The Application Of Kalman Filter In Term Structure Of Interest Rate

Under the condition of interest rates'liberalization, treasury bonds'term structure of interest rates is the basis of the pricing of all financial products, including stock,bonds and related derivatives. It is also one of the most important tools for investment analysis and management,interest rate risk management and monetary policy making and analysis. So the research of this field is of important theoretical and practical significance.Firstly, this thesis introduces some basic concepts of treasury bonds'yield and gives some reasons of using spot rates during constructing the models. And then, we introduce the research evolution of latent factors models, including Nelson-Siegel model and Affine model, and macro factors models in term structure of interest rates. Secondly, we describe the main work and innovation of this thesis. Finally, we introduce the main method of this thesis: Kalman filter.The main part of this thesis is the application of Kalman filter with VAR in term structure of interest rates. We use this method into applications at two different angles. First, in Nelson-Siegel model, we do unit root test and cointegration test aboutβ₀,β₁ ,β₂ series. And then we know that we can make a Vector Auto Regression Model(VAR) aboutβ₀,β₁ ,β₂ series. We estimate treasury bonds'spot rates curves in shanghai security exchange using Kalman filter with VAR. We compare the result of Gauss-Newton method with the result of Kalman filter with VAR. We find that the result of Kalman filter with VAR is better. Second, in macro factors model, we use least squares method and Kalman filter with VAR to estimate macro factors model. At the end, we compare the residual sum of squares of least squares method, 17×10^-4, with the residual sum of squares of Kalman filter with VAR, 2.477×10^-4. We find that the result of Kalman filter with VAR is obviously better than the result of least squares method and the error has been diminished about 85.34%.Next, we do unit root test about interest rates series and find that first order difference of every series will tend to be stable. We use variation of interest rates as the object of principal component analysis and know that two factors dynamic model of interest rates can describe the variation of treasury bonds'spot rate very well. We introduce the definition of Affine model and related assumptions. About Gaussian Essential Affine Model, we apply Ito lemma and equilibrium pricing theory to stochastic differential equations. First, we transform stochastic differential equations into partial differential equations. Second, we transform partial differential equations into ordinary differential equations. Then, we get an equation of spot rates. Through solving conditional mean and conditional variance of state variable in stochastic differential equations, we get a state equation. And then, we get a state-space model. So we can use Kalman filter to estimate state variable. In the end, we conclude the innovations and limitations of this thesis.