IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v08y2005i08ns0219024905003396.html
   My bibliography  Save this article

Closed Form Solutions For Quadratic And Inverse Quadratic Term Structure Models

Author

Listed:
  • PETER LAURENCE

    (Department of Mathematics, Università di Roma 1, Piazz. Aldo Moro 2, 00185 Rome, Italy)

  • TAI-HO WANG

    (National Chung-Cheng University, 160, San-Hsing, Min-Hsiung, Chia-Yi621, Taiwan, ROC)

Abstract

We find fundamental solutions in closed form for a family of parabolic equations with two spatial variables, whose symmetry groups had been determined in an earlier paper by Finkel [12]. We show how these results can be applied in finance to yield closed form solutions for special affine and quadratic two factor term structure models as well as a new class of models with inverse square behavior. The latter can be considered a partial extension to two factors of pricing models related to the Bessel process devised by Albanese and Campolieti [3] and Albanese et al. [2]. A by-product of our results is that Lie's reduction method in this setting leads only to fundamental solutions that can be factorized as products of functions that depend jointly on time and on one spatial coordinate. Thus all the results in this paper extend immediately to n factor models.

Suggested Citation

  • Peter Laurence & Tai-Ho Wang, 2005. "Closed Form Solutions For Quadratic And Inverse Quadratic Term Structure Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(08), pages 1059-1083.
    • Handle: RePEc:wsi:ijtafx:v:08:y:2005:i:08:n:s0219024905003396
    DOI: 10.1142/S0219024905003396
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024905003396
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219024905003396?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jan Baldeaux, 2011. "Exact Simulation of the 3/2 Model," Papers 1105.3297, arXiv.org, revised May 2011.
    2. Wenqing Bao & ChunLi Chen & Jin E. Zhang, 2013. "Option Pricing with Lie Symmetry Analysis and Similarity Reduction Method," Papers 1311.4074, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Alexander Lipton, 2020. "Old Problems, Classical Methods, New Solutions," Papers 2003.06903, arXiv.org.
    2. Andrey Itkin, 2023. "The ATM implied skew in the ADO-Heston model," Papers 2309.15044, arXiv.org.
    3. A. Itkin & A. Lipton & D. Muravey, 2021. "Multilayer heat equations: application to finance," Papers 2102.08338, arXiv.org.
    4. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    5. Simon J. A. Malham & Anke Wiese, 2013. "Chi-Square Simulation Of The Cir Process And The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(03), pages 1-38.
    6. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2019. "Gauge transformations in the dual space, and pricing and estimation in the long run in affine jump-diffusion models," Papers 1912.06948, arXiv.org, revised Dec 2019.
    7. Alexander Lipton, 2024. "Hydrodynamics of Markets:Hidden Links Between Physics and Finance," Papers 2403.09761, arXiv.org.
    8. Amengual, Dante & Xiu, Dacheng, 2018. "Resolution of policy uncertainty and sudden declines in volatility," Journal of Econometrics, Elsevier, vol. 203(2), pages 297-315.
    9. Andrey Itkin & Dmitry Muravey, 2023. "American options in time-dependent one-factor models: Semi-analytic pricing, numerical methods and ML support," Papers 2307.13870, arXiv.org.
    10. Andrey Itkin, 2015. "HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-24.
    11. P. Carr & A. Itkin & D. Muravey, 2022. "Semi-analytical pricing of barrier options in the time-dependent Heston model," Papers 2202.06177, arXiv.org.
    12. Alexander Lipton, 2023. "Kelvin Waves, Klein-Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results," Papers 2309.04547, arXiv.org.
    13. Sam Howison & Avraam Rafailidis & Henrik Rasmussen, 2004. "On the pricing and hedging of volatility derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 11(4), pages 317-346.
    14. Leif Andersen & Alexander Lipton, 2013. "Asymptotics For Exponential Lévy Processes And Their Volatility Smile: Survey And New Results," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(01), pages 1-98.
    15. Susanne Griebsch & Uwe Wystup, 2011. "Quantitative Finance, Vol. 11, No. 5, May 2011, 693-709 On the valuation of fader and discrete barrier options in Heston's stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1271-1271.
    16. Alexander Lipton & Marcos Lopez de Prado, 2020. "A closed-form solution for optimal mean-reverting trading strategies," Papers 2003.10502, arXiv.org.
    17. A. Itkin & A. Lipton & D. Muravey, 2020. "From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy," Papers 2006.11976, arXiv.org, revised Jan 2021.
    18. Sudip Ratan Chandra & Diganta Mukherjee, 2016. "Barrier Option Under Lévy Model : A PIDE and Mellin Transform Approach," Mathematics, MDPI, vol. 4(1), pages 1-18, January.
    19. Likuan Qin & Vadim Linetsky, 2016. "Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing," Operations Research, INFORMS, vol. 64(1), pages 99-117, February.
    20. Peter Buchen & Hamish Malloch, 2014. "CLA's, PLA's and a new method for pricing general passport options," Quantitative Finance, Taylor & Francis Journals, vol. 14(7), pages 1201-1209, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wsi:ijtafx:v:08:y:2005:i:08:n:s0219024905003396. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/ijtaf/ijtaf.shtml .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.