Investment Theory and Risk Management, + Website

Interest Rate Dynamics

The goal is to derive a model that is consistent with the observed interest rate dynamics and matches the term structure. We will look at the Ho-Lee and the Black-Derman-Toy (BDT) models. Ho and Lee was one of the early models that matched the term structure. I illustrate how this is accomplished in the chapter spreadsheet, using Excel's Solver.

Ho and Lee propose a linear short-rate model , where b is a constant volatility parameter, specifically b = 2σ, where σ is the annual volatility of the short rate. The annual drift rate is given by which, following Luenberger (Chapter 14) is treated as a variable to be solved in such a way that the short-rate lattice constructed is matched to the current term structure. The continuous time counterpart to the Ho and Lee model is given by , where dz is a standard Wiener process. Therefore, the parameter is a drift component, while (b × s) is the volatility factor. In our lattice, s is the state variable that measures the number of up-movements in rates (similar to the binomial lattice). It therefore represents the volatility ...