Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, 2001
Lecture Notes in Mathematics Series, Vol. 1760

Bond markets differ in one fundamental aspect from standard stock markets. While the latter are built up to a finite number of trade assets, the underlying basis of a bond market is the entire term structure of interest rates: an infinite-dimensional variable which is not directly observable. On the empirical side, this necessitates curve-fitting methods for the daily estimation of the term structure. Pricing models, on the other hand, are usually built upon stochastic factors representing the term structure in a finite-dimensional state space. Written for readers with knowledge in mathematical finance (in particular interest rate theory) and elementary stochastic analysis, this research monograph has threefold aims: to bring together estimation methods and factor models for interest rates, to provide appropriate consistency conditions and to explore some important examples.

Introduction. Bond Markets. Forward Curve-Fitting Methods and Factor Models. The HJM Methodology. Invariant Manifolds. Outline. Remark on Notation.-
Stochastic Equations in Infinite Dimension. Infinite- Dimensional Brownian Motion. The Stochastic Integral. Fundamental Tools. Ito's Formula. The Stochastic Fubini Theorem. Girsanov's Theorem. Stochastic Equations. Mild, Weak and Strong Solutions. Existence and Uniqueness.-
Consistent State Space Processes. Ito Process Factor Models. Exponential-Polynomial Families. Auxiliary Results. The Case BEP (1,n). The General Case BEP (K,n). The Diffusion Case. Applications. The Nelson-Siegel Family. The Svensson Family. Conclusions.-
The HJM Methodology Revisited. Term Structure Movements. The Musiela Parametrization. Arbitrage-free Term Structure Movements. Contingent Claim Valuation. When is Z (.,T) a True Q-Martingale? The Forward Measure. Forward LIBOR Rates. Caps. What is a Model?-
The Forward Curve Spaces H w. Definition of H w. Volatility Specification. The Yield Curve. Local State Dependent Volatility. Functional Dependent Volatility. The BGM Model.- Invariant Manifolds for Stochastic Equations. Finite-Dimensional Submanifolds in Banach Spaces. Invariant Manifolds. Proof of Theorems 6.2.1-6.2.4. Consistency Conditions in Local Coordinates.-
Consistent HJM Models. Consistency Problems. A Simple Regularity Criterion for G. Regular Exponential-Polynomial Families. The Nelson-Siegel Family. The Regular Svensson Family. Affine Term Structure. The Cox-Ingersoll-Ross (CIR) Model. The Vasicek Model.-
Appendix: A Summary of Conditions. Axioms for the Forward Curve Space. Conditions on the Forward Curve Movements. Conditions for HJM Models. Assumptions for Characterizing Invariant Manifolds.