Consistency Problems for Heath-Jarrow-Morton Interest Rate Models

ISBN-13: 9783540414933 / Angielski / Miękka / 2001 / 138 str.

Damir Filipovic

Opis

Written for readers with knowledge in mathematical finance 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.

Spis treści

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.

Blurb

The book is written for a reader with knowledge in mathematical finance (in particular interest rate theory) and elementary stochastic analysis, such as provided by Revuz and Yor (Continuous Martingales and Brownian Motion, Springer 1991). It gives a short introduction both to interest rate theory and to stochastic equations in infinite dimension. The main topic is the Heath-Jarrow-Morton (HJM) methodology for the modelling of interest rates. Experts in SDE in infinite dimension with interest in applications will find here the rigorous derivation of the popular "Musiela equation" (referred to in the book as HJMM equation). The convenient interpretation of the classical HJM set-up (with all the no-arbitrage considerations) within the semigroup framework of Da Prato and Zabczyk (Stochastic Equations in Infinite Dimensions) is provided. One of the principal objectives of the author is the characterization of finite-dimensional invariant manifolds, an issue that turns out to be vital for applications. Finally, general stochastic viability and invariance results, which can (and hopefully will) be applied directly to other fields, are described.