Discretization of deflated bond prices

This paper proposes and analyzes discrete-time approximations to a class of diffusions, with an emphasis on preserving certain important features of t he continuous-time processes in the approximations. We start with multivari ate diffusions having three features in particular: they are martingales, e ach of their components evolves within the unit interval, and the component s are almost surely ordered. In the models of the term structure of interes t rates that motivate our investigation, these properties have the importan t implications that the model is arbitrage-free and that interest rates rem ain positive. In practice, numerical work with such models often requires M onte Carlo simulation and thus entails replacing the original continuous-ti me model with a discrete-time approximation. It is desirable that the appro ximating processes preserve the three features of the original model just n oted, though standard discretization methods do not. We introduce new discr etizations based on first applying nonlinear transformations from the unit interval to the real line (in particular, the inverse normal and inverse le git), then using an Euler discretization, and finally applying a small adju stment to the drift in the Euler scheme. We verify that these methods enfor ce important features in the discretization with no loss in the order of co nvergence tweak or strong). Numerical results suggest that these methods ca n also yield a better approximation to the law of the continuous-time proce ss than does a more standard discretization.