On quadratic models of yield in risk-neutral world

Instead of "usual used" quadratic models of the interest rate it is possible to obtain the equivalent description of the interest rate through process X (t) unobserved (latent) state variables by relations dX(t) = K(0-X(t)) dt + S dW(t), t > t_Q, X(t_Q) = X_Q, r(t) = r_mln + X^^X^), X(t) e R, r e R, Ф e R, Where K, 0, S, X_Q are corresponding constant vectors and matrixes, and Ф is a diagonal matrix on which diagonal positive eigenvalues of initial matrix of "usual used" model are located. The condition of absence of arbitration leads to the equation in partial derivatives for the price of zero coupon bond which is obtained in form P(r(X), x) = = exp[- XA(x)X - XB(x) - C(x)]. For functions of term structure A(x), B(x) and C(x) there is a system of the nonlinear ordinary differential equations which generally in an explicit analytical form is unsolved. In paper assumptions are made: 1) process of latent variables X(t) is normal process with a stationary expectation 0 = 0. 2) probability properties of the interest rate r(X) submit to a risk neutral probability measure. At these assumptions it is found out that function B (x) = 0 for all values x. 3) the latent variables constituting vector X(t), are independent stochastic processes. At these assumptions the functions of term structure A(x) and C(x) are obtained in the analytical form and the yield curve Y(x) and the for ·Jk + > k, 1 < i < n: ward rate curve F(x) are derived in an explicit form for z_t = ф_iX > 0, v_t i k 1 E - t!* n f ·+E i=1 2z, ch(v, x) +-- sh(v, x) - + ln - k, x Y(x) = rmin + v V 1 i z v, cth(v,x) + k, 2 , 2 v, - k, 1 F(x) = r (V ch(vx) + k, sh(v,x)) 2 v, cth(vx) + k, Both curves Y(t) and F(t) at change т from 0 to т от 0 до ю, starting from the fixed value of the interest rate r, determined by current « v _- k values of state variables X(t), converge to a general limit y (ю) = r_mi_n + ^ --- that is independent on current state X(t) and is de- i=1 pending only from model parameters. As seen, in quadratic models the current interest rate r at yield curve deriving in an explicit form is not used, instead of it values of latent variables X which at known parameters models uniquely determine the interest rate r are used. However in quadratic models the same value of the interest rate r can be derived for some set of various state variables X. In paper it is found out, haw is the family of yield curve at a fixed interest rate determined by a variety of state variables, corresponding to this fixed interest rate. Besides, the width of a strip in which lie all possible yield curves (or forward curves), corresponding to various starting vectors X and to matrixes Ф, that determine the rate r, is found. All received analytical results are illustrated by a numerical example.

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