Yves Achdou

VOLATILITY SMILE BY MULTILEVEL LEAST SQUARE

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. Dupires 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.

Tuning the mesh of a mixed method for the stream function Vorticity formulation of the Navier-Stokes equations

SummaryIn this article we study a new mixed method for the Stokes and Navier-Stokes equations. The method uses two meshes, one very fine for ω and a coarser one for ψ. Error estimates show that boundary layers do not require to refine the mesh for the stream function ψ as much as for the vorticity ω when the Reynolds number is large. We prove estimates and study implementation problems.