 
Summary: A Numerical Procedure for the Calibration of the Volatility with
American Options
Yves Achdou #
Olivier Pironneau +
August 25, 2004
Abstract
In finance, the price of an American option is obtained from the price of the underlying asset by
solving a parabolic variational inequality. The calibration of volatility from the prices of a family
of American options yields an inverse problem involving the solution of the previously mentioned
parabolic variational inequality. In this paper, the discretization of the variational inequality by
finite elements is studied in detail. Then, a calibration procedure, where the volatility belongs to a
finitedimensional space (finite element or bicubic splines) is described. A least square method, with
suitable regularization terms is used. Necessary optimality conditions involving adjoint states are
given and the di#erentiability of the cost function is studied. A parallel algorithm is proposed and
numerical experiments, on both academic and realistic cases, are presented.
1 Introduction
The volatility is the di#cult parameter of the BlackScholes model of finance (see Wilmott [23], Lamberton
and Lapeyre [20] 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 financial data. Therefore, much
research is being conducted to solve e#ciently the inverse problem whereby one adjusts the local volatility
