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Explosion in the quasi-Gaussian HJM model

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  • Dan Pirjol
  • Lingjiong Zhu

Abstract

We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We discuss the implications of these results for the pricing of the zero coupon bonds and Eurodollar futures under this model.

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  • Dan Pirjol & Lingjiong Zhu, 2019. "Explosion in the quasi-Gaussian HJM model," Papers 1908.07102, arXiv.org.
    • Handle: RePEc:arx:papers:1908.07102
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    1. Wissel, Johannes, 2007. "Some results on strong solutions of SDEs with applications to interest rate models," Stochastic Processes and their Applications, Elsevier, vol. 117(6), pages 720-741, June.
    2. Xing, Jiamin & Li, Yong, 2016. "Nonlinear Lyapunov criteria for stochastic explosive solutions," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 63-67.
    3. 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.
    4. Chow, Pao-Liu & Khasminskii, Rafail, 2014. "Almost sure explosion of solutions to stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 639-645.
    5. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    6. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    7. Leif Andersen & Jesper Andreasen, 2000. "Volatility skews and extensions of the Libor market model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(1), pages 1-32.
    8. Klaus Sandmann & Dieter Sondermann, 1997. "A Note on the Stability of Lognormal Interest Rate Models and the Pricing of Eurodollar Futures," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 119-125, April.
    9. Nusret Cakici & Jintao Zhu, 2001. "Pricing Eurodollar Futures Options with the Heath—Jarrow—Morton Model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 21(7), pages 655-680, July.
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