{"id":{"repo_id":"rosario","oai_identifier":"oai:repository.urosario.edu.co:10336/14430"},"canonical_url":"https://search.dev.ndltd.org/etd/rosario/oai:repository.urosario.edu.co:10336/14430","repository":{"repo_id":"rosario","name":"Universidad del Rosario","base_url":"https://repository.urosario.edu.co/oai/request"},"display":{"title":"Numerical Solutions to PDE Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs","abstract":"The purpose of this paper is to present numerical solutions to PDE representations for derivatives pricing including bilateral credit valuation adjustments and funding costs valuation adjustment as presented in Burgard and Kjaer (2011). In particular, we use Crank-Nicolson finite-difference scheme to solve Black-Scholes risk-free PDE, for European and American options, and show how this numerical solution approach is extendable to solve the risky PDE for the value of the same derivative using the same finite-difference scheme and algorithm. Also, we present numerical solutions to valuation adjustments derived from PDE representations for European options through Monte Carlo simulation and numerical integration and we explore an empirical approach for American options through Monte Carlo simulation, least-squares and numerical integration.","abstract_html":"The purpose of this paper is to present numerical solutions to PDE representations for derivatives pricing including bilateral credit valuation adjustments and funding costs valuation adjustment as presented in Burgard and Kjaer (2011). In particular, we use Crank-Nicolson finite-difference scheme to solve Black-Scholes risk-free PDE, for European and American options, and show how this numerical solution approach is extendable to solve the risky PDE for the value of the same derivative using the same finite-difference scheme and algorithm. Also, we present numerical solutions to valuation adjustments derived from PDE representations for European options through Monte Carlo simulation and numerical integration and we explore an empirical approach for American options through Monte Carlo simulation, least-squares and numerical integration.","abstract_has_math":false,"creators":["Torres Laserna, Nicolas"],"institution":"Universidad del Rosario","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-02-15","date_published":"2018-02-15","updated_at":"2026-07-27T20:46:13Z","subjects":["Counterparty risk","Funding Costs","CVA","FVA","PDEs","Finite-Differences","Monte Carlo","Numerical Integration","Least-Squares","Derivatives","Options","Collateral Agreements","Análisis","Ecuaciones diferenciales","Preci::Soluciones numéricas"],"languages":["spa"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":["http://creativecommons.org/licenses/by-nc-sa/2.5/co/"],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["http://repository.urosario.edu.co/handle/10336/14430"],"render_values":[{"text":"http://repository.urosario.edu.co/handle/10336/14430","href":"http://repository.urosario.edu.co/handle/10336/14430","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.48713/10336_14430","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Torres Laserna, Nicolas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-02-15","2018-02-22T12:13:34Z"]},{"key":"dc:publisher","label":"Institution","values":["Universidad del Rosario","Facultad de Economía","Maestría en Finanzas Cuantitativas"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/masterThesis","info:eu-repo/semantics/acceptedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Counterparty risk","Funding Costs","CVA","FVA","PDEs","Finite-Differences","Monte Carlo","Numerical Integration","Least-Squares","Derivatives","Options","Collateral Agreements","Análisis","Ecuaciones diferenciales","Preci::Soluciones numéricas"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["spa"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess","http://creativecommons.org/licenses/by-nc-sa/2.5/co/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://doi.org/10.48713/10336_14430","http://repository.urosario.edu.co/handle/10336/14430"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The purpose of this paper is to present numerical solutions to PDE representations for derivatives pricing including bilateral credit valuation adjustments and funding costs valuation adjustment as presented in Burgard and Kjaer (2011). In particular, we use Crank-Nicolson finite-difference scheme to solve Black-Scholes risk-free PDE, for European and American options, and show how this numerical solution approach is extendable to solve the risky PDE for the value of the same derivative using the same finite-difference scheme and algorithm. Also, we present numerical solutions to valuation adjustments derived from PDE representations for European options through Monte Carlo simulation and numerical integration and we explore an empirical approach for American options through Monte Carlo simulation, least-squares and numerical integration."