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

Claude BardosRaphaël DouadyAndrei Fursikov — 2002

ESAIM: Control, Optimisation and Calculus of Variations

Let L be a parabolic second order differential operator on the domain Π ¯ = 0 , T × . Given a function u ^ : R and x ^ > 0 such that the support of u ^ is contained in ( - , - x ^ ] , we let y ^ : Π ¯ be the solution to the equation: L y ^ = 0 , y ^ | { 0 } × = 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 } ...

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

Claude BardosRaphaël DouadyAndrei Fursikov — 2010

ESAIM: Control, Optimisation and Calculus of Variations

Let be a parabolic second order differential operator on the domain Π ¯ = 0 , T × . Given a function u ^ : R and x ^ > 0 such that the support of û is contained in ( - , - x ^ ] , we let y ^ : Π ¯ be the solution to the equation: L y ^ = 0 , y ^ | { 0 } × = u ^ . Given positive bounds 0 < x 0 < x 1 , we seek a function with support in x 0 , x 1 such that the corresponding solution satisfies: y ( t , 0 ) = y ^ ( t , 0 ) t 0 , T . We prove in this article that, under some regularity conditions on the coefficients of continuous solutions are unique and dense in the sense that y ^ | [ 0 , T ] × { 0 } can be -approximated, but an exact...

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