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 because the analytical decisions can be often got and a simple computing analysis can be done. However one-factor models have certain lacks. Basic of them consists in fact that all the term structure is determined by only the unique value of the short-term rate which is fixed at the initial moment of construction of term structure. And it seems to be 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 the loss of possibility of getting of analytical decisions, obtaining of the equations with partial derivatives with the raised dimension and difficulties in procedure of results obtaining. 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 by other parameter changing in time are investigated. There are considered two cases. In the first as an additional state variable is accepted local on time average value of the short-term interest rate. In the second case as an additional state variable the instant variance of the interest rate is accepted. Two-factor models are constructed 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 for these functions do not suppose analytical decisions, it is proposed to find their approximations. In view of that in real cases the volatility is usually small, the method of small parameter of Poincare is used for this purpose.

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