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:source","label":"Dc Source","values":["Alavian, S., Ding, j., Whitehead, P., and Laudicina, L. (2008). Credit Valuation Ad- justment (CVA). https://papers.ssrn.com/sol3/papers.cfm?abstractid = 1310226:","Brigo, D. and Capponi, A. (2009). Bilateral counterparty risk valuation with stochastic dynamical models and application to Credit Default Swaps. https://arxiv.org/pdf/0812.3705.pdf.","Brigo, D., Pallavicini, A., and Papatheodorou, V. (2009). Bilateral counterparty risk valuation for interest rate products: impact of volatilities and correlations. https://arxiv.org/pdf/0812.3705.pdf.","Burgard, C. and Kjaer, M. (2011a). In The Balance. https://papers.ssrn.com/sol3/papers.cfm?abstractid = 1785262:","Burgard, C. and Kjaer, M. (2011b). Partial di erential equation representations of deriva- tives with bilateral counterparty risk and funding costs. The Journal of Credit Risk, 7(3):75{93.","Burgard, C. and Kjaer, M. (2012). CVA and FVA with funding aware close outs. https://papers.ssrn.com/sol3/papers.cfm?abstractid = 2157631:","Burgard, C. and Kjaer, M. (2017). Derivatives Funding, Netting and Accounting. https://papers.ssrn.com/sol3/papers.cfm?abstractid = 2534011:","Crank, J. (1984). Free and Moving Boundary Problems. Clarendon Press, Oxford.","Cryer, C. (1979). Successive overrelaxation methods for solving linear complementarity problems arising from free boundary value problems. Presented at a seminar held in Pavia (Italy), SeptemberOctober 1979, Roma 1980.","Du y, D. (2004). A Critique of the Crank-Nicolson Scheme: Strengths and Weaknesses for Financial Instrument Pricing. WILMOTT magazine.","Du y, D. (2006). Finite Di erence Methods for Financial Engineering: A Partial Di eren- tial Equation Approach. John Wiley Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England.","Facchini, M. (2013). Pricing derivatives under CVA, DVA and funding costs. Master's thesis, University of Amsterdam - Faculty of Science - Sthochastics and Financial Math- ematics.","Feynman-Kac formula (2017). Feynman-Kac formula | Hochschule RheinMain Wies- baden, University of Applied Sciences, Lectures, Chapter 17. [Online; accessed 14- January-2017].","Fitch Ratings (2017). Structured Finance and Covered Bonds Counterparty Criteria: Derivative Addendum. https://www. tchratings.com/site/re/898538.","Green, A. (2016). XVA: Credit, Funding and Capital Valuation Adjustments. John Wiley Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England.","Gregory, J. (2009). Being Two-faced over Counterparty Credit Risk. Risk.","Gregory, J. (2015). The XVA Challenge: Counterparty Credit Risk, Funding, Collateral and Capital. John Wiley Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, United Kingdom.","Johnson, P. (2013). Finite Di erence - Crank-Nicolson. http://www.maths.manchester.ac.uk/ pjohnson/resources/math60082/lecture- nite- di erence-crank.pdf.","Karatzas, I. and Shreve, S. (1998). Brownian Motion and Stochastic Calculus. Springer Science+Business Media,LLC, 233 Spring ST New York, NY 10013, USA.","LeVeque, R. (2007). Finite di erence methods for ordinary and partial di erential equa- tions: steady state and time-dependent problems. Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA,19104, USA.","Longsta , F. and Schwartz, E. (2001). Valuing American Options by Simulation: A Simple Least-Squares Approach. The Review of Financial Studies, 14(1):113{147.","Moreno, M. and Navas, J. (2003). On the Robustness of Least-Squares for Pricing American Derivatives. Review of Derivatives Research, 6(2).","Piessens, R., deDonckerKapenga, E., Uberhuber, C., and Kahane, D. (1983). Quadpack: a Subroutine Package for Automatic Integration. R package version 3.2.5 | For new features, see the 'Changelog' le (in the package source).","Piterbarg, V. (2010). Funding beyond discounting: Collateral agreements and