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Self Exciting Threshold Interest Rates Models

Author

Listed:
  • MARC DECAMPS

    (K. U. Leuven, FETEW, Naamsestraat 69, B-3000 Leuven, Belgium)

  • MARC GOOVAERTS

    (K. U. Leuven and U. v. Amsterdam, FETEW, Naamsestraat 69, B-3000 Leuven, Belgium)

  • WIM SCHOUTENS

    (K. U. Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium)

Abstract

In this paper, we study a new class of tractable diffusions suitable for model's primitives of interest rates. We consider scalar diffusions with scale s′(x) and speed m(x) densities discontinuous at the level x*. We call that family of processes Self Exciting Threshold (SET) diffusions. Following Gorovoi and Linetsky [18], we obtain semi-analytical expressions for the transition density of SET (killed) diffusions. We propose several applications to interest rates modeling. We show that SET short rate processes do not generate arbitrage possibilities and we adapt the HJM procedure to forward rates with discontinuous scale density. We also extend the CEV and the shifted-lognormal LIBOR market models. Finally, the models are calibrated to the US market. SET diffusions can also be used to model stock price, stochastic volatility, credit spread, etc.

Suggested Citation

  • Marc Decamps & Marc Goovaerts & Wim Schoutens, 2006. "Self Exciting Threshold Interest Rates Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(07), pages 1093-1122.
    • Handle: RePEc:wsi:ijtafx:v:09:y:2006:i:07:n:s0219024906003937
    DOI: 10.1142/S0219024906003937
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    References listed on IDEAS

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    1. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
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    Cited by:

    1. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    2. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    3. Antoine Lejay & Sara Mazzonetto, 2024. "Maximum likelihood estimator for skew Brownian motion: The convergence rate," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(2), pages 612-642, June.
    4. Tian, Yingxu & Zhang, Haoyan, 2018. "Skew CIR process, conditional characteristic function, moments and bond pricing," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 230-238.
    5. Shiyu Song & Yongjin Wang, 2017. "Pricing double barrier options under a volatility regime-switching model with psychological barriers," Review of Derivatives Research, Springer, vol. 20(3), pages 255-280, October.
    6. Alexander Gairat & Vadim Shcherbakov, 2014. "Density of Skew Brownian motion and its functionals with application in finance," Papers 1407.1715, arXiv.org, revised Mar 2015.
    7. Zaniar Ahmadi & Xiaowen Zhou, 2024. "A note on Refracted Skew Brownian Motion with an application," Papers 2407.09321, arXiv.org, revised Apr 2025.
    8. Antoine Lejay & Paolo Pigato, 2020. "Maximum likelihood drift estimation for a threshold diffusion," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 609-637, September.
    9. Guangli Xu & Shiyu Song & Yongjin Wang, 2016. "The Valuation Of Options On Foreign Exchange Rate In A Target Zone," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-19, May.
    10. Yury Kutoyants, 2012. "On identification of the threshold diffusion processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(2), pages 383-413, April.
    11. Dingwen Zhang, 2024. "Determining the Number and Values of Thresholds for Multi-regime Threshold Ornstein–Uhlenbeck Processes," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3581-3626, November.
    12. Andrey Itkin & Alexander Lipton & Dmitry Muravey, 2021. "Multilayer heat equations and their solutions via oscillating integral transforms," Papers 2112.00949, arXiv.org, revised Dec 2021.

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