이산시간 이자율기간구조모형 (A Review on Discrete Time Term Structure Models of Interest Rates)

서지정보

· 발행기관 : 한국금융학회
· 수록지 정보 : 금융연구 / 26권 / 3호 / 93 ~ 153페이지
· 저자명 : 박정민, 조재호

초록

본 논문은 이산시간 이자율기간구조모형에 관한 연구를 정리․소개하고 그 발전과정을살펴본다. 이자율기간구조에 관한 많은 연구에서 연속시간모형을 제시하고 있지만, 현실에서관측되는 이자율 자료가 이산시간 단위이므로 이를 반영할 수 있는 다양한 이산시간모형에 대한관심 또한 적지 않다. 연속시간모형은 채권가격의 해석해를 구하기는 쉽지만 실증분석에서연속시간 확률과정을 이산화하는 과정을 거쳐야 하는 반면, 이산시간모형은 상태변수의 해석적인전이확률밀도함수를 얻을 수 있는 이점 등으로 인해 현실의 자료를 사용하는 실증분석에 유용할수 있다. 본 연구에서는 이산시간모형을 크게 두 가지 범주로 나눈다. 즉, 연속시간모형을 단순히이산화한 ‘연속시간 상사 이산시간모형(continuous time model analog)’과 이산시간 지수선형과정인 Car 과정에 바탕을 둔 ‘정확한 이산시간모형(exact discrete time model)’으로 구분하여각각의 내용을 요약하고 설명한다. 특히, 연속시간모형과의 비교를 통해 이산시간모형의 특성과한계를 이해함으로써 이자율기간구조이론에서 이산시간모형이 가지는 의미를 조명해 보고자한다.

영어초록

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 multi-variate 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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