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

• Volume: 8, page 127-142
• ISSN: 1292-8119

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## Abstract

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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:$L\stackrel{^}{y}=0,\phantom{\rule{1.0em}{0ex}}\stackrel{^}{y}{|}_{\left\{0\right\}×ℝ}=\stackrel{^}{u}.$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:$y\left(t,0\right)=\stackrel{^}{y}\left(t,0\right)\phantom{\rule{1.0em}{0ex}}\phantom{\rule{1.0em}{0ex}}\forall t\in \left[0,T\right].$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].

## How to cite

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Bardos, Claude, Douady, Raphaël, and Fursikov, Andrei. "Static hedging of barrier options with a smile : an inverse problem." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 127-142. <http://eudml.org/doc/245707>.

@article{Bardos2002,
abstract = {Let $L$ be a parabolic second order differential operator on the domain $\bar\{\Pi \}=\left[ 0,T\right] \times \{\mathbb \{R\}\}.$ Given a function $\hat\{u\}:\{\mathbb \{R\}\rightarrow R\}$ and $\hat\{x\}&gt;0$ such that the support of $\hat\{u\}$ is contained in $(-\infty ,-\hat\{x\}]$, we let $\hat\{y\}:\bar\{\Pi \}\rightarrow \{\mathbb \{R\}\}$ be the solution to the equation:$L\hat\{y\}=0,\quad \hat\{y\}|\_\{\lbrace 0\rbrace \times \{\mathbb \{R\}\}\}=\hat\{u\} .$Given positive bounds $0&lt;x_\{0\}&lt;x_\{1\},$ we seek a function $u$ with support in $\left[ x_\{0\},x_\{1\}\right]$ such that the corresponding solution $y$ satisfies:$y(t,0)=\hat\{y\}(t,0)\quad \quad \forall t\in \left[ 0,T\right] .$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 $\hat\{y\}|_\{[0,T]\times \lbrace 0\rbrace \}$ 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].},
author = {Bardos, Claude, Douady, Raphaël, Fursikov, Andrei},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {inverse problems; Carleman estimates; barrier option hedging; replication},
language = {eng},
pages = {127-142},
publisher = {EDP-Sciences},
title = {Static hedging of barrier options with a smile : an inverse problem},
url = {http://eudml.org/doc/245707},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Bardos, Claude
AU - Douady, Raphaël
AU - Fursikov, Andrei
TI - Static hedging of barrier options with a smile : an inverse problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 127
EP - 142
AB - Let $L$ be a parabolic second order differential operator on the domain $\bar{\Pi }=\left[ 0,T\right] \times {\mathbb {R}}.$ Given a function $\hat{u}:{\mathbb {R}\rightarrow R}$ and $\hat{x}&gt;0$ such that the support of $\hat{u}$ is contained in $(-\infty ,-\hat{x}]$, we let $\hat{y}:\bar{\Pi }\rightarrow {\mathbb {R}}$ be the solution to the equation:$L\hat{y}=0,\quad \hat{y}|_{\lbrace 0\rbrace \times {\mathbb {R}}}=\hat{u} .$Given positive bounds $0&lt;x_{0}&lt;x_{1},$ we seek a function $u$ with support in $\left[ x_{0},x_{1}\right]$ such that the corresponding solution $y$ satisfies:$y(t,0)=\hat{y}(t,0)\quad \quad \forall t\in \left[ 0,T\right] .$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 $\hat{y}|_{[0,T]\times \lbrace 0\rbrace }$ 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].
LA - eng
KW - inverse problems; Carleman estimates; barrier option hedging; replication
UR - http://eudml.org/doc/245707
ER -

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