IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v20y2017i02ns0219024917500212.html
   My bibliography  Save this article

CLASSIFICATION OF TWO- AND THREE-FACTOR TIME-HOMOGENEOUS SEPARABLE LMMs

Author

Listed:
  • JAKA GOGALA

    (Department of Statistics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK)

  • JOANNE E. KENNEDY

    (Department of Statistics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK)

Abstract

The flexibility of parametrizations of the LIBOR market model (LMM) comes at a cost, namely the LMM is high-dimensional, which makes it cumbersome to use when pricing derivatives with early exercise features. One way to overcome this issue for short- and medium-term time horizons is by imposing the separability condition on the volatility functions and approximating the model using a single time-step approximation.In this paper, we examine the flexibility of separable LMMs under the relaxed assumption that the driving Brownian motions can be correlated. In particular, we are interested in how the separability condition interacts with time-homogeneity, a desirable property of a LMM. We show that the two concepts can be related using a Levi-Civitá equation and provide a characterization of two- and three-factor separable and time-homogeneous LMMs and show that they are of practical interest. The results presented in this paper are also applicable to local-volatility LMMs. These separable volatility structures can be used for the driver of a two- or three-dimensional Markov-functional model — in which case no (single time-step) approximation is needed and the resultant model is both time-homogeneous and arbitrage-free.

Suggested Citation

  • Jaka Gogala & Joanne E. Kennedy, 2017. "CLASSIFICATION OF TWO- AND THREE-FACTOR TIME-HOMOGENEOUS SEPARABLE LMMs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(02), pages 1-44, March.
    • Handle: RePEc:wsi:ijtafx:v:20:y:2017:i:02:n:s0219024917500212
    DOI: 10.1142/S0219024917500212
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024917500212
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219024917500212?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. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    2. Roger Lord & Antoon Pelsser, 2007. "Level-Slope-Curvature - Fact or Artefact?," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(2), pages 105-130.
    3. Joanne Kennedy & Phil Hunt & Antoon Pelsser, 2000. "Markov-functional interest rate models," Finance and Stochastics, Springer, vol. 4(4), pages 391-408.
    4. 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..
    5. 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.
    6. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. "Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-430, March.
    7. Marek Rutkowski & Marek Musiela, 1997. "Continuous-time term structure models: Forward measure approach (*)," Finance and Stochastics, Springer, vol. 1(4), pages 261-291.
    8. Andrew Carverhill, 1994. "When Is The Short Rate Markovian?," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 305-312, October.
    9. Raoul Pietersz & Antoon Pelsser & Marcel van Regenmortel, 2005. "Fast drift approximated pricing in the BGM model," Finance 0502005, University Library of Munich, Germany.
    10. Alan Brace & Dariusz G¸atarek & Marek Musiela, 1997. "The Market Model of Interest Rate Dynamics," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 127-155, April.
    11. Mark Joshi & Alan Stacey, 2008. "New and robust drift approximations for the LIBOR market model," Quantitative Finance, Taylor & Francis Journals, vol. 8(4), pages 427-434.
    12. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
    Full references (including those not matched with items on IDEAS)

    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. Raoul Pietersz & Marcel Regenmortel, 2006. "Generic market models," Finance and Stochastics, Springer, vol. 10(4), pages 507-528, December.
      • Pietersz, R. & van Regenmortel, M., 2005. "Generic Market Models," ERIM Report Series Research in Management ERS-2005-010-F&A, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
      • Raoul Pietersz & Marcel van Regenmortel, 2005. "Generic Market Models," Finance 0502009, University Library of Munich, Germany.
    2. Nicolas Merener & Paul Glasserman, 2003. "Numerical solution of jump-diffusion LIBOR market models," Finance and Stochastics, Springer, vol. 7(1), pages 1-27.
    3. Paul Glasserman & S. G. Kou, 2003. "The Term Structure of Simple Forward Rates with Jump Risk," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 383-410, July.
    4. Glasserman, P. & Zhao, X., 1998. "Arbitrage-Free Discretization of Lognormal Forward Libor and Swap Rate Models," Papers 98-09, Columbia - Graduate School of Business.
    5. S. Galluccio & J.‐M. Ly & Z. Huang & O. Scaillet, 2007. "Theory And Calibration Of Swap Market Models," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 111-141, January.
    6. Raoul Pietersz & Antoon Pelsser, 2010. "A comparison of single factor Markov-functional and multi factor market models," Review of Derivatives Research, Springer, vol. 13(3), pages 245-272, October.
    7. Svenstrup, Mikkel, 2005. "On the suboptimality of single-factor exercise strategies for Bermudan swaptions," Journal of Financial Economics, Elsevier, vol. 78(3), pages 651-684, December.
    8. Heidari, Massoud & Wu, Liuren, 2009. "A Joint Framework for Consistently Pricing Interest Rates and Interest Rate Derivatives," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 44(3), pages 517-550, June.
    9. Christiansen, Charlotte & Strunk Hansen, Charlotte, 2000. "Implied Volatility of Interest Rate Options: An Empirical Investigation of the Market Model," Finance Working Papers 00-1, University of Aarhus, Aarhus School of Business, Department of Business Studies.
    10. Simon H. Babbs, 2002. "Conditional Gaussian models of the term structure of interest rates," Finance and Stochastics, Springer, vol. 6(3), pages 333-353.
    11. K. F. Pilz & E. Schlögl, 2013. "A hybrid commodity and interest rate market model," Quantitative Finance, Taylor & Francis Journals, vol. 13(4), pages 543-560, March.
    12. Zühlsdorff, Christian, 2002. "Extended Libor Market Models with Affine and Quadratic Volatility," Bonn Econ Discussion Papers 6/2002, University of Bonn, Bonn Graduate School of Economics (BGSE).
    13. Christina Nikitopoulos-Sklibosios, 2005. "A Class of Markovian Models for the Term Structure of Interest Rates Under Jump-Diffusions," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2005.
    14. Antonis Papapantoleon, 2010. "Old and new approaches to LIBOR modeling," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(s1), pages 257-275.
    15. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    16. Frank De Jong & Joost Driessen & Antoon Pelsser, 2001. "Libor Market Models versus Swap Market Models for Pricing Interest Rate Derivatives: An Empirical Analysis," Review of Finance, European Finance Association, vol. 5(3), pages 201-237.
    17. Linlin Xu & Giray Ökten, 2015. "High-performance financial simulation using randomized quasi-Monte Carlo methods," Quantitative Finance, Taylor & Francis Journals, vol. 15(8), pages 1425-1436, August.
    18. Massoud Heidari & Liuren Wu, 2002. "Term Structure of Interest Rates, Yield Curve Residuals, and the Consistent Pricing of Interest Rates and Interest Rate Derivatives," Finance 0207010, University Library of Munich, Germany, revised 10 Sep 2002.
    19. Takashi Yasuoka, 2001. "Mathematical Pseudo-Completion Of The Bgm Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(03), pages 375-401.
    20. Christina Nikitopoulos-Sklibosios, 2005. "A Class of Markovian Models for the Term Structure of Interest Rates Under Jump-Diffusions," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 6, July-Dece.

    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:20:y:2017:i:02:n:s0219024917500212. 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.