Heath–jarrow–morton Interest Rate Dynamics and Approximately Consistent Forward Rate Curves

Abstract

We study a finite-dimensional approach to the Heath-Jarrow-Morton model for interest rate and introduce a notion of approximate consistency for a family of functions in a deterministic and stochastic framework. This amounts to asking the decrease of the minimum distance in least squares sense. We start from a general linearly parameterized set of functions and extend the theory to a nonlinear Nelson-Siegel family. Necessary and sufficient condition to have approximately consistency are given as well as a criterion of stability for the approximation.

Shape factors and cross-sectional risk

“…Our approach substantially differs from the alternative strategy followed by La Chioma and Piccoli (2007). These authors consider a finite dimensional Musiela's model and let the family of cross-sections vary in a way that the resulting pair is approximately cross-sectionally consistent.…”

“…The term structure of interest rates is identified with a single point in a functional space and then represented as a superposition of fundamental shapes. Several studies have been pursued in this framework, including Brace and Musiela (1994), Musiela (1995), Filipović (2000, Björk and Christensen (1999), Bayraktar et al (2006), and La Chioma and Piccoli (2007). It is worth noting that all these papers work under the somehow simplified assumption of dynamics driven by a standard n-dimensional Wiener process.…”

Approximation and application of the Musiela stochastic PDE in forward rate models

“…To see its convergence to this exact expected value, we computed the average value of E[û (1,5)] in the domain [0, 10] × [0, 10] with the same initial condition u 0 and σ 0 as in the previous section. Here, we set t = τ for efficiency.…”

“…The average plot and the bond pricing band plot are similar to the ones in the previous section. Table 1 provides estimates of the expected value E[û (1,5)] for different grid sizes. The exact value of E[û (1,5)] is 1.525.…”

“…It is considered in [5] for the study of HJM interest rate dynamics. They investigate the problem of approximating infinite dimensional forward rate model by a finite dimensional stochastic system.…”