 
Summary: Volatility Smile by Multilevel Least Square
Yves Achdou
Olivier Pironneau y
October 1, 2001
Abstract
The aim of this paper is to propose several algorithms for nding 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 nite 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
timestepping algorithm is considered. For all the proposed algorithms, numerical tests are
supplied.
1 Introduction
The volatility, , is the diÆcult parameter of the BlackScholes model of nance (see Wilmott
[23] for an introduction). It is convenient but unrealistic to take it to be constant but on the
other hand little is known to relate it explicitly to any observable nancial data. Therefore much
research is being conducted to solve eÆciently the inverse problem whereby one adjusts so
that the solution of the BlackScholes model ts nancial observations. Such nancial parameter
