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].

Classification : 93C20,  65M32,  62P05,  91B28
Mots clés : inverse problems, Carleman estimates, barrier option hedging, replication
     author = {Bardos, Claude and Douady, Rapha\"el and Fursikov, Andrei},
     title = {Static hedging of barrier options with a smile : an inverse problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {127--142},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002040},
     zbl = {1063.91028},
     mrnumber = {1932947},
     language = {en},
     url = {}
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.

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