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

 $L\stackrel{^}{y}=0,\phantom{\rule{1em}{0ex}}\stackrel{^}{y}{|}_{\left\{0\right\}×ℝ}=\stackrel{^}{u}.$
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:
 $y\left(t,0\right)=\stackrel{^}{y}\left(t,0\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall t\in \left[0,T\right].$
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 $\stackrel{^}{y}{|}_{\left[0,T\right]×\left\{0\right\}}$ 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].

DOI : https://doi.org/10.1051/cocv:2002040
Classification : 93C20,  65M32,  62P05,  91B28
Mots clés : inverse problems, Carleman estimates, barrier option hedging, replication
@article{COCV_2002__8__127_0,
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 = {http://www.numdam.org/item/COCV_2002__8__127_0/}
}
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. http://www.numdam.org/item/COCV_2002__8__127_0/

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