Static Hedging of Barrier Options with a Smile: An Inverse Problem

Claude Bardos; Raphaël Douady; Andrei Fursikov

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

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Let L be a parabolic second order differential operator on the domain Π ¯ = 0 , T × ℝ . Given a function u ^ : ℝ → R and x ^ > 0 such that the support of û is contained in ( - ∞ , - x ^ ] , we let y ^ : Π ¯ → ℝ be the solution to the equation: L y ^ = 0 , y ^ | { 0 } × ℝ = u ^ . Given positive bounds 0 < x 0 < x 1 , we seek a function u with support in x 0 , x 1 such that the corresponding solution y satisfies: y ( t , 0 ) = y ^ ( 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 y ^ | [ 0 , T ] × { 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].


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 - 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
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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