Robust binomial lattices for univariate and multivariate applications: choosing probabilities to match local densities

A wide variety of diffusions used in financial economics are mean-reverting and many have state- and time-dependent volatilities. For processes with the latter property, a transformation along the lines suggested by Nelson and Ramaswamey can be used to give a diffusion with constant volatility and thus a computationally simple binomial lattice. Drift terms in mean-reverting and transformed processes frequently result in either ill-defined probabilities or complex grids. We develop closed-form, legitimate probabilities on a simple grid for univariate and multivariate lattices for well-posed smooth diffusions. The probabilities are based on conditional normal density functions with parameters determined by the diffusion. We demonstrate convergence in distribution under mild restrictions and provide numerical comparisons with other univariate and multivariate approaches.