On Term Structure of Yield Rates. 4. The Duffie – Kan Two Factor Model

Interest rate models in which the short-term rate is a unique state variable are usually considered. These models are attractive that analytical decisions often give the chance to receive and provide concerning the simple computing analysis. However one-factor models have certain lacks. Basic of them consists that all term structure determines by only the unique value of the short-term rate fixed at the initial moment of construction of term structure. And it is represented unreasonable from the economic point of view. To avoid this lack authors suggest to use for modeling of dynamics of the interest rate more than one state variable. At transition from the unique factor to several there should be an improvement of approximation of term structure. In general the price for it is loss of possibility of receiving of analytical decisions, receiving of the equations with partial derivatives with the raised dimension and complication of procedure of receiving of results. In the paper the Duffie-Kan models, describing dynamics of the short-term interest rate in a case when the state of the financial market is characterized not only by level of the interest rate, but also one more other parameter changing in time are investigated. It is considered two cases. In the first in quality of an additional state variable local on time the average value of the short-term interest rate is accepted. In the second case as an additional state variable the instant variance of the interest rate is accepted. Two-factor models are under construction so that they led to affine term structure of yield. The basic attention is given to definition of functions of term structure. As the equations turning out for these functions do not suppose analytical decisions, it is offered to find their approximations. In view of that in real cases the volatility is usually small, the method of small parameter of Poincare is for this purpose used