 
Summary: AN INVERSE PROBLEM FOR A PARABOLIC VARIATIONAL
INEQUALITY ARISING IN THE CALIBRATION OF AMERICAN
OPTIONS
YVES ACHDOU #
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 free boundary associated with
this variational inequality can be interpreted as the price for which the option should be exercised.
The calibration of volatility from the observations of the prices of an American option yields an
inverse problem for the previously mentioned parabolic variational inequality. After studying the
variational inequality and the exercise price, we give results concerning the sensitivity of the option
price and of the exercise price with respect to the variations of the volatility. The inverse problem
is addressed by a least square method, with suitable regularization terms. We give necessary
optimality conditions involving an adjoint state for a simplified inverse problem and we study
the di#erentiability of the cost function. Optimality conditions are also given for the genuine
calibration problem.
1. Introduction. A European vanilla call (respectively put) option is a con
tract giving its owner the right to buy (respectively sell) a share of a specific common
stock at a fixed price K at a certain date T . The specific stock is called the under
lying asset. The fixed price K is termed the strike and T is termed the maturity.
The term vanilla is used to notify that this kind of option is the simplest one: in
