# Implementing the Hull-White Trinomial Term Structure Model

## Stuart McCrary

August 31, 2015

Stuart McCrary writes about the Hull-White trinomial interest rate tree, calibrating interest-rate term structure, and unconditional probabilities.

The Hull-White model (HW) is a trinomial tree of forward short rates, where each rate on the tree can jump to three possible states or nodes on the tree one step later. The median rates on the trinomial illustration are on the horizontal row at the center of the tree, but the numerical values for these median rates generally drift upward from left to right as they approximately equal expected forward rates for the same period. Rates are separated from each other vertically by a constant yield spread used throughout the tree. Therefore, all rates on the tree can be defined as a median rate plus or minus a number of increments above or below that median. This is true both for individual nodes and, importantly, the entire tree. Although forward drift will actually twist the tree, it is easy to see the median rates at the center of the symmetric tree.

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