On the Geometry of Interest Rate Models
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.
|Date of creation:||24 Nov 2003|
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|Note:||To apppear in "Springer Lecture Notes in Mathematics"|
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- Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-92.
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- Carl Chiarella & Oh Kang Kwon, 2001.
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Springer, vol. 5(2), pages 237-257.
- Carl Chiarella & Oh-Kang Kwon, 1999. "Forward Rate Dependent Markovian Transformations of the Heath-Jarrow-Morton Term Structure Model," Research Paper Series 5, Quantitative Finance Research Centre, University of Technology, Sydney.
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- Björk, Tomas & Landén, Camilla & Svensson, Lars, 2002. "Finite dimensional Markovian realizations for stochastic volatility forward rate models," SSE/EFI Working Paper Series in Economics and Finance 498, Stockholm School of Economics, revised 06 May 2002.
- Ho, Thomas S Y & Lee, Sang-bin, 1986. " Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance, American Finance Association, vol. 41(5), pages 1011-29, December.
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