{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/33917"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/33917","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Numerical Analysis of Jump-Diffusion Models for Option Pricing","abstract":"Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized.","abstract_html":"Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized.","abstract_has_math":true,"creators":["Strauss, Arne Karsten"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Sachs, Ekkehard W."],"committee_members":["Beattie, Christopher A.","Adjerid, Slimane"],"year":2006,"date_issued":"2006-07-07","date_published":"2006-07-07","updated_at":"2026-07-22T22:19:10Z","subjects":["Jump-diffusion processes","Option pricing","Finite differences","Fast Fourier Transform","Conjugate Gradient method"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-07072006-154259"],"render_values":[{"text":"etd-07072006-154259","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/33917","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Sachs, Ekkehard W."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Beattie, Christopher A.","Adjerid, Slimane"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Strauss, Arne Karsten"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T20:41:10Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T20:41:10Z","2006-09-15"]},{"key":"dc:date.issued","label":"Date","values":["2006-07-07"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Jump-diffusion processes","Option pricing","Finite differences","Fast Fourier Transform","Conjugate Gradient method"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-07072006-154259"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/33917"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:title","label":"Title","values":["Numerical Analysis of Jump-Diffusion Models for Option Pricing"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Sachs, Ekkehard W."],"dc:contributor.committeemember":["Beattie, Christopher A.","Adjerid, Slimane"],"dc:contributor.department":["Mathematics"],"dc:creator":["Strauss, Arne Karsten"],"dc:date.accessioned":["2014-03-14T20:41:10Z"],"dc:date.available":["2014-03-14T20:41:10Z","2006-09-15"],"dc:date.issued":["2006-07-07"],"dc:description.abstract":["Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized."],"dc:description.degree":["Master of Science"],"dc:identifier.other":["etd-07072006-154259"],"dc:identifier.uri":["http://hdl.handle.net/10919/33917"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Jump-diffusion processes","Option pricing","Finite differences","Fast Fourier Transform","Conjugate Gradient method"],"dc:title":["Numerical Analysis of Jump-Diffusion Models for Option Pricing"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:10Z"}