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

Author

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  • Björk, Tomas

    (Department of Finance)

  • Landen, Camilla

    (Department of Mathematics)

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.

Suggested Citation

  • 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.
    • Handle: RePEc:hhs:hastef:0420
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    1. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53, January.
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    4. 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.
    5. Tomas Björk & Lars Svensson, 2001. "On the Existence of Finite‐Dimensional Realizations for Nonlinear Forward Rate Models," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 205-243, April.
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    9. Peter Ritchken & L. Sankarasubramanian, 1995. "Volatility Structures Of Forward Rates And The Dynamics Of The Term Structure1," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 55-72, January.
    10. Jeffrey, Andrew, 1995. "Single Factor Heath-Jarrow-Morton Term Structure Models Based on Markov Spot Interest Rate Dynamics," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 30(4), pages 619-642, December.
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    Cited by:

    1. Björk, Tomas, 2003. "On the Geometry of Interest Rate Models," SSE/EFI Working Paper Series in Economics and Finance 545, Stockholm School of Economics.
    2. Eckhard Platen & Stefan Tappe, 2011. "Affine Realizations for Levy Driven Interest Rate Models with Real-World Forward Rate Dynamics," Research Paper Series 289, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Tomas Björk & Lars Svensson, 2001. "On the Existence of Finite‐Dimensional Realizations for Nonlinear Forward Rate Models," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 205-243, April.
    4. Tomas Björk & Magnus Blix & Camilla Landén, 2006. "On Finite Dimensional Realizations For The Term Structure Of Futures Prices," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(03), pages 281-314.
    5. Jing Yuan & Yan Peng & Zongwu Cai & Zhengyi Zhang, 2021. "A Quantitative Evaluation to Interest Rate Marketization Reform in China," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 202122, University of Kansas, Department of Economics.
    6. Fred Espen Benth & Paul Kruhner, 2015. "Approximation of forward curve models in commodity markets with arbitrage-free finite dimensional models," Papers 1512.05983, arXiv.org.
    7. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    8. Fred Benth & Jukka Lempa, 2014. "Optimal portfolios in commodity futures markets," Finance and Stochastics, Springer, vol. 18(2), pages 407-430, April.
    9. Samuel Chege Maina, 2011. "Credit Risk Modelling in Markovian HJM Term Structure Class of Models with Stochastic Volatility," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2011, January-A.
    10. Carl Chiarella & Samuel Chege Maina & Christina Nikitopoulos-Sklibosios, 2010. "Markovian Defaultable HJM Term Structure Models with Unspanned Stochastic Volatility," Research Paper Series 283, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. Christina Nikitopoulos-Sklibosios, 2005. "A Class of Markovian Models for the Term Structure of Interest Rates Under Jump-Diffusions," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 6, July-Dece.
    12. Gaspar, Raquel M., 2004. "On Finite Dimensional Realizations of Forward Price Term Structure Models," SSE/EFI Working Paper Series in Economics and Finance 569, Stockholm School of Economics.
    13. Mikael Elhouar, 2008. "Finite-dimensional Realizations of Regime-switching HJM Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(4), pages 331-354.
    14. Yassine El Qalli, 2010. "Recursive Bayesian Estimation In Forward Price Models Implied By Fair Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(02), pages 301-333.
    15. Christina Nikitopoulos-Sklibosios, 2005. "A Class of Markovian Models for the Term Structure of Interest Rates Under Jump-Diffusions," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2005, January-A.
    16. Maria B. Chiarolla & Tiziano De Angelis, 2012. "Analytical Pricing of American Bond Options in the Heath-Jarrow-Morton Model," Papers 1212.0781, arXiv.org, revised Mar 2014.
    17. Alberto Ohashi & Alexandre B Simas, 2015. "Principal Components Analysis for Semimartingales and Stochastic PDE," Papers 1503.05909, arXiv.org, revised Mar 2016.
    18. Samuel Chege Maina, 2011. "Credit Risk Modelling in Markovian HJM Term Structure Class of Models with Stochastic Volatility," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 5, July-Dece.

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    More about this item

    Keywords

    forward rate; HJM models; term structure; factor models; state space models; Markovian realizations;
    All these keywords.

    JEL classification:

    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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