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A reduced basis for option pricing

Author

Listed:
  • Rama Cont

    () (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

  • Nicolas Lantos

    () (LJLL - Laboratoire Jacques-Louis Lions - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

  • Olivier Pironneau

    () (LJLL - Laboratoire Jacques-Louis Lions - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

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.

Suggested Citation

  • Rama Cont & Nicolas Lantos & Olivier Pironneau, 2011. "A reduced basis for option pricing," Post-Print hal-00522410, HAL.
  • Handle: RePEc:hal:journl:hal-00522410
    DOI: 10.1137/10079851X
    Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00522410
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    Cited by:

    1. Kathrin Glau, 2015. "Feynman-Kac formula for L\'evy processes with discontinuous killing rate," Papers 1502.07531, arXiv.org, revised Nov 2015.
    2. Karakaya, Emrah, 2016. "Finite Element Method for forecasting the diffusion of photovoltaic systems: Why and how?," Applied Energy, Elsevier, vol. 163(C), pages 464-475.
    3. Maciej Balajewicz & Jari Toivanen, 2016. "Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models," Papers 1612.00402, arXiv.org.
    4. Kathrin Glau, 2016. "A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates," Finance and Stochastics, Springer, vol. 20(4), pages 1021-1059, October.
    5. Maximilian Ga{ss} & Kathrin Glau & Maximilian Mair, 2015. "Magic points in finance: Empirical integration for parametric option pricing," Papers 1511.00884, arXiv.org, revised Nov 2016.
    6. Karakaya, Emrah, 2014. "Finite Element Model of the Innovation Diffusion: An Application to Photovoltaic Systems," INDEK Working Paper Series 2014/6, Royal Institute of Technology, Department of Industrial Economics and Management.
    7. Stefano, Pagliarani & Pascucci, Andrea & Candia, Riga, 2011. "Expansion formulae for local Lévy models," MPRA Paper 34571, University Library of Munich, Germany.

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