| The term structure theory of interest rates has long been an important subject in economics and finance as it is widely used for diverse purposes such as valuing bonds and interest rate derivative securities, managing bond portfolios, and making monetary policies. Since Vasicek (1977) and Cox, Ingersoll, and Ross (1985), numerous studies have introduced dynamic term structure models (DTSMs) in various settings. DTSMs developed so far may be classified into continuous vs discrete time models on the one hand, and equilibrium vs arbitrage models on the other. This paper aims at a review of existing equilibrium DTSMs in discrete time in order to discuss some recent issues and to shed light on the direction of future research in this area. Although, since Vasicek (1977) and Cox, Ingersoll, and Ross (1985), continuous time models seem to have been more popular in both theoretical and empirical studies, a variety of discrete time models have also been developed. While continuous time models may be advantageous for obtaining closed-form solutions of bond prices, discrete time models can be more useful for empirical analyses to reflect the fact that bond prices (or interest rates) in reality are observed at discrete time intervals. The difficulty in estimating continuous time models for practical purposes arises from the facts, in particular, that it is necessary to discretize continuous time stochastic processes of latent state variables, and that it may not be possible to obtain analytical transition probability density functions of the discretized stochastic processes. In this paper, we classify discrete time DTSMs into two categories: one is the ``continuous-time model analog`` in which continuous time stochastic processes are Euler-discretized, and the other is the ``exact discrete-time model`` in which state variables follow discrete time exponential affine processes, of which the Car (compound autoregressive) process is typical. The continuous time model analogs result from approximations of stochastic processes of latent state variables via a simple discretization at the same frequency with other observable variables available. While these discretized processes might appear to have the same form as the corresponding continuous time processes, they do not preserve all the properties of their counterparts. For instance, state variables in discretized processes can be negative even if the Feller condition, which guarantees the non-negativity of continuous time state variables, is met. Moreover, the non-negativity may be obtained in discretized processes although the Feller condition is violated. Also, it may not be possible to obtain the closed-form transition probability density functions of state variables. The exact discrete time models are specified and characterized by a special class of discrete time exponential affine processes called Car processes. These processes fulfill the properties of their continuous time counterparts and make it possible to obtain closed-form transition probability density functions. Thus, the exact discrete time models resolve the problems of the continuous time model analogs. We discuss characteristics and limitations of respective models in each category by comparing with those of their continuous-time counterparts in this vein. The paper is organized as follows. Section 2 reviews some basic knowledge on the dynamic term structure theory of interest rates. Section 3 introduces continuous time model analogs, dividing them into uni-variate vs multivariate models and linear vs nonlinear models. Section 4 introduces the properties and representatives of Car processes that constitute exact discrete-time models. Also, this section surveys related studies put forth recently and compares them with continuous time model analogs. Finally, section 5 summarizes and concludes. |
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