Static Hedging of Barrier Options with a Smile: An Inverse Problem
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 {\ma...
Gespeichert in:
| Veröffentlicht in: | ESAIM. Control, optimisation and calculus of variations optimisation and calculus of variations, 2002-01, Vol.8, p.127-142 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 142 |
|---|---|
| container_issue | |
| container_start_page | 127 |
| container_title | ESAIM. Control, optimisation and calculus of variations |
| container_volume | 8 |
| creator | Bardos, Claude Douady, Raphaël Fursikov, Andrei |
| description | 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 |
| doi_str_mv | 10.1051/cocv:2002040 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_hal_p</sourceid><recordid>TN_cdi_hal_primary_oai_HAL_hal_01477102v1</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2511339191</sourcerecordid><originalsourceid>FETCH-LOGICAL-c372t-5024331e6dc3aa82318491a45d30ee9086792ae640ae83e39deac51627e779f3</originalsourceid><addsrcrecordid>eNo90E1Lw0AQBuAgCtbqzR-weBOMzu4m2Wxvbf1ooVCxQY_LmkzarWlSd9Oq_96ElJ5mGB5ehtfzrincUwjpQ1ql-wEDYBDAidejLGI-50KctrtkfkypPPcunFsD0IgHQc8bLWpdm5RMMFuackmqnIy0tQYtmW9rU5WO_Jh6RTRZbEyBAzIsybTco3VIXm31WeDm0jvLdeHw6jD7XvL8lIwn_mz-Mh0PZ37KBav9EFjAOcUoS7nWMeM0DiTVQZhxQJQQR0IyjVEAGmOOXGao05BGTKAQMud977aLXelCba3ZaPunKm3UZDhT7Q1oIAQFtqeNvens1lbfO3S1Wlc7WzbfKQk8lmFjG3TXodRWzlnMj6kUVNunavtUhz4b7nfcuBp_j1bbLxUJLkIVw4dio-Tx7T1cqIT_A210dHM</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>903895014</pqid></control><display><type>article</type><title>Static Hedging of Barrier Options with a Smile: An Inverse Problem</title><source>EZB-FREE-00999 freely available EZB journals</source><source>Alma/SFX Local Collection</source><source>NUMDAM</source><creator>Bardos, Claude ; Douady, Raphaël ; Fursikov, Andrei</creator><creatorcontrib>Bardos, Claude ; Douady, Raphaël ; Fursikov, Andrei</creatorcontrib><description>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].</description><identifier>ISSN: 1292-8119</identifier><identifier>EISSN: 1262-3377</identifier><identifier>DOI: 10.1051/cocv:2002040</identifier><language>eng</language><publisher>Les Ulis: EDP Sciences</publisher><subject>62P05 ; 65M32 ; 91B28 ; 93C20 ; barrier option hedging ; Boundary conditions ; Carleman estimates ; Computational Finance ; Hedging ; Interest rates ; Inverse problems ; Pricing of Securities ; Quantitative Finance ; replication</subject><ispartof>ESAIM. Control, optimisation and calculus of variations, 2002-01, Vol.8, p.127-142</ispartof><rights>EDP Sciences, SMAI, 2002</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c372t-5024331e6dc3aa82318491a45d30ee9086792ae640ae83e39deac51627e779f3</citedby><cites>FETCH-LOGICAL-c372t-5024331e6dc3aa82318491a45d30ee9086792ae640ae83e39deac51627e779f3</cites><orcidid>0000-0003-4931-1806 ; 0000-0002-1890-3801</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,780,784,885,27924,27925</link.rule.ids><backlink>$$Uhttps://hal.science/hal-01477102$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Bardos, Claude</creatorcontrib><creatorcontrib>Douady, Raphaël</creatorcontrib><creatorcontrib>Fursikov, Andrei</creatorcontrib><title>Static Hedging of Barrier Options with a Smile: An Inverse Problem</title><title>ESAIM. Control, optimisation and calculus of variations</title><description>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].