# 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 û 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 C0-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 (2010): 127-142. <http://eudml.org/doc/90642>.

@article{Bardos2010,
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 $\hat\{x\}>0$ such that the support of û is contained in $(-\infty ,-\hat\{x\}]$, we let $\hat\{y\}:\bar\{\Pi\}\rightarrow \{\mathbb R\}$ be the solution to the equation: $L\hat\{y\}=0,\text\{\quad \}\hat\{y\}|\_\{\\{0\\}\times \{\mathbb R\}\}=\hat\{u\} .$ 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: $y(t,0)=\hat\{y\}(t,0)\quad \quad \forall t\in \left[ 0,T\right] .$ 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 $\hat\{y\}|_\{[0,T]\times \\{0\\}\}$ can be C0-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.; inverse problems; replication},
language = {eng},
month = {3},
pages = {127-142},
publisher = {EDP Sciences},
title = {Static Hedging of Barrier Options with a Smile: An Inverse Problem},
url = {http://eudml.org/doc/90642},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Bardos, Claude
AU - Fursikov, Andrei
TI - Static Hedging of Barrier Options with a Smile: An Inverse Problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
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 $\hat{x}>0$ such that the support of û is contained in $(-\infty ,-\hat{x}]$, we let $\hat{y}:\bar{\Pi}\rightarrow {\mathbb R}$ be the solution to the equation: $L\hat{y}=0,\text{\quad }\hat{y}|_{\{0\}\times {\mathbb R}}=\hat{u} .$ 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: $y(t,0)=\hat{y}(t,0)\quad \quad \forall t\in \left[ 0,T\right] .$ 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 $\hat{y}|_{[0,T]\times \{0\}}$ can be C0-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.; inverse problems; replication
UR - http://eudml.org/doc/90642
ER -

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