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The Reduction of Forward Rate Dependent Volatility HJM Models to Markovian Form: Pricing European Bond Option

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Abstract

We consider a single factor Heath-Jarrow-Morton model with a forward rate volatility function depending upon a function of time to maturity, the instantaneous spot rate of interest and a forward rate to a fixed maturity. With this specification the stochastic dynamics determining the prices of interest rate derivatives may be reduced to Markovian form. Furthermore, the evolution of the forward rate curve is completely determined by the two rates specified in the volatility function and it is thus possible to obtain a closed form expression for bond prices. The prices of bond options are determined by a partial differential equation involving two spatial variables. We discuss the evaluation of European bond options in this framework by use of the ADI method.

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  • Ram Bhar & Carl Chiarella & Nadima El-Hassan & Xiaosu Zheng, 2000. "The Reduction of Forward Rate Dependent Volatility HJM Models to Markovian Form: Pricing European Bond Option," Research Paper Series 36, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:36
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    1. 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, March.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    3. 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.
    4. Robert A. Jarrow & Arkadev Chatterjea, 2019. "Interest Rates," World Scientific Book Chapters, in: An Introduction to Derivative Securities, Financial Markets, and Risk Management, chapter 2, pages 22-52, World Scientific Publishing Co. Pte. Ltd..
    5. Andrew Carverhill, 1994. "When Is The Short Rate Markovian?," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 305-312, October.
    6. 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 1011-1029, December.
    7. 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.
    8. 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.
    9. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-592.
    10. Andrew Mark Jeffrey, 1995. "Single Factor Heath-Jarrow-Morton Term Structure Models Based on Markov Spot Interest Rate Dynamics," Yale School of Management Working Papers ysm46, Yale School of Management.
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    Cited by:

    1. Carl Chiarella & Xue-Zhong He & Christina Sklibosios Nikitopoulos, 2015. "Derivative Security Pricing," Dynamic Modeling and Econometrics in Economics and Finance, Springer, edition 127, number 978-3-662-45906-5, July-Dece.
    2. Y. D'Halluin & P. A. Forsyth & K. R. Vetzal & G. Labahn, 2001. "A numerical PDE approach for pricing callable bonds," Applied Mathematical Finance, Taylor & Francis Journals, vol. 8(1), pages 49-77.
    3. Carl Chiarella & Oh-Kang Kwon, 2000. "A Class of Heath-Jarrow-Morton Term Structure Models with Stochastic Volatility," Research Paper Series 34, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Carl Chiarella & Sara Pasquali & Wolfgang Runggaldier, 2001. "On Filtering in Markovian Term Structure Models (An Approximation Approach)," Research Paper Series 65, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Chiarella, Carl & Clewlow, Les & Musti, Silvana, 2005. "A volatility decomposition control variate technique for Monte Carlo simulations of Heath Jarrow Morton models," European Journal of Operational Research, Elsevier, vol. 161(2), pages 325-336, March.
    6. Carl Chiarella & Sara Pasquali & Wolfgang J. Runggaldier, 2001. "On Filtering in Markovian Term Structure Models," World Scientific Book Chapters, in: Jiongmin Yong (ed.), Recent Developments In Mathematical Finance, chapter 12, pages 139-150, World Scientific Publishing Co. Pte. Ltd..
    7. Fima Klebaner & Truc Le & Robert Liptser, 2006. "On Estimation of Volatility Surface and Prediction of Future Spot Volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(3), pages 245-263.

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