derivatives pricing. Risk, 2:97{102.","R (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. R version 3.2.5 (2016-04-14).","Salsa, S. (2008). Partial Di erential Equations in Action: From Modelling to Theory. Springer Science+Business Media,LLC, Springer-Verlag Italia, Milano, Italy.","Shreve, S. (2004). Stochastic Calculus for Finance: Continuous Time Models, Vol II. Springer Science+Business Media,LLC, 233 Spring ST New York, NY 10013, USA.","Smith, G. (1985). Numerical Solution of Partial Di erential Equations: Finite Di erence Methods. Oxford University Press, UK.","Thomas, J. (1998). Numerical Partial Di erential Equations, Volume I. Finite Di erence Methods. Springer, 233 Spring ST New York, NY 10013, USA.","Thomas, J. (1999). Numerical Partial Di erential Equations,Volume II. Conversation Laws and Elliptic Equations. Springer, 233 Spring ST New York, NY 10013, USA.","Venegas, F. (2008). Riesgos Financieros y Economicos: Productos derivados y decisiones economicas bajo incertidumbre. Cencage Learning Editores S.A. de C.V., Corporativo Santa Fe, Av Santa Fe 505, P12 Col. Cruz Manca, Santa Fe, C.P.05349, Mexico, D.F.","Wilmott, P. (2006). Paul Wilmott on Quantitative Finance. John Wiley Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England.","Wilmott, P., Howinson, S., and Dewynne, J. (1995). The Mathematics of Financial Deriva- tives. Press Syndicate of the University of Cambridge, The Pitt Building, Trumpington Street, Cambridge CB2 1RP, UK.","instname:Universidad del Rosario","reponame:Repositorio Institucional EdocUR"]},{"key":"dc:title","label":"Title","values":["Numerical Solutions to PDE Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs"]}]}],"canonical_facts":{"dc:creator":["Torres Laserna, Nicolas"],"dc:date":["2018-02-15","2018-02-22T12:13:34Z"],"dc:description":["The purpose of this paper is to present numerical solutions to PDE representations for derivatives pricing including bilateral credit valuation adjustments and funding costs valuation adjustment as presented in Burgard and Kjaer (2011). In particular, we use Crank-Nicolson finite-difference scheme to solve Black-Scholes risk-free PDE, for European and American options, and show how this numerical solution approach is extendable to solve the risky PDE for the value of the same derivative using the same finite-difference scheme and algorithm. Also, we present numerical solutions to valuation adjustments derived from PDE representations for European options through Monte Carlo simulation and numerical integration and we explore an empirical approach for American options through Monte Carlo simulation, least-squares and numerical integration."],"dc:format":["application/pdf"],"dc:identifier":["https://doi.org/10.48713/10336_14430","http://repository.urosario.edu.co/handle/10336/14430"],"dc:language":["spa"],"dc:publisher":["Universidad del Rosario","Facultad de Economía","Maestría en Finanzas Cuantitativas"],"dc:rights":["info:eu-repo/semantics/openAccess","http://creativecommons.org/licenses/by-nc-sa/2.5/co/"],"dc:source":["Alavian, S., Ding, j., Whitehead, P., and Laudicina, L. (2008). Credit Valuation Ad- justment (CVA). https://papers.ssrn.com/sol3/papers.cfm?abstractid = 1310226:","Brigo, D. and Capponi, A. (2009). Bilateral counterparty risk valuation with stochastic dynamical models and application to Credit Default Swaps. https://arxiv.org/pdf/0812.3705.pdf.","Brigo, D., Pallavicini, A., and Papatheodorou, V. (2009). Bilateral counterparty risk valuation for interest rate products: impact of volatilities and correlations. https://arxiv.org/pdf/0812.3705.pdf.","Burgard, C. and Kjaer, M. (2011a). In The Balance. https://papers.ssrn.com/sol3/papers.cfm?abstractid = 1785262:","Burgard, C. and Kjaer, M. (2011b). Partial di erential equation representations of deriva- tives with bilateral counterparty risk and funding costs. The Journal of Credit Risk, 7(3):75{93.","Burgard, C. and Kjaer, M. (2012). CVA and FVA with funding aware close outs. https://papers.ssrn.com/sol3/papers.cfm?abstractid = 2157631:","Burgard, C. and Kjaer, M. (2017). Derivatives Funding, Netting and Accounting. https://papers.ssrn.com/sol3/papers.cfm?abstractid = 2534011:","Crank, J. (1984). Free and Moving Boundary Problems. Clarendon Press, Oxford.","Cryer, C. (1979). Successive overrelaxation methods for solving linear complementarity problems arising from free boundary value problems. Presented at a seminar held in