一类高度非线性即期利率模型及其E-M近似解

发布时间：2018-06-06 15:52

本文选题：即期利率 + 随机微分方程　；参考：《华中科技大学》2012年硕士论文

【摘要】：本文对即期利率的基本含义、研究的重要性、发展历程及其取得的重要成果进行了概述，详细介绍了利用随机微分方程建立的利率模型。其中的众多模型是线性的形式，具有良好的性质，满足线性增长条件和(局部)Lipschitz条件，，方程满足解的存在唯一性条件。虽然有些方程并非满足线性增长条件或(局部)Lipschitz条件，如CIR模型，但是其非负解的存在唯一性等解析性质也得到了证明。Wu还证明了均值回归-过程的非负解的存在唯一性、有界性、EM数值解的收敛性。另外，也有非线性的形式，比较有代表性的就是Ait-Sahalia模型，方程的漂移系数与扩散系数均为非线性，该模型的有关解析性质至今也得到了较好的证明。在CIR模型、均值回归-过程的基础上，本文建立了一个更一般化的高度非线性随机微分方程形式的即期利率模型：显然，这个方程是包含CIR模型、均值回归-γ过程的。在此模型的基础上，本文主要做了以下两方面的工作：第一，选取美国联邦储备系统公布的2002年1月2日至2012年2月17日的短期国债收益率估计模型中的参数，估计结果显示参数α㧐1，β㧐1。第二，在上述估计的参数结果下，证明了方程的非负解的存在唯一性、解的随机有界性，还特别指出了方程的EM近似解在时间步长充分小时是依概率1收敛于方程的真解，收敛的结果还说明了基于EM方法的蒙特卡洛模拟可以用来计算期权等金融产品的预期收益问题。
[Abstract]:In this paper, the basic meaning of spot interest rate, the importance of research, the development of spot interest rate and its important achievements are summarized, and the interest rate model based on stochastic differential equation is introduced in detail. Many of the models are linear, have good properties, satisfy the linear growth condition and (local Lipschitz condition), the equation satisfies the existence and uniqueness conditions of solution. Although some equations do not satisfy the linear growth condition or (local Lipschitz condition, such as CIR model), the existence and uniqueness of the nonnegative solution are also proved. Wu also proves the existence and uniqueness of the non-negative solution of the mean regression-process. Boundedness and convergence of EM numerical solutions. In addition, there is also a nonlinear form. The Ait-Sahalia model is more representative. The drift coefficient and diffusion coefficient of the equation are both nonlinear. The analytical properties of the model have been well proved. On the basis of mean regression-process, a more general model of spot interest rate in the form of highly nonlinear stochastic differential equation is established: obviously, this equation includes CIR model, mean regression-纬 process. On the basis of this model, this paper mainly does the following two aspects of work: first, select the parameters of the short-term Treasury bond yield estimation model published by the United States Federal Reserve system from January 2, 2002 to February 17, 2012. The estimation results show that the parameters 伪 1, 尾 1. Secondly, the existence and uniqueness of the nonnegative solution of the equation and the stochastic boundedness of the solution are proved under the above estimated parameter results. In particular, it is pointed out that the EM approximate solution of the equation converges to the true solution of the equation according to probability 1 when the sufficient time step is small. The convergence results also show that Monte Carlo simulation based on EM method can be used to calculate the expected returns of financial products such as options.
【学位授予单位】：华中科技大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F224;F820

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