On term structure of yield rates. 6. The three factor model

Models of Duffie - Kan, describing dynamics of a short-term interest rate in a case when the condition of the financial market is characterized not only level of the most interest rate, but also two more the time variable parameters are investigated. Three versions of expansion of one-factor model to three-factor, allowing to get affine term structure of yield are considered. These versions assume that parameters of one-factor model - level of return of an interest rate and its volatility -are not constants, and diffusion processes. In the first version volatility of process of level of return of an interest rate doesn't depend on the level and is stochastic. In the second version process of level of return of an interest rate is process «with a square root». In the third version volatility of process of level of return of an interest rate doesn't depend on the level and is determined. The main attention is given to properties of the yield curve and the forward curve when dynamics of a short-term interest rate is described by the described three-factor models. With increase in number of factors of model, their analysis essentially become complicated, and receiving of results in an analytical form becomes impossible. The numerical analysis also becomes complicated, as the number of parameters of models grows. Therefore it is not possible to carry out all-round comparison of models, their advantages and lacks in a volume of article. Properties of yields for one set of the parameters found D. An and B. Gao at processing of real financial data is given only. Wider comparison of models should be made in the future. Data on what and how many parameters are used for creation of the considered models are provided. In the interval of time to maturity change from zero to indefinitely the yield curves and the forward curves for all models start from the single points - the current value a spot rate and converge to the corresponding limits depending on parameters of model, but not depending on values of the current level of state variables. These limiting values generally are defined not only model parameters, but also sets of weight factors and parameters of the prices of risk that considerably complicates formulas. However if to consider that the short-term rate of yield of an asset is defined only a spot rate, stochastic processes of a rate and its instant variance are neutral to risk, and the bottom borders for an interest rate and its variance are equal to zero, formulas for calculation of yield strongly become simpler. Their explicit analytical expressions are given at these assumptions. Limiting values of yield can be considered as yield of long-term securities. They don't depend on the current value of state variables, and depend only on model parameters. For the considered numerical example the limiting values of yields decrease with increase in number of factors. More valid conclusions can be made after research of yield in all admissible area of ten-measured space of parameters. Comparative research of mutual behavior of yield curves and forward curves in all intervals of terms to maturities asset in all admissible area of parameters is necessary.

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