methods. In order to describe the random behavior of interest rate, the essay discusses a series of characteristics about level models based on Chan etc. These models show the relationship between conditional mean and variance of the volatility of interest rate and the level of interest rate. The regime-switching model about interest rate extends Vasicek and CIR models. The essay compares them and finds that the former is more suitable to the behavior of interest rate in China financial market than others. The stochastic process with jumps considers those rare events which have affect on stock price, the pricing of option and so on, the dynamic state of interest rate is
divided into continuous and discontinuous (jumP) processes. The article discusses the dynamic model of interest rate with Poisson process and Levy process in theory, and contrasting with those models under diffusion process. The article gets the former only adds a part with jump, and the model is made up of Ornstein - Uhlenbek process with the characteristic of mean reversion and Poisson process.
Fourth, the fundamental thought of martingale method is how to elect a suitable stochastic discounted factor, such that the price of other derivatives divided by discounted factor equals the relative price. According to representation of martingale method of zero-coupon bond, the paper discusses the consistency of term structure of interest rate and how to use them to solve the financing and policy decision problem in enterprise.
第一，介绍利率期限结构的有关概念、定义和相关的金融背景。
第二，传统的利率期限结构模型主要集中于研究收益率曲线的形状及其形成的原因。本文介绍了传统利率期限结构的几种理论，并对各种理论的特点做了比较。现代利率期限结构的研究与利率衍生证券的定价是密不可分的。因此，本文对常用离散时间和连续时间的利率期限结构模型分别进行了分析，并结合债券的定价问题，得出了各模型的优缺点。
第三，新型的利率期限结构模型的目的是将依赖于状态变量数的短期利率动态化所造成的不确定的多源因素包涵到模型中，用现代数学的方法对利率行为加以研究。本文在Chan，Karolyi，Longstaff and Sanders等人提出的连续时间模型统一框架的基础上，讨论了一系列水平模型的特点，这些模型显示了利率变动的条件均值和条件方差与利率水平之间的关系。利率的结构转换模型是对Vasicek模型和CIR模型的推广。本文对这三种模型进行了分析和比较，利用中国金融市场上银行间同业拆借利率的数据，对加入结构转换变量的GARCH利率期限结构模型做了实证分析，发现利率结构转换模型更适合中国金融市场上利率行为的特点。随机跳跃过程模型考虑了金融市场上稀有事件对股票价格、期权定价等问题的影响，将利率的动态过程分为连续部分和跳跃部分，用Brown运动来描述金融资产的价格过程是连续的，而用跳跃部分来描述不可预测的随机事件对这种连续性的破坏。本文从理论上探讨了用Poisson过程和Levy过程描述跳跃部分的利率动态模型，并和扩散过程下的模型作了比较，得到随机跳跃过程模型只是多了一个跳跃项，且这一模型是由一个具有均值回复特性的Ornstein-Uhlenbek过程和Poisson过程混合而成的。
第四，鞅方法的基本思想是先选取适当的随机贴现因子，使利率衍生产品
的价格除以贴现因子后得到相对价格，而这一相对价格过程在风险中性测度尸’
下是一个鞍过程.本文从零息债券的鞍表示法出发，探讨了在确定短期利率的模
型后，偏微分方程法(尸Z)E法)和教方法结果的一致性，但鞍方法更直观，更
容易求解，也更便于应用.从而得到了一个更一般化的结论，它是对Hull一White
模型的推广.并利用其结论从理论上讨论了如何将利率期限结构模型用于企业的
融资决策问题.
Finance mathematics is a new bi-disciplinary science, and is a intersection between mathematics and finance. Their key problems are the choice theory of optimal strategy for investment and the pricing theory of assets. And short interest rate has a determinative importance to the pricing of financial assets and risk managements. So term structure models of interest rate and characteristics of the behavior of interest rate are always a focal point to financial researches. In this paper, recalling the development process about the term structure of interest rate in twenty years, the author wants to get a clear sequence of ideas in all kinds of complicate mathematics models, and gives a brief summary to the major researches achievements in this field. Mainly there are the following contents:
First, introduce some conceptions, definitions and relational backgrounds in finance.
Second, traditional term structure models of interest rate focus on what the yield curve's shape is and how it forms. The article introduce four theories and compares their characteristics. The research on the modern term structure of interest rate is relative to the pricing of derivatives. Therefore, the paper analyses these models which are often used discrete time and continuous time, combining the problem about the pricing of bond, and gets their advantages and shortcomings in theory.
