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A Markovian Defaultable Term Structure Model with State Dependent Volatilities

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Abstract

The defaultable forward rate is modeled as a jump diffusion process within the Schonbucher (2000, 2003) general Heath, jarrow and Morton (1992) framework where jumps in the defaultable term structure f d(t, T) cause jumps and defaults to the defaultable bond prices P d(t, T). Within this framework, we investigate an appropriate forward rate volatility structure that results in Markovian defaultable spot rate dynamics. In particular, we consider state dependent Wiener volatility functions and time dependent Poisson volatility functions. The corresponding term structures of interest rates are expressed as finite dimensional affine realisations in terms of benchmark defaultable forward rates. In addition, we extend this model to incorporate stochastic spreads by allowing jump intensities to follow a square-root diffusion process. In that case the dynamics become non-Markovian and to restore path independence we propose either an approximate Markovian scheme or, alternatively, constant Poisson volatility functions. We also conduct some numerical simulations to gauge the effect of the stochastic intensity and the distributional implications of various volatility specifications.

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Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 135.

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Length: 40
Date of creation: 01 Oct 2004
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Handle: RePEc:uts:rpaper:135

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Keywords: defaultable HJM model; strochastic credit spreads; defaultable bond prices;

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References

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  1. Carl Chiarella & Christina Sklibosios, 2003. "A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework," Asia-Pacific Financial Markets, Springer, vol. 10(2), pages 87-127, September.
  2. R. Bhar & C. Chiarella, 1997. "Transformation of Heath?Jarrow?Morton models to Markovian systems," The European Journal of Finance, Taylor & Francis Journals, vol. 3(1), pages 1-26.
  3. Björk, Tomas & Gombani, Andrea, 1997. "Minimal Realizations of Forward Rates," Working Paper Series in Economics and Finance 182, Stockholm School of Economics.
  4. Olivier Scaillet & Olivier Renault & Jean-Luc Prigent, 2000. "An Empirical Investigation in Credit Spread Indices," FMG Discussion Papers dp363, Financial Markets Group.
  5. Bystrom, Hans & Kwon, Oh Kang, 2007. "A simple continuous measure of credit risk," International Review of Financial Analysis, Elsevier, vol. 16(5), pages 508-523.
  6. Tomas Björk & Yuri Kabanov & Wolfgang Runggaldier, 1997. "Bond Market Structure in the Presence of Marked Point Processes," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 7(2), pages 211-239.
  7. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, Econometric Society, vol. 53(2), pages 385-407, March.
  8. Sarig, Oded & Warga, Arthur, 1989. " Some Empirical Estimates of the Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 44(5), pages 1351-60, December.
  9. Jean -Luc Prigent & Olivier Renault & Olivier Scaillet, 2000. "An Empirical Investigation in Credit Spread Indices," Working Papers 2000-59, Centre de Recherche en Economie et Statistique.
  10. To, Thuy Duong & Carl Chiarella, 2003. "The Jump Component of the Volatility Structure of Interest Rate Futures Markets: An International Comparison," Royal Economic Society Annual Conference 2003, Royal Economic Society 205, Royal Economic Society.
  11. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, Econometric Society, vol. 60(1), pages 77-105, January.
  12. Carl Chiarella & Oh Kwon, 2003. "Finite Dimensional Affine Realisations of HJM Models in Terms of Forward Rates and Yields," Review of Derivatives Research, Springer, vol. 6(2), pages 129-155, May.
  13. Hiroshi Shirakawa, 1991. "Interest Rate Option Pricing With Poisson-Gaussian Forward Rate Curve Processes," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 1(4), pages 77-94.
  14. Inui, Koji & Kijima, Masaaki, 1998. "A Markovian Framework in Multi-Factor Heath-Jarrow-Morton Models," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 33(03), pages 423-440, September.
  15. Das, Sanjiv R., 2002. "The surprise element: jumps in interest rates," Journal of Econometrics, Elsevier, vol. 106(1), pages 27-65, January.
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Cited by:
  1. Nicola Bruti-Liberati & Christina Nikitopoulos-Sklibosios & Eckhard Platen & Erik Schlogl, 2009. "Alternative Defaultable Term Structure Models," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 242, Quantitative Finance Research Centre, University of Technology, Sydney.
  2. Carl Chiarella & Samuel Chege Maina & Christina Nikitopoulos Sklibosios, 2013. "Credit Derivatives Pricing With Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(04), pages 1350019-1-1.
  3. Laura Morino & Wolfgang J. Ruggaldier, 2014. "On multicurve models for the term structure," Papers 1401.5431, arXiv.org.
  4. Carl Chiarella & Samuel Chege Maina & Christina Nikitopoulos-Sklibosios, 2010. "Markovian Defaultable HJM Term Structure Models with Unspanned Stochastic Volatility," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 283, Quantitative Finance Research Centre, University of Technology, Sydney.

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