The high interest rate volatility, especially since October 6, 1979 [401], calls for stochastic interest rate models. Models are also needed in managing interest rate risks of securities with interest-rate-sensitive cash flows. This chapter investigates stochastic term structure modeling with the binomial interest rate tree [779]. Simple as the model is, it illustrates most of the basic ideas underlying the models to come. The applications are also generic in that the pricing and hedging methodologies can be easily adapted to other models. Although the idea is similar to the one previously used in option pricing, the current task is complicated by two facts. First, the evolution of an entire term structure, not just a single stock price, is to be modeled. Second, interest rates of various maturities cannot evolve arbitrarily or arbitrage profits may result. The multitude of interest rate models is in sharp contrast to the single dominating model of Black and Scholes in option pricing.

Introduction

A stochastic interest rate model performs two tasks. First, it provides a stochastic process that defines future term structures. The ensuing dynamics must also disallow arbitrage profits. Second, the model should be “consistent” with the observed term structure [457]. Merton's work in 1970 marked the starting point of the continuous time methodology to term structure modeling [493, 660].