Third, new term structure models of interest rate include some uncertain multi-factors, which the dynamic state of short interest rate depends on the variable about state, and researching the behavior of interest rate by modern mathematical methods. In order to describe the random behavior of interest rate, the essay discusses a series of characteristics about level models based on Chan etc. These models show the relationship between conditional mean and variance of the volatility of interest rate and the level of interest rate. The regime-switching model about interest rate extends Vasicek and CIR models. The essay compares them and finds that the former is more suitable to the behavior of interest rate in China financial market than others. The stochastic process with jumps considers those rare events which have affect on stock price, the pricing of option and so on, the dynamic state of interest rate is
divided into continuous and discontinuous (jumP) processes. The article discusses the dynamic model of interest rate with Poisson process and Levy process in theory, and contrasting with those models under diffusion process. The article gets the former only adds a part with jump, and the model is made up of Ornstein - Uhlenbek process with the characteristic of mean reversion and Poisson process.
Fourth, the fundamental thought of martingale method is how to elect a suitable stochastic discounted factor, such that the price of other derivatives divided by discounted factor equals the relative price. According to representation of martingale method of zero-coupon bond, the paper discusses the consistency of term structure of interest rate and how to use them to solve the financing and policy decision problem in enterprise.
引文
[1] Vasicek .O. An Equilibrium Characterization of the Term structure [J].Journal of Financial Economics, 1977, 5:177-188
[2] Cox, J.C., Ingersoll, J.E.and Ross, S.A. A Theory of the Term Structure of Interest Rates [J].Econometrica, 1985, 53:385-407
[3] Hull.J., White, A. Pricing interest rate derivative securities [J], ReviewofFinancial studies, 1990, 3:573-592
[4] Ho T S Y, S B Lee.Term structure Movements and Pricing Interest Rate Contingent Claims [J].Journal of Financial, 1986, (41):1011-1029
[5] Black F., Derman E., Toyw(1990) A one factor model of interest rates and its application to Tresury bond option, Financial Annals Vol.46 p33-49
[6] Hull J. White.A(1993)One factor interest rate models and the valuation of interest rate derivate securities, Journal of Financial and Quantitative Analysis, Vol.28 No.2 June
[7] Heath D., Jarrow R., Morton A.(1992)Bond pricing and the term structure of interest rates:a new methodology for contingent claims valuation, Econometrica Vol.60
[8] Merton R.C.(1973)Theory of rational option pricing, Bell Journal of Economics and mangement Science, Vol.4
[9] Brennan M.J.Schwartz E.S(1979)A cotinous time approach to the pricing of bonds, Journal of Banking and Finance , Vol.3
[10] Schaefer S.M. Schwartz E.S(1987)Time-dependent variance and the pricing of bond options, Journal of Finance Vol.42
[11] 郑小迎等，关于利率衍生工具定价的研究，系统工程学报，2001年第3期，172-175
[12] 杨智元，利率衍生证券定价研究评述，集美大学学报(哲学社会科学版)，2001年第1期，62-66
[13] 郑小迎，陈军，陈金贤，可转换债券定价模型探讨，系统工程理论与实践，2000年第8期，24-28
[14] Liu Jianhua，Yah Xiaobin，Discussions on Term Structure of Interest Rate，长沙电力学院学报，1998年第2期，136-139
