(A) study on stuchastic interest rate models

Abstract

We study some stochastic models for interest rates. We classify models into spot rate models and forward rate models. We give some examples such as Rendleman-Bartter model, Vasicek model, Cox-Ingersoll-Ross(CIR) model, and Heath-Jarrow-Morton(HJM) model. Also, we propose an interest rate model of the form $dR(t) = (\alpha_{-1}R^{-1}(t) - \alpha_0- \alpha_2 R^2(t))dt + \sigma R^{\gamma}(t)dW(t),$ where $\alpha_{-1}, \alpha_0, \alpha_2, \sigma >0,\gamma>1$ and $W(t)$ is a Brownian motion. This model is a slight modification of the model proposed by Ait-Sahalia [1] in 1992. In this model, it is non-trivial to show that there exist some analytic properties such as the existence and the uniqueness of its solution, boundedness of moments, and convergence of a numerical solution. We verify that our new model satisfies these analytic properties by modifying the proofs in papers of Wu et al. [11] and Cheng [2] in 2008.