On the valuation of interest rate products under multi-factor HJM term-structures
In the present paper the valuation of interest rate products with effected cash flow under a multifactor Heath-Jarrow-Morton model of the term-structure interest rates by hierarchical approximation is considered. In the Introduction a short presentation of facts on general infinite dimensional multifactor models of the terms-structure of interest rates is made. In Section 2.1 the mathematical formulation of the valuation problem by infinite dimensional stochastic differential equations is considered. The conditions on the volatility and the drift are given. The market completeness conditions are presented. In Section 2.2 the forward rate curve is presented in terms of its Fourier-Legendre expansion. The finite dimensional process, associated with the curve, is developed. In Section 2.3 the infinite dimensional valuation problem is approximated by the finite dimensional analogue. In Section 2.4 the penalty technique is applied to regularize the valuation problem. Here the regularized variational inequality is approximated by a non-linear partial differential equation through the use of the penalization technique. In Section 3 the method-of-lines finite element method is presented and the lower-level approximation of the higher-level spectral expansion is constituted. The projection of the forward rates dynamics onto the finite span of the first \((p+1)\)-Legendre polynomials is considered. In Section 3.1 the problem of the penalized equation is formulated as a variational initial-value problem. In Section 3.2 the spatial semi-discretization is introduced in terms of the finite element basis. In Section 3.3 an adaptive extrapolation method is developed from the full-discretization by the backward Euler method. In Section 4 numerical experiments are detailed and an interest rate cap, and an American put option are valued. In Section 4.1 the initial forward rate curve and the factor loadings for a three-shock model are specified. The results about factor loadings versus time to maturity are graphically illustrated. In Section 4.2 the forward rate model to the valuation of interest rate caps is applied. In Section 4.3 an American put option on a discount bond is valued. In Section 4.4 a discussion on the methods is presented. In Section 5 a summary and conclusions for the paper are provided.
stochastic partial differential equations
scientific article; zbMATH DE number 5611784
scientific article