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 the continuous-time processes in the approximations. We start with multivariate diffusions having three features in particular: they are martingales, each of their components evolves within the unit interval, and the components are almost surely ordered. In the models of the term structure of interest rates that motivate our investigation, these properties have the important implications that the model is arbitrage-free and that interest rates remain positive. In practice, numerical work with such models often requires Monte Carlo simulation and thus entails replacing the original continuous-time model with a discrete-time approximation. It is desirable that the approximating processes preserve the three features of the original model just noted, though standard discretization methods do not. We introduce new discretizations based on first applying nonlinear transformations from the unit interval to the real line (in particular, the inverse normal and inverse legit), then using an Euler discretization, and finally applying a small adjustment to the drift in the Euler scheme. We verify that these methods enforce important features in the discretization with no loss in the order of convergence tweak or strong). Numerical results suggest that these methods can also yield a better approximation to the law of the continuous-time process than does a more standard discretization.