On the Existence 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 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.
|Date of creation:||22 Oct 1999|
|Date of revision:|
|Publication status:||Published in Mathematical Finance, 2001, pages 205-243.|
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- Björk, Tomas & Landen, Camilla, 2000. "On the construction of finite dimensional realizations for nonlinear forward rate models," SSE/EFI Working Paper Series in Economics and Finance 420, Stockholm School of Economics.
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Working Paper Series
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- Carl Chiarella & Oh-Kang Kwon, 1999.
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Research Paper Series
5, Quantitative Finance Research Centre, University of Technology, Sydney.
- Carl Chiarella & Oh Kang Kwon, 2001. "Forward rate dependent Markovian transformations of the Heath-Jarrow-Morton term structure model," Finance and Stochastics, Springer, vol. 5(2), pages 237-257.
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- Robert A. Jarrow, 2009. "The Term Structure of Interest Rates," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 69-96, November.
- Björk, Tomas & Gombani, Andrea, 1997. "Minimal Realizations of Forward Rates," SSE/EFI Working Paper Series in Economics and Finance 182, Stockholm School of Economics.
- Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-92.
- Nelson, Charles R & Siegel, Andrew F, 1987. "Parsimonious Modeling of Yield Curves," The Journal of Business, University of Chicago Press, vol. 60(4), pages 473-89, October.
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