[15] 李楠，利率期限结构及利率风险控制，2001年硕士生论文，华侨大学
[16] 吴恒煜，陈金贤，利率期限结构理论研究，西安交通大学学报，2001年增刊，19-22
[17] 吴林祥，利率期限结构理论的最新发展，国外经济与管理，1998年第2期，20-22
[18] 孙传忠等，ARCH模型及其应用与发展，数理统计与应用概率，1995年第4期，62-70
[19] 张汉江等，金融市场预测决策的有利工具：ARCH模型，系统工程，1997年第1期
[20] 丁华，股价指数波动中的ARCH现象，数量经济技术经济研究，1999年第9期
[21] 王军波，邓述慧，利率、成交量对股价波动的影响-GARCH修正模型的应用，系统工程理论与实践，1999年第9期
[22] 吴其明等，自回归条件异方差(ARCH)模型及应用，预测，1998年第4期，47-48，54
[23] 魏巍贤，周晓明，中国股票市场波动的非线性GARCH预测模型，预测，1999年第5期
[24] 吴长风，利用回归—GARCH模型对我国沪深股市的分析，预测，1999年第4期
[25] 李汉东，张世英，自回归条件异方差的持续性研究，预测，2000年第1期，51-54
[26] Longstaff F.Schwarta E(1992)Interest rate volatility and the term structure:a two factor genernal equilibrium model, Journal of Finance Vol 47
[27] Chen L.(1996)Stochastic mean and stochastic volatility: a three factor model of the term structure of Interest rates and its application of the pricing of interest rate derivatives Blackwell publishers
[28] Chan K C, Karolyi G A, Longstaff F A, etal.An empirical omparison of alternative models of the short-term interest rates[J].Journal of Finance, 1992, 47:1209-1227
[29] Engle R F, Ng V.Time-varing volatility and the dynamic behavior of the term structure [J].Journal of Models, Credit and Banking, 1993, 25:336-349
[30] Brenner R J, Harjies R H, Kroner K F.Another look at models of the short-term interest rate [J].Journal of Financial and Quantitative Analysis, 1996, 31:85-107
[31] Engle R.J., Hg, V.and Rothschild, M.Asset pricing with a Factor ARCH
Covariance structure: Empirical Estinates for Tresasury Bills[J]. Journal of Economentrics, 1990, 45:213-238
[32] Koedijk.K, G., Nissen F., Schotman, P. C and Wolff, C.The Dynamics of short-Term Interest Rate Volatility Reconsidered [J].European Finance Review, 1997, 1:105-130
[33] Nelson, D.B Conditional Hetroskedsticity in Asset Returns:A New Approach [J].Econometrica, 1991, 59:347-370
[34] 王宇，中国货币政策调控机制发生历史性变革[J]．国研信息，2001，3：20
[35] 吴雄伟，利率期限结构模型的估计及其在利率行为描述中的应用研究，2001年硕士生论文，湖南大学
[36] Dalquist M.On alternative interest rate process [J].Journal of Banking and Finance, 1996, 20:1093-1119
[37] 于瑾，利率期限结构研究，2002年博士生论文，对外经济贸易大学
[38] (美)约翰·出版社，2000．1
[39] Duffle D.Dynamic asset theory (2nd ed)[M].Princeton, New Jersey:Princeton University press, 1996.
[40] 陈典发，利率期限结构的一致性，系统工程，2002年第1期：17-19
[41] 郑振龙，林海，中国市场利率期限结构的静态估计，武汉金融，2003年第3期：33-36
[42] 叶中行，林建忠编著，数理金融—资产定价与金融决策理论，北京：科学出版社，2000．7
[43] 姜礼尚著，期权定价的数学模型和方法，北京：高等教育出版社，2003．1
[44] 王小群，金融数学介绍，系统工程，1999年第17卷第6期，1-6，11
[45] Chapman & Hall, Introduction au calcul stochastique appliqué à la finance, 1996.
[46] 武宝亭，李庆士，杨跃武编著，随机过程与随机微分方程，电子科技大学出版社，1994．2
[47] (美)蒋中一著，王永宏译，动态最优化基础，北京：商务印书馆，1999．11
[2] Cox, J.C., Ingersoll, J.E.and Ross, S.A. A Theory of the Term Structure of Interest Rates [J].Econometrica, 1985, 53:385-407
[3] Hull.J., White, A. Pricing interest rate derivative securities [J], ReviewofFinancial studies, 1990, 3:573-592
[4] Ho T S Y, S B Lee.Term structure Movements and Pricing Interest Rate Contingent Claims [J].Journal of Financial, 1986, (41):1011-1029
[5] Black F., Derman E., Toyw(1990) A one factor model of interest rates and its application to Tresury bond option, Financial Annals Vol.46 p33-49
[6] Hull J. White.A(1993)One factor interest rate models and the valuation of interest rate derivate securities, Journal of Financial and Quantitative Analysis, Vol.28 No.2 June
