In this dissertation, I consider a new nonlinear stochastic interest rate model that is adapted from a stochastic population growth model and exhibits the desirable properties of positivity of interest rates and mean reversion. We show that in the constant parameter case this model falls within the paradigm of the Rogers approach for generating positive interest rate models.

Moreover, motivated by a procedure initiated by Hull and White, we also

oer a variant of the model with a time-dependent parameter that allows calibration of the model to a specified initial term structure when a trinomial-tree method is implemented to obtain discrete approximations of the distributions of the interest-rate process. Although nonlinear, our model has a closed form solution, which facilitates the generation of sample paths by standard numerical

methods. This allows us to carry out the trinomial-tree method to obtain

approximate distribution of the interest-rate process and compare that result to the approximate distributions obtained by Monte Carlo simulation. We incorporated the positive interest rate to derive the firm's default probability, which thereby extends Qian's work from a linear interest rate model to a non-linear interest rate model. In the research, first comparing to Qian's method, we used the rst passage time method based on the Fortet integral equation to derive the firm's default probability as driven by the Vasicek interest rate model.

As an alternative, we also proposed the coupled trinomial tree method to derive the default probability. With the comparison of the numerical results among the three methods, we successfully extended the coupled trinomial tree algorithm for default probability from the linear model to a nonlinear model and obtained reasonably consistent results.