Static hedging of barrier options with a smile : an inverse problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 127-142.

Let L be a parabolic second order differential operator on the domain Π ¯=0,T×. Given a function u ^:R and x ^>0 such that the support of u ^ is contained in (-,-x ^], we let y ^:Π ¯ be the solution to the equation:

Ly ^=0,y ^| {0}× =u ^.
Given positive bounds 0<x 0 <x 1 , we seek a function u with support in x 0 ,x 1 such that the corresponding solution y satisfies:
y(t,0)=y ^(t,0)t0,T.
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 y ^| [0,T]×{0} can be C 0 -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 [6].

DOI : 10.1051/cocv:2002040
Classification : 93C20, 65M32, 62P05, 91B28
Mots clés : inverse problems, Carleman estimates, barrier option hedging, replication
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Bardos, Claude; Douady, Raphaël; Fursikov, Andrei. Static hedging of barrier options with a smile : an inverse problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 127-142. doi : 10.1051/cocv:2002040. http://www.numdam.org/articles/10.1051/cocv:2002040/

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