The Art of Term Structure Models: Drift
Chapters 7 and 8 show that assumptions about the true and risk-neutral short-term rate processes determine the term structure of interest rates and the prices of fixed income derivatives. The goal of this chapter is to describe the most common building blocks of short-term rate models. Selecting and rearranging these building blocks to create suitable models for the purpose at hand is the art of term structure modeling.
This chapter begins with an extremely simple model with no drift and normally distributed rates. The next sections add and discuss the implications of alternate specifications of the drift: a constant drift, a time-deterministic shift, and a mean-reverting drift.
MODEL 1: NORMALLY DISTRIBUTED RATES AND NO DRIFT
The particularly simple model of this section will be called Model 1. The continuously compounded, instantaneous rate rt is assumed to evolve according to the following equation:
The quantity dr denotes the change in the rate over a small time interval, dt, measured in years; σ denotes the annual basis-point volatility of rate changes; and dw denotes a normally distributed random variable with a mean of zero and a standard deviation of .1
Say, for example, that the current value of the short-term rate ...