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

Bardos, Claude; Douady, Raphaël; Fursikov, Andrei

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 $\overline{\Pi }=\left[0,T\right]\times ℝ.$ Given a function $\widehat{u}:ℝ\to R$ and $\widehat{x}>0$ such that the support of $\widehat{u}$ is contained in $(-\infty ,-\widehat{x}]$, we let $\widehat{y}:\overline{\Pi }\to ℝ$ be the solution to the equation:

$$L\widehat{y}=0,\widehat{y}{|}_{\left\{0\right\}\times ℝ}=\widehat{u}.$$

Given positive bounds $0<x_0<x_1,$ we seek a function $u$ with support in $\left[x_0,x_1\right]$ such that the corresponding solution $y$ satisfies:

$$y(t,0)=\widehat{y}(t,0)\forall t\in \left[0,T\right].$$

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 $\widehat{y}{|}_{[0,T]\times \left\{0\right\}}$ 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].

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