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

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.