On a quadratic model of yield term structure

Within framework of the theory of diffusion processes there are various versions of an evolution of short-term yield interest rates. Nevertheless till now there was no such model which could be a suitable basis for construction of term structure of yield close to existing one on real financial market. The models of interest rates leading to affine term structures of yield are simple, most known and imply a solution in an analytical form. However, reproduction of real term structures by means of affine models are inexact. Recently development of models goes in two directions: increase dimension of models and refusal of affine properties. As representatives of such development are most popular now so-called quadratic models of interest rate processes in which interest rate process r(t) is set by the equations dX(t) = £,(X(t)) dt + a(X(t)) dW(t), t > t₀, X(t₀) = Х₀, r(t) = a + X(t)TX(t), X(t) e R, a e R, T e R. Usually a > 0, T is a symmetrical positive definite matrix. When the vector S,(X) linearly depends onX, and the matrix ст(Х) does not depend on X, process X(t) is Gaussian and in a stationary conditions has, say, expectation ц and a matrix of a covariance V. If T and V are diagonal matrixes T = \\I, V = vI, and ц = 0 the shifted gamma distribution with shift parameter a, scale parameter 1/2v\ and form parameter n/2 will be marginal distribution of process r(t). The shifted gamma distribution characterizes also the short-term interest rate in affine model of Duffie-Kan. Thus, the Duffie-Kan model and quadratic model generate the stochastic processes r(t) with identical distribution. In the paper, the explicit expressions for the term structure of zero-coupon yield to maturity and forward interest rate curve for both models are obtained. Also, the differences between the yield term structures of the models considered in the risk-neutral setting are discussed, when the market price of risk is zero It is shown that if in quadratic model of any dimension n latent state variable X are independent and identical distributed under the normal law with a zero expectation the term structure of interest rates of yield does not depend on concrete values of variables X, and is depended only on starting value r of the current short-term interest rate in the same way as in affine models. Thus long-term limiting rates turn out the same, as in model of Duffie-Kan. Comparative properties of affine Duffie-Kan model and quadratic model of yield are illustrated by a numerical example.

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