## On the geometry of interest rate models

- Date: 10/05/2007

Thomas Bjork (Stockholm School of Economics)

University of British Columbia

The purpose of this talk is to give an overview of some recent work

concerning the structural and geometric properties of the evolution of

the forward rate curve in an arbitrage free bond market. The main

problems to be discussed are as follows.

1. When is a given forward rate model consistent with a given family of forward rate curves?

2. When can the inherently infinite dimensional forward rate process be

realized by means of a finite dimensional state space model?

We consider interest rate models of Heath-Jarrow-Morton type, where the

forward rates are driven by a multidimensional Wiener process, and

where the volatility is allowed to be an arbitrary smooth functional of

the present forward rate curve. Within this framework we give necessary

and sufficient conditions for consistency, as well as for the existence

of a finite dimensional realization, in terms of the forward rate

volatilities. We also study stochastic volatility HJM models, and we

provide a systematic method for the construction of concrete

realizations.

UBC Math Dept. Colloquium 2007