Approximating Diffusions with Trees

Abstract

In this appendix, we show how to approximate a diffusion process with a tree. The general procedure we outline is used throughout the book in the tree construction for both one-factor and two-factor short-rate models. In the one-factor case, the tree is constructed by imposing that the conditional local mean and variances at each node are equal to those of the basic continuous-time process. The geometry of the tree is then designed so as to ensure the positivity of all branching probabilities. In the two-factor case, instead, we first construct the trees for the two factors along the procedure that applies to one-factor diffusions. We then construct a two-dimensional tree by imposing that the tree marginal distributions match those of the two factors’ trees and by imposing the correct local correlation structure so as to preserve the positivity of all branching probabilities as well.

The Holy One directed his steps to that blessed Bodhi-tree beneath whose shade he was to accomplish his search. Paul Carus, “The Gospel of Buddha”, 1894.