VOLATILITY SMILE BY MULTILEVEL LEAST SQUARE

Yves Achdou () and Olivier Pironneau ()
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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

International Journal of Theoretical and Applied Finance (IJTAF), 2002, vol. 05, issue 06, 619-643

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.

Keywords: Calibration; European options; Dupire equation; least square methods; finite element methods (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1142/S0219024902001602

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