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International audienceLet L be a parabolic second order differential operator on the domain ¯ Π = [0, T ] ×  ℝ. Given a function û :  ℝ → R and ^x > 0 such that the support of of û is contained in (−∞, −ˆx], we let ˆy  : ¯ Π → Ê be the solution to the equation: Lˆy= 0,  ^ y| {0}× ℝ = û. Given positive bounds 0 < x0 < x1, we seek a function u with support in [x0, x1] such that the corresponding solution y satisfies: y(t, 0) = ˆy(t, 0) ∀t ∈ [0, 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 ˆy| [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]

Static Hedging of Barrier Options with a Smile: An Inverse Problem