[7] Heath D., Jarrow R., Morton A.(1992)Bond pricing and the term structure of interest rates:a new methodology for contingent claims valuation, Econometrica Vol.60
[8] Merton R.C.(1973)Theory of rational option pricing, Bell Journal of Economics and mangement Science, Vol.4
[9] Brennan M.J.Schwartz E.S(1979)A cotinous time approach to the pricing of bonds, Journal of Banking and Finance , Vol.3
[10] Schaefer S.M. Schwartz E.S(1987)Time-dependent variance and the pricing of bond options, Journal of Finance Vol.42
[11] 郑小迎等，关于利率衍生工具定价的研究，系统工程学报，2001年第3期，172-175
[12] 杨智元，利率衍生证券定价研究评述，集美大学学报(哲学社会科学版)，2001年第1期，62-66
[13] 郑小迎，陈军，陈金贤，可转换债券定价模型探讨，系统工程理论与实践，2000年第8期，24-28
[14] Liu Jianhua，Yah Xiaobin，Discussions on Term Structure of Interest Rate，长沙电力学院学报，1998年第2期，136-139
[15] 李楠，利率期限结构及利率风险控制，2001年硕士生论文，华侨大学
[16] 吴恒煜，陈金贤，利率期限结构理论研究，西安交通大学学报，2001年增刊，19-22
[17] 吴林祥，利率期限结构理论的最新发展，国外经济与管理，1998年第2期，20-22
[18] 孙传忠等，ARCH模型及其应用与发展，数理统计与应用概率，1995年第4期，62-70
[19] 张汉江等，金融市场预测决策的有利工具：ARCH模型，系统工程，1997年第1期
[20] 丁华，股价指数波动中的ARCH现象，数量经济技术经济研究，1999年第9期
[21] 王军波，邓述慧，利率、成交量对股价波动的影响-GARCH修正模型的应用，系统工程理论与实践，1999年第9期
[22] 吴其明等，自回归条件异方差(ARCH)模型及应用，预测，1998年第4期，47-48，54
[23] 魏巍贤，周晓明，中国股票市场波动的非线性GARCH预测模型，预测，1999年第5期
[24] 吴长风，利用回归—GARCH模型对我国沪深股市的分析，预测，1999年第4期
[25] 李汉东，张世英，自回归条件异方差的持续性研究，预测，2000年第1期，51-54
[26] Longstaff F.Schwarta E(1992)Interest rate volatility and the term structure:a two factor genernal equilibrium model, Journal of Finance Vol 47
[27] Chen L.(1996)Stochastic mean and stochastic volatility: a three factor model of the term structure of Interest rates and its application of the pricing of interest rate derivatives Blackwell publishers
[28] Chan K C, Karolyi G A, Longstaff F A, etal.An empirical omparison of alternative models of the short-term interest rates[J].Journal of Finance, 1992, 47:1209-1227
[29] Engle R F, Ng V.Time-varing volatility and the dynamic behavior of the term structure [J].Journal of Models, Credit and Banking, 1993, 25:336-349
[30] Brenner R J, Harjies R H, Kroner K F.Another look at models of the short-term interest rate [J].Journal of Financial and Quantitative Analysis, 1996, 31:85-107
[31] Engle R.J., Hg, V.and Rothschild, M.Asset pricing with a Factor ARCH
Covariance structure: Empirical Estinates for Tresasury Bills[J]. Journal of Economentrics, 1990, 45:213-238
[32] Koedijk.K, G., Nissen F., Schotman, P. C and Wolff, C.The Dynamics of short-Term Interest Rate Volatility Reconsidered [J].European Finance Review, 1997, 1:105-130
[33] Nelson, D.B Conditional Hetroskedsticity in Asset Returns:A New Approach [J].Econometrica, 1991, 59:347-370
[34] 王宇，中国货币政策调控机制发生历史性变革[J]．国研信息，2001，3：20
[35] 吴雄伟，利率期限结构模型的估计及其在利率行为描述中的应用研究，2001年硕士生论文，湖南大学
[36] Dalquist M.On alternative interest rate process [J].Journal of Banking and Finance, 1996, 20:1093-1119
[37] 于瑾，利率期限结构研究，2002年博士生论文，对外经济贸易大学
[38] (美)约翰·出版社，2000．1
[39] Duffle D.Dynamic asset theory (2nd ed)[M].Princeton, New Jersey:Princeton University press, 1996.
[40] 陈典发，利率期限结构的一致性，系统工程，2002年第1期：17-19
[41] 郑振龙，林海，中国市场利率期限结构的静态估计，武汉金融，2003年第3期：33-36
[42] 叶中行，林建忠编著，数理金融—资产定价与金融决策理论，北京：科学出版社，2000．7
[43] 姜礼尚著，期权定价的数学模型和方法，北京：高等教育出版社，2003．1
[44] 王小群，金融数学介绍，系统工程，1999年第17卷第6期，1-6，11
[45] Chapman & Hall, Introduction au calcul stochastique appliqué à la finance, 1996.
[46] 武宝亭，李庆士，杨跃武编著，随机过程与随机微分方程，电子科技大学出版社，1994．2
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