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Theory and Calibration of Swap Market Models

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  • S.Galluccio

    ()

  • Z. Huang
  • J.-M. Ly
  • O. Scaillet

Abstract

This paper introduces a general framework for market models, named Market Model Approach, through the concept of admissible sets of for-ward swap rates spanning a given tenor structure. We relate this concept to results in graph theory by showing that a set is admissible if and only if the associated graph is a tree. This connection enables us to enumerate all admissible models for a given tenor structure. Three main classes are identified within this framework, and correspond to the co-terminal, co-initial, and co-sliding model. We prove that the LIBOR market model is the only admissible model of a co-sliding type. By focusing on the co-terminal model in a lognormal setting, we develop and compare several approximating analytical formulae for caplets, while swaptions can be priced by a simple Black-type formula. A novel calibration technique is introduced to allow simultaneous calibration to caplet and swaption prices. Empirical calibration of the co-terminal model is shown to be faster, more robust and more efficient than the same procedure applied to the LIBOR market model. We then argue that the co-terminal approach is the simplest and most convenient market model for pricing and hedging a large variety of exotic interest-rate derivatives.

Suggested Citation

  • S.Galluccio & Z. Huang & J.-M. Ly & O. Scaillet, 2005. "Theory and Calibration of Swap Market Models," FAME Research Paper Series rp107, International Center for Financial Asset Management and Engineering.
  • Handle: RePEc:fam:rpseri:rp107
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    References listed on IDEAS

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    1. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    2. Raoul Pietersz & Marcel Regenmortel, 2006. "Generic market models," Finance and Stochastics, Springer, vol. 10(4), pages 507-528, December.
    3. A. D'Aspremont, 2003. "Interest rate model calibration using semidefinite Programming," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(3), pages 183-213.
    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. 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.
    6. de Jong, F.C.J.M. & Driessen, J.J.A.G. & Pelsser, A., 2000. "Libor and Swap Market Models for the Pricing of Interest Rate Derivatives : An Empirical Analysis," Discussion Paper 2000-35, Tilburg University, Center for Economic Research.
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    Citations

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    Cited by:

    1. Joshi, Mark & Yang, Chao, 2011. "Fast delta computations in the swap-rate market model," Journal of Economic Dynamics and Control, Elsevier, vol. 35(5), pages 764-775, May.
    2. Thomas Lux, 2006. "Applications of Statistical Physics in Finance and Economics," Working Papers wpn06-07, Warwick Business School, Finance Group.
    3. Raoul Pietersz & Marcel Regenmortel, 2006. "Generic market models," Finance and Stochastics, Springer, vol. 10(4), pages 507-528, December.
    4. Ferdinando Ametrano & Mark Joshi, 2011. "Smooth simultaneous calibration of the LMM to caplets and co-terminal swaptions," Quantitative Finance, Taylor & Francis Journals, vol. 11(4), pages 547-558.
    5. Leippold, Markus & Strømberg, Jacob, 2014. "Time-changed Lévy LIBOR market model: Pricing and joint estimation of the cap surface and swaption cube," Journal of Financial Economics, Elsevier, vol. 111(1), pages 224-250.

    More about this item

    Keywords

    Swap Market Model; Cap; Swaption; Calibration; Graph Theory;

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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