Static hedging of barrier options with a smile: An inverse problem

Abstract

Let L be a parabolic second order differential operator on the domain Π̄ = [0,T] × R. Given a function û: ℝ → R and x̄ > 0 such that the support of ûis contained in (-∞, -x̂], we let ŷ: Π̄ → ℝ be the solution to the equation: Lŷ = 0, ŷ|{o} ×ℝ = û. 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)=ŷ(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 ŷ| [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]. © EDP Sciences, SMAI 2002.