Dynamic Models of the Term Structure

In the past 25 years, tremendous progress has been made in modeling the dynamics of the term structure of interest rates, which play an instrumental role in determining prices and hedging portfolios of fixed-income derivative securities. This article reviews the theoretical development of the dynamic models of the default-free term structure and their applications in pricing interest rate options. Classic models, sometimes termed equilibrium models, and their multifactor extensions are outlined. These models provide clear economic intuitions connecting the term structure with economic fundamentals. They also lay a foundation for the framework of the arbitrage models that price interest rate derivatives on the basis of the market prices of bonds. This framework has been expanded and enriched by recent advances in directly modeling observable market rates through the market models and in incorporating an internally consistent correlation structure through the “infinite-dimensional” models. In the past 25 years, tremendous progress has been made in modeling the dynamics of the term structure of interest rates, which play an instrumental role in determining prices and hedging portfolios of fixed-income derivative securities. This article reviews the theoretical development of the dynamic models of the default-free term structure and their applications in pricing interest rate options.Classic models specify a stochastic process for the instantaneous short rate and a functional form for the market price of risk. These models and their multifactor extensions depict an interest rate process that reverts to its long-run mean, which may itself be a stochastic variable. Although the models differ in modeling how the expected changes in interest rates and the volatility of these rate changes are related to the levels of interest rates, they are all supported by a general equilibrium in a specific economy. This condition allows them to provide clear economic intuitions connecting the term structure with economic fundamentals. At the same time, the affine structure (signified by the linearity of the expected changes in interest rates and the variance of the rate changes in the state variables) allows near-analytic formulas for option prices. So, even though the models are not directly used by practitioners, they form the foundation for the development of the arbitrage models that do find widespread applications in pricing interest rate derivatives.The arbitrage models are used to evaluate interest rate derivative securities based on the prevailing market prices of bonds. The methodology is similar in spirit to the Black–Scholes model for pricing equity options. It specifies the dynamics of the evolution of the instantaneous forward rates from the current market forward curve and imposes the no-arbitrage condition to pin down the relationship between the drift and volatility of the forward-rate process. The assumption of a complete bond market (which states that all payoffs in the market can be replicated by combinations of traded securities) allows a practitioner to obtain the derivative prices through arbitrage pricing by replication techniques.The arbitrage pricing framework is very general and embeds many popular models as special cases. However, not all specifications that fit in this framework are useful. One of the desirable features to have is a Markov structure that will maintain a recombining tree for evaluating derivative prices. In addition, the need to ensure that the modeled nominal rates remain positive leads to a positive interest rate approach that is innovative but consistent with the original framework.Recent advances to efficiently implement models of interest rate dynamics and pricing of interest rate derivatives have expanded and enriched the original framework. On the one hand, market models directly describe rates observable in the LIBOR or swap markets by lognormal processes. This specification simplifies the formulation for option prices and justifies the industry practice of using a simple model to price caps or swaptions with guidance on the volatility input. On the other hand, “infinite dimensional” models incorporate an internally consistent correlation structure among the shocks to each point of the forward curve. This approach allows the models to match the forward curve with the market curve at all times while still maintaining a parsimonious parameterization. Further characterization and integration of these models may provide more intuitive and efficient ways to price fixed-income derivative products.