On the geometry of interest rate models

Abstract: 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.

Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).