Analytical implementation of the Ho and Lee model for the short interest rate

Abstract

Ho and Lee introduced the first no-arbitrage model of the evolution of the short interest rate. When expositing the Ho and Lee model, other authors used the method of numerical solutions and forward induction, an approach pioneered by Black, Derman, and Toy for their own model much later. This standard method of implementation is relatively complex and time consuming when applied to scenarios that enable the use of an interest-rate lattice. Under many assumptions, however, the Ho and Lee model will generate an interest-rate tree. Under these circumstances, implementation via numerical methods and forward induction appears to be impractical, if not impossible. In this paper, we show how to implement the model analytically. We demonstrate that it is relatively straightforward to identify at the initial date analytical expressions for all interest rates at all dates. Once these expressions are evaluated, the calculations to obtain interest rates are arithmetic operations. Our recommended method of implementation applies equally effortlessly to interest-rate trees and Monte Carlo simulation.