**SSE/EFI Working Paper Series in Economics and Finance**
# No 420:

On the construction of finite dimensional realizations for nonlinear forward rate models

*Tomas Björk *()

* and Camilla Landen *()

**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 structure is allowed to be a smooth
functional of the present forward rate curve.

In a recent paper (to
appear in "Mathematical Finance" ) Björk and Svensson give necessary and
sufficient conditions for the existence of a finite dimensional Markovian
state space realization (FDR) for such a forward rate model, and in the
present paper we provide a general method for the actual construction of an
FDR.

The method works as follows: From the results of Björk and
Svensson we know that there exists an FDR if and only if a certain Lie
algebra is finite dimensional. Given a set of generators for this Lie
algebra we show how to construct an FDR by solving a finite number of
ordinary differential equations in Hilbert space. We illustrate the method
by constructing FDR:s for a number of concrete models. These FDR:s
generalize previous results by allowing for a more general volatility
structure. Furthermore, the dimension of the realizations obtained by using
our method is typically smaller than that of the corresponding previously
known realizations. We also show how to obtain realizations in terms of
benchmarforward rates from the realizations obtained using our method, and
finally we present a bond pricing formula for the realizations we have
obtained.

**Keywords:** forward rate; HJM models; term structure; factor models; state space models; Markovian realizations; (follow links to similar papers)

**JEL-Codes:** E43; G13; (follow links to similar papers)

32 pages, December 20, 2000

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**Publication Data:**
- Forthcoming in Finance and Stochastics.

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