Pavia (Italy), SeptemberOctober 1979, Roma 1980.","Du y, D. (2004). A Critique of the Crank-Nicolson Scheme: Strengths and Weaknesses for Financial Instrument Pricing. WILMOTT magazine.","Du y, D. (2006). Finite Di erence Methods for Financial Engineering: A Partial Di eren- tial Equation Approach. John Wiley Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England.","Facchini, M. (2013). Pricing derivatives under CVA, DVA and funding costs. Master's thesis, University of Amsterdam - Faculty of Science - Sthochastics and Financial Math- ematics.","Feynman-Kac formula (2017). Feynman-Kac formula | Hochschule RheinMain Wies- baden, University of Applied Sciences, Lectures, Chapter 17. [Online; accessed 14- January-2017].","Fitch Ratings (2017). Structured Finance and Covered Bonds Counterparty Criteria: Derivative Addendum. https://www. tchratings.com/site/re/898538.","Green, A. (2016). XVA: Credit, Funding and Capital Valuation Adjustments. John Wiley Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England.","Gregory, J. (2009). Being Two-faced over Counterparty Credit Risk. Risk.","Gregory, J. (2015). The XVA Challenge: Counterparty Credit Risk, Funding, Collateral and Capital. John Wiley Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, United Kingdom.","Johnson, P. (2013). Finite Di erence - Crank-Nicolson. http://www.maths.manchester.ac.uk/ pjohnson/resources/math60082/lecture- nite- di erence-crank.pdf.","Karatzas, I. and Shreve, S. (1998). Brownian Motion and Stochastic Calculus. Springer Science+Business Media,LLC, 233 Spring ST New York, NY 10013, USA.","LeVeque, R. (2007). Finite di erence methods for ordinary and partial di erential equa- tions: steady state and time-dependent problems. Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA,19104, USA.","Longsta , F. and Schwartz, E. (2001). Valuing American Options by Simulation: A Simple Least-Squares Approach. The Review of Financial Studies, 14(1):113{147.","Moreno, M. and Navas, J. (2003). On the Robustness of Least-Squares for Pricing American Derivatives. Review of Derivatives Research, 6(2).","Piessens, R., deDonckerKapenga, E., Uberhuber, C., and Kahane, D. (1983). Quadpack: a Subroutine Package for Automatic Integration. R package version 3.2.5 | For new features, see the 'Changelog' le (in the package source).","Piterbarg, V. (2010). Funding beyond discounting: Collateral agreements and derivatives pricing. Risk, 2:97{102.","R (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. R version 3.2.5 (2016-04-14).","Salsa, S. (2008). Partial Di erential Equations in Action: From Modelling to Theory. Springer Science+Business Media,LLC, Springer-Verlag Italia, Milano, Italy.","Shreve, S. (2004). Stochastic Calculus for Finance: Continuous Time Models, Vol II. Springer Science+Business Media,LLC, 233 Spring ST New York, NY 10013, USA.","Smith, G. (1985). Numerical Solution of Partial Di erential Equations: Finite Di erence Methods. Oxford University Press, UK.","Thomas, J. (1998). Numerical Partial Di erential Equations, Volume I. Finite Di erence Methods. Springer, 233 Spring ST New York, NY 10013, USA.","Thomas, J. (1999). Numerical Partial Di erential Equations,Volume II. Conversation Laws and Elliptic Equations. Springer, 233 Spring ST New York, NY 10013, USA.","Venegas, F. (2008). Riesgos Financieros y Economicos: Productos derivados y decisiones economicas bajo incertidumbre. Cencage Learning Editores S.A. de C.V., Corporativo Santa Fe, Av Santa Fe 505, P12 Col. Cruz Manca, Santa Fe, C.P.05349, Mexico, D.F.","Wilmott, P. (2006). Paul Wilmott on Quantitative Finance. John Wiley Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England.","Wilmott, P., Howinson, S., and Dewynne, J. (1995). The Mathematics of Financial Deriva- tives. Press Syndicate of the University of Cambridge, The Pitt Building, Trumpington Street, Cambridge CB2 1RP, UK.","instname:Universidad del Rosario","reponame:Repositorio Institucional EdocUR"],"dc:subject":["Counterparty risk","Funding Costs","CVA","FVA","PDEs","Finite-Differences","Monte Carlo","Numerical Integration","Least-Squares","Derivatives","Options","Collateral Agreements","Análisis","Ecuaciones diferenciales","Preci::Soluciones numéricas"],"dc:title":["Numerical Solutions to PDE Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs"],"dc:type":["info:eu-repo/semantics/masterThesis","info:eu-repo/semantics/acceptedVersion"]},"updated_at":"2026-07-27T20:46:13Z"}