Consistency problems for Heath-Jarrow-Morton interest rate models.

*(English)*Zbl 1008.91038
Lecture Notes in Mathematics. 1760. Berlin: Springer. viii, 134 p. DM 44.00; öS 322.00; sFr 40.00; £15.00; $ 29.80 (2001).

The Heath-Jarrow-Morton approach to the modeling of the forward curve (the term structure of forward rates) was a significant progress in the pricing of fixed-income products because, basically, it unifies all continuous interest rate models. The forward curve cannot be observed on the market and therefore has to be estimated. The most common curve-fitting methods can be represented as a parametrized family of smooth curves \({\mathcal{G}}=\{ G(. ,z): z \in {\mathcal{Z}}\}\) with some finite dimensional parameter set \({\mathcal{Z}}\).

This monograph is an excellent and thorough review of the current work in the study of consistent (arbitrage free) stochastic models for \(z\). The author presents a systematic study of this consistency problem in a very general framework that includes many known results as well as some new insights and challenging open questions.

The book provides a brief introduction to infinite dimensional stochastic analysis in order to present the Heath-Jarrow-Morton methodology in a functional analytic framework that incorporates an infinite-dimensional driving Brownian motion. The Musiela parametrization leads to a stochastic equation in a Hilbert space \(H\) that describes the arbitrage-free evolution of the forward curve. In this setting, the family \({\mathcal G}\) is just a subset of \(H\) and the consistency considerations are transformed into a stochastic invariance problem. Some general results on stochastic invariance for finite dimensional submanifolds in a Hilbert space are deduced. They are then applied to the classical estimation methods.

This monograph is an excellent and thorough review of the current work in the study of consistent (arbitrage free) stochastic models for \(z\). The author presents a systematic study of this consistency problem in a very general framework that includes many known results as well as some new insights and challenging open questions.

The book provides a brief introduction to infinite dimensional stochastic analysis in order to present the Heath-Jarrow-Morton methodology in a functional analytic framework that incorporates an infinite-dimensional driving Brownian motion. The Musiela parametrization leads to a stochastic equation in a Hilbert space \(H\) that describes the arbitrage-free evolution of the forward curve. In this setting, the family \({\mathcal G}\) is just a subset of \(H\) and the consistency considerations are transformed into a stochastic invariance problem. Some general results on stochastic invariance for finite dimensional submanifolds in a Hilbert space are deduced. They are then applied to the classical estimation methods.

Reviewer: Miguel Ángel Mirás Calvo (Vigo)

##### MSC:

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

91G30 | Interest rates, asset pricing, etc. (stochastic models) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

93B29 | Differential-geometric methods in systems theory (MSC2000) |