A Yield - Factor Model of Interest Rates

This paper presents a consistent and arbitrage-free multifactor model of the term structure of interest rates in which yields at selected fixed maturities follow a parametric multivariate Markov diffusion process with " stochastic volatility. " The yield of any zero-coupon bond is taken to be a maturity-dependent affine combination of the selected " basis " set of yields. We provide necessary and sufficient conditions on the stochastic model for this affine representation. We include numerical techniques for solving the model, as wcll as numerical techniques for calculating the prices of term-structure derivative prices. The case of jump diffusions i\ also considered. I. INTRODUCTION This paper defines and analyzes a simple multifactor model of the term structure of interest rates. The factors of the model are the yields X = the current five-year (zero-coupon) yield as a factor. The yield factors form a Markov process, to be described below, that can be thought of as a multivariate version of the single-factor model of Cox, Ingersoll, and Ross (1 98%). As opposed to most multifactor term structure models, the factors (Markov state variables) are observable from the current yield curve and their increments can have an arbitrarily specified correlation matrix. The model includes stochastic volatility factors that are specified linear combinations of yield factors. Discount bond prices at any maturity are given as solutions to Ricatti (ordinary differential) equations, and path-independent derivative prices can be solved by, among other methods, an alternating-direction implicit finite-difference solution of the " usual " partial differential equation (PDE). Fully workcd examples of solutions to these Ricatti equations and PDEs are included. Our yield model is " affine " in the sense that there is, for each maturity t, an affine function Y,: R " + R such that, at any time t , the yield of any zero-coupon bond of maturity t is Y , (X ,). Indeed, ruling out singularities, cssentially any n yields would serve as the factors, and given the imperfections of any model, it is an empirical issue as to which IZ yields will serve best as such. Likewise, because of linearity, the Markov state variables can be taken to be forward rates at given maturities, so that the model can be viewed as a multifactor Markov parameterization of the Heath, Jarrow, and Morton (HJM) (1992) model. In fact, Frachot and Lesne (1993) have extended our model to the HJM setting. One …