AbstractsMathematics

Interest rate models are the building blocks of financial market and the interest rate derivatives market is the largest derivatives market in the world. In this dissertation, we shall focus on numerical pricing of interest rate derivatives, estimating model parameters by Kalman filter, and studying various models empirically. We shall propose a front-fixing finite element method to price the American put option under the quadratic term structure framework and compare it with a trinomial tree method and common finite element method. Numerical test results show the superiority of our front-fixing finite element method in the aspects of computing the option and free boundary simultaneously with high accuracy. We shall also employ the Kalman filter and its variant techniques to estimate parameters of the affine term structure models as well as quadratic term structure models. Various comparisons of different Kalman filter performance and both the in-sample fit and out-sample fit for Monte Carlo simulations as well as real treasury yield data are presented. In addition, we shall propose a general one-factor interest rate model and apply a homotopy perturbation method to valuate bond prices. One of the attractive qualities of the approximated solution of homotopy perturbation method is its fast speed of achieving the same accuracy compared to the tree method.