Scandinavian Working Papers in Economics

SSE/EFI Working Paper Series in Economics and Finance,
Stockholm School of Economics

No 545: On the Geometry of Interest Rate Models

Tomas Björk ()
Additional contact information
Tomas Björk: Dept. of Finance, Stockholm School of Economics, Postal: Stockholm School of Economics, P.O. Box 6501, SE-113 83 Stockholm, Sweden

Abstract: In this paper, which is a substantial extension of the earlier essay Björk (2001), we give an overview of some recent work on the 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 Markovian 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 he 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.

Keywords: Forward rate curves; interest rate models; factor models; state space models; Markovian realizations

JEL-codes: E43; G13

87 pages, November 24, 2003

Note: To apppear in "Springer Lecture Notes in Mathematics"

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