The probability density of the processes of yield interest rates

The processes of short-term interest rates rise to many changes in market indices, as well as form the basis of determining the value of market assets and commercial contracts. A special role they play in calculating the term structure of the yield. Therefore, the development of mathematical models of these processes is extremely interesting for financial analysts and market research issues. There are many versions of change of short-term risk-free interest rates in the framework of the theory of diffusion processes. However, there is still no such a model, which would be the basis for building a term structure of yields close to existing on a real financial market. It is interesting to analyze the existing models in order to clarify their features in a probabilistic sense, in more detail than has been done by their creators and users. To this end, the paper examines the marginal probability density of the diffusion processes generated by sixteen models of short-term interest rates, that allow obtain densities in an analytical form. Here will be made such analysis for the family of models used by the authors of papers that are widely known in their fit to the actual time series of yield. All considered models belong to the class of diffusion that generate processes X(t), described by the equation dX(t) = |X(t)) dt + a(X(t)) dW(t), t > t₀, X(t₀) = Х₀, where a specific setting of drift |(x) and volatility ст(х) defines one or another particular model. Some models, such as the Vasicek model, Cox - Ingersoll - Ross, geometric Brownian motion, Ahn - Gao, are well documented in the literature, but nevertheless their properties are listed here for convenience of comparison with other, less well-known or unstudied models. Other densities are described for the first time. The proposed analysis will be useful to the reader to determine the most appropriate models of short-term rates in the determination of the term structure of zero-coupon yield approximating actually observed, as far as possible, by the best way. Analysis scheme reduces to solution of the direct Kolmogorov equation for the stationary probability density and, if necessary, discuss its features and the first four moments are calculated, usually of interest in practice. It is shown that for the models the coefficients of skewness and kurtosis, defined moments third and fourth order, depend on a single parameter, that called as the form parameter density, which, in turn, is determined only by the ratio of the variance to the square of the expectation (this corresponds to the square of the so-called coefficient of variation).

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