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Volatility Smile By Multilevel Least Square

Author

Listed:
  • YVES ACHDOU

    (UFR Mathématiques, Université Paris 7, Case 7012, 75251 Paris Cedex 05, France)

  • OLIVIER PIRONNEAU

    (Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris Cedex 05, France)

Abstract

The aim of this paper is to propose several algorithms for finding the local volatility from partial observations of the price of an European vanilla option. Dupire's equation is used. The local volatility and the price of the option are discretized by finite elements with highly non uniform meshes and with a coarser mesh for the local volatility. The inverse problem is formulated as a least square problem and the minimization is done by an interior point method. The gradient of the cost function is computed exactly by solving an adjoint problem. A multilevel approach is proposed for accelerating the computations. Also, a suboptimal time-stepping algorithm is considered. For all the proposed algorithms, numerical tests are supplied.

Suggested Citation

  • Yves Achdou & Olivier Pironneau, 2002. "Volatility Smile By Multilevel Least Square," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(06), pages 619-643.
    • Handle: RePEc:wsi:ijtafx:v:05:y:2002:i:06:n:s0219024902001602
    DOI: 10.1142/S0219024902001602
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    Cited by:

    1. B. Düring & A. Jüngel & S. Volkwein, 2008. "Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing," Journal of Optimization Theory and Applications, Springer, vol. 139(3), pages 515-540, December.

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