STATIC HEDGING OF BARRIER OPTIONS WITH A SMILE: AN INVERSE PROBLEM

Claude Bardos (1) , Raphaël Douady (2) , Andrei Fursikov (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CES - Centre d'économie de la Sorbonne
3 Moscow State University

Let 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].

Mots clés: Barrier options; inverse problem

Domaines: Finance quantitative [q-fin.CP]; Pricing [q-fin.PR]

Claude Bardos, Raphaël Douady, Andrei Fursikov. STATIC HEDGING OF BARRIER OPTIONS WITH A SMILE: AN INVERSE PROBLEM. ESAIM: Control, Optimisation and Calculus of Variations, 2002, 8, pp.127-142. ⟨10.1051/cocv:2002040⟩. ⟨hal-01477102⟩