Summary: AN INVERSE PROBLEM FOR A PARABOLIC VARIATIONAL
INEQUALITY WITH AN INTEGRO-DIFFERENTIAL OPERATOR
Abstract. We consider the calibration of a L´evy process with American vanilla options.
The price of an American vanilla option as a function of the maturity and the strike satisfies a
forward in time linear complementarity problem involving a partial integro-differential operator.
It leads to a variational inequality in a suitable weighted Sobolev space. Calibrating the L´evy
process amounts to solving an inverse problem where the state variable satisfies the previously
mentioned variational inequality. We propose a regularized least square method. After studying
the variational inequality carefully, we find necessary optimality conditions for the least square
problem. In this work, we focus on the case when the volatility is bounded away from zero.
1. Introduction. Consider an arbitrage-free market described by a probabil-
ity measure P on a scenario space (, A). There is a risk-free asset whose price at
time is er
, r 0 and a risky asset whose price at time is S . Specifying an
arbitrage-free option pricing model necessitates the choice of a risk-neutral measure,
i.e. a probability P
equivalent to P such that the discounted price (e-r
S )[0,T ]
is a martingale under P