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Picard Approximation of Stochastic Differential Equations and Application to Libor Models

  • Antonis Papapantoleon

    ()

    (Institute of Mathematics, TU Berlin and Quantitative Products Laboratory, Deutsche Bank AG)

  • David Skovmand

    ()

    (Aarhus School of Business and CREATES)

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    The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our contribution is twofold. Firstly, we propose an alternative approximation scheme based on Picard iterations. This approach is similar in accuracy to the Euler discretization, but with the feature that each rate is evolved independently of the other rates in the term structure. This enables simultaneous calculation of derivative prices of different maturities using parallel computing. Secondly, the product terms occurring in the drift of a LIBOR market model driven by a jump process grow exponentially as a function of the number of rates, quickly rendering the model intractable. We reduce this growth from exponential to quadratic using truncated expansions of the product terms. We include numerical illustrations of the accuracy and speed of our method pricing caplets, swaptions and forward rate agreements.

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    File URL: ftp://ftp.econ.au.dk/creates/rp/10/rp10_40.pdf
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    Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2010-40.

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    Length: 21
    Date of creation: 16 Jul 2010
    Date of revision:
    Handle: RePEc:aah:create:2010-40
    Contact details of provider: Web page: http://www.econ.au.dk/afn/

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    1. Miltersen, K. & K. Sandmann & D. Sondermann, 1994. "Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Discussion Paper Serie B 308, University of Bonn, Germany.
    2. 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.
    3. Erik Schlögl, 2002. "A multicurrency extension of the lognormal interest rate Market Models," Finance and Stochastics, Springer, vol. 6(2), pages 173-196.
    4. Ernst Eberlein & Jean Jacod & Sebastian Raible, 2005. "Lévy term structure models: No-arbitrage and completeness," Finance and Stochastics, Springer, vol. 9(1), pages 67-88, January.
    5. Ernst Eberlein & Fehmi Özkan, 2005. "The Lévy LIBOR model," Finance and Stochastics, Springer, vol. 9(3), pages 327-348, 07.
    6. 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.
    7. 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.
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