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A reduced basis for option pricing

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  • Rama Cont

    ()
    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - CNRS : UMR7599 - Université Pierre et Marie Curie - Paris VI - Université Paris-Diderot - Paris VII)

  • Nicolas Lantos

    ()
    (LJLL - Laboratoire Jacques-Louis Lions - CNRS : UMR7598 - Université Pierre et Marie Curie - Paris VI)

  • Olivier Pironneau

    ()
    (LJLL - Laboratoire Jacques-Louis Lions - CNRS : UMR7598 - Université Pierre et Marie Curie - Paris VI)

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    Abstract

    We introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations which arise in option pricing theory. Our method uses a basis of functions constructed from a sequence of Black-Scholes solutions with different volatilities. We show that this choice of basis leads to a sparse representation of option pricing functions, yielding an approximation whose precision is exponential in the number of basis functions. A Galerkin method using this basis for solving the pricing PDE is presented. Numerical tests based on the CEV diffusion model and the Merton jump diffusion model show that the method has better numerical performance relative to commonly used finite-difference and finite-element methods. We also compare our method with a numerical Proper Orthogonal Decomposition (POD). Finally, we show that this approach may be used advantageously for the calibration of local volatility functions.

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    Bibliographic Info

    Paper provided by HAL in its series Post-Print with number hal-00522410.

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    Date of creation: 21 Mar 2011
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    Publication status: Published, SIAM Journal on Financial Mathematics, 2011, 2, 1, 287-316
    Handle: RePEc:hal:journl:hal-00522410

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00522410/en/
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    Cited by:
    1. Stefano, Pagliarani & Pascucci, Andrea & Candia, Riga, 2011. "Expansion formulae for local Lévy models," MPRA Paper 34571, University Library of Munich, Germany.
    2. Karakaya, Emrah, 2014. "Finite Element Model of the Innovation Diffusion: An Application to Photovoltaic Systems," INDEK Working Paper Series, Department of Industrial Economics and Management, Royal Institute of Technology 2014/6, Department of Industrial Economics and Management, Royal Institute of Technology.

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