Scandinavian Working Papers in Economics

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

No 338: On the Existence of Finite Dimensional Realizations for Nonlinear Forward Rate Models

Tomas Björk () and Lars Svensson ()
Additional contact information
Tomas Björk: Dept. of Finance, Stockholm School of Economics, Postal: P.O. Box 6501, S-113 83 Stockholm, Sweden
Lars Svensson: Department of Mathematics, Postal: Royal Institute of Technology, S-100 44 Stockholm, Sweden

Abstract: 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. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite dimensional realization. A number of concrete applications are given, and most previously known realization results for time homogenous Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov porcess. In particular we show that the only forward rate models, with short rate dependent volatility structures, which generically give rise to a Markovian short rate are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.

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

JEL-codes: E43; G13

46 pages, October 22, 1999

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Published as
Tomas Björk and Lars Svensson, (2001), 'On the Existence of Finite Dimensional Realizations for Nonlinear Forward Rate Models', Mathematical Finance, vol 11, no 2, pages 205-243

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