</description><subject>62P05</subject><subject>65M32</subject><subject>91B28</subject><subject>93C20</subject><subject>barrier option hedging</subject><subject>Boundary conditions</subject><subject>Carleman estimates</subject><subject>Computational Finance</subject><subject>Hedging</subject><subject>Interest rates</subject><subject>Inverse problems</subject><subject>Pricing of Securities</subject><subject>Quantitative Finance</subject><subject>replication</subject><issn>1292-8119</issn><issn>1262-3377</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2002</creationdate><recordtype>article</recordtype><recordid>eNo90E1Lw0AQBuAgCtbqzR-weBOMzu4m2Wxvbf1ooVCxQY_LmkzarWlSd9Oq_96ElJ5mGB5ehtfzrincUwjpQ1ql-wEDYBDAidejLGI-50KctrtkfkypPPcunFsD0IgHQc8bLWpdm5RMMFuackmqnIy0tQYtmW9rU5WO_Jh6RTRZbEyBAzIsybTco3VIXm31WeDm0jvLdeHw6jD7XvL8lIwn_mz-Mh0PZ37KBav9EFjAOcUoS7nWMeM0DiTVQZhxQJQQR0IyjVEAGmOOXGao05BGTKAQMud977aLXelCba3ZaPunKm3UZDhT7Q1oIAQFtqeNvens1lbfO3S1Wlc7WzbfKQk8lmFjG3TXodRWzlnMj6kUVNunavtUhz4b7nfcuBp_j1bbLxUJLkIVw4dio-Tx7T1cqIT_A210dHM</recordid><startdate>20020101</startdate><enddate>20020101</enddate><creator>Bardos, Claude</creator><creator>Douady, Raphaël</creator><creator>Fursikov, Andrei</creator><general>EDP Sciences</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0003-4931-1806</orcidid><orcidid>https://orcid.org/0000-0002-1890-3801</orcidid></search><sort><creationdate>20020101</creationdate><title>Static Hedging of Barrier Options with a Smile: An Inverse Problem</title><author>Bardos, Claude ; Douady, Raphaël ; Fursikov, Andrei</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c372t-5024331e6dc3aa82318491a45d30ee9086792ae640ae83e39deac51627e779f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2002</creationdate><topic>62P05</topic><topic>65M32</topic><topic>91B28</topic><topic>93C20</topic><topic>barrier option hedging</topic><topic>Boundary conditions</topic><topic>Carleman estimates</topic><topic>Computational Finance</topic><topic>Hedging</topic><topic>Interest rates</topic><topic>Inverse problems</topic><topic>Pricing of Securities</topic><topic>Quantitative Finance</topic><topic>replication</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bardos, Claude</creatorcontrib><creatorcontrib>Douady, Raphaël</creatorcontrib><creatorcontrib>Fursikov, Andrei</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>ESAIM. Control, optimisation and calculus of variations</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bardos, Claude</au><au>Douady, Raphaël</au><au>Fursikov, Andrei</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Static Hedging of Barrier Options with a Smile: An Inverse Problem</atitle><jtitle>ESAIM. Control, optimisation and calculus of variations</jtitle><date>2002-01-01</date><risdate>2002</risdate><volume>8</volume><spage>127</spage><epage>142</epage><pages>127-142</pages><issn>1292-8119</issn><eissn>1262-3377</eissn><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].</abstract><cop>Les Ulis</cop><pub>EDP Sciences</pub><doi>10.1051/cocv:2002040</doi><tpages>16</tpages><orcidid>https://orcid.org/0000-0003-4931-1806</orcidid><orcidid>https://orcid.org/0000-0002-1890-3801</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1292-8119 |
| ispartof | ESAIM. Control, optimisation and calculus of variations, 2002-01, Vol.8, p.127-142 |
| issn | 1292-8119 1262-3377 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_01477102v1 |
| source | EZB-FREE-00999 freely available EZB journals; Alma/SFX Local Collection; NUMDAM |
| subjects | 62P05 65M32 91B28 93C20 barrier option hedging Boundary conditions Carleman estimates Computational Finance Hedging Interest rates Inverse problems Pricing of Securities Quantitative Finance replication |
| title | Static Hedging of Barrier Options with a Smile: An Inverse Problem |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-27T11%3A23%3A19IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_hal_p&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Static%20Hedging%20of%20Barrier%20Options%20with%20a%20Smile:%20An%20Inverse%20Problem&rft.jtitle=ESAIM.%20Control,%20optimisation%20and%20calculus%20of%20variations&rft.au=Bardos,%20Claude&rft.date=2002-01-01&rft.volume=8&rft.spage=127&rft.epage=142&rft.pages=127-142&rft.issn=1292-8119&rft.eissn=1262-3377&rft_id=info:doi/10.1051/cocv:2002040&rft_dat=%3Cproquest_hal_p%3E2511339191%3C/proquest_hal_p%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=903895014&rft_id=info:pmid/&rfr_iscdi=true |