THIS CHAPTER REVIEWS models of the term structure of interest rates that are used for the pricing and hedging of fixed-income securities, those whose future payoffs are contingent on future interest rates. Termstructure modeling is one of the most active and sophisticated areas of application of financial theory to everyday business problems, ranging from managing the risk of a bond portfolio to the design and pricing of collateralized mortgage obligations.
Included in this chapter are such standard examples as the Merton, Ho-Lee, Dothan, Brennan-Schwartz, Vasicek, Black-Derman-Toy, BlackKarasinski, and Cox-Ingersoll-Ross models, and variations of these “singlefactor” term-structure models, so named because they treat the entire term structure of interest rates at any time as a function of a single state variable, the short rate of interest. We will also review multifactor models, including multifactor affine models, extending the Cox-Ingersoll-Ross and Vasicek models.
All of the named single-factor and multifactor models can be viewed in terms of marginal forward rates rather than directly in terms of interest rates, within the Heath-Jarrow-Morton (HJM) term-structure framework. The HJM framework allows, under technical conditions, any initial term structure of forward interest rates and any process for the conditional volatilities and correlations of these forward rates.
Numerical tractability is essential for practical applications. The “calibration” of model parameters and the pricing of term-structure derivatives are typically done by such numerical methods as “binomial trees” (Chapter 3), Fourier transform methods (Chapter 8), Monte Carlo simulation (Chapter 12), and finite-difference solution of PDEs (Chapter 12).
This chapter makes little direct use of the pricing theory developed in Chapter 6 beyond the basic idea of an equivalent martingale measure,