Using Hull - White Interest - Rate Trees

The Hull-White tree-building procedure was first outlined in the Fall 1994 issue of Journal of Derivatives. It is becoming widely used by practitioners. This procedure is appropriate for models where there is some function x = f(r) of the short rate r that follows a meanreverting arithmetic process. It can be used to implement the Ho-Lee model, the HullWhite model, and the Black-Karasinski model. Also, it is a tool that can be used for developing a wide range of new models. In this article we provide more details on the way in which Hull-White trees can be used. We discuss the analytic results available when x = r and make the point that it is important to distinguish between the ∆ t-period rate on the tree and the instantaneous short rate that is used in some of these analytic results. We provide an example of the implementation of the model using market data. We show how the model can be implemented so that it provides an exact fit to the initial volatility environment while at the same time explaining why we do not recommend this approach. We also discuss how to deal with such issues as variable time steps, cash flows that occur between nodes, barrier options, and path dependence.