Static Hedging of Barrier Options with a Smile: An Inverse Problem
Université de Paris VII and C.M.L.A.,
URA 1611 du CNRS, École Normale
Supérieure de Cachan, France.
2 C.M.L.A. 1611 URA du CNRS, École Normale Supérieure de Cachan, France; Raphael@clearingrisk.com.
3 Moscow State University. Partially supported by CNRS while visiting the Emile Borel Center, and by R.F.B.I. under grant 96-01-00947.
Let L be a parabolic second order differential operator on the domain Given a function and such that the support of û is contained in , we let be the solution to the equation: Given positive bounds we seek a function u with support in such that the corresponding solution y satisfies: We prove in this article that, under some regularity conditions on the coefficients of L, continuous solutions are unique and dense in the sense that can be C0-approximated, but an exact solution does not exist in general. This result solves the problem of almost replicating a barrier option in the generalised Black–Scholes framework with a combination of European options, as stated by Carr et al. in .
Mathematics Subject Classification: 93C20 / 65M32 / 62P05 / 91B28
Key words: Inverse problems / Carleman estimates / barrier option hedging / replication.
© EDP Sciences, SMAI, 2002