Comptes Rendus
no. 3
Partial differential equations/Mathematical economics
Mathematical analysis of a nonlinear PDE model for European options with counterparty risk

Presented by: Olivier Pironneau

Iñigo Arregui1, 2; Beatriz Salvador1, 2; Daniel Ševčovič3; Carlos Vázquez1, 2, 4
1 Department of Mathematics, University of A Coruña, Campus de Elviña, 15071 A Coruña, Spain
2 CITIC, Campus de Elviña, 15071 A Coruña, Spain
3 Department of Applied Mathematics and Statistics, Comenius University, Mlynska Dolina, 84248 Bratislava, Slovakia
4 ITMATI, Campus Vida, 15782 Santiago de Compostela, Spain
Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 252-257.

In this work, we analyze a nonlinear partial differential equation (PDE) model for the total value adjustment on European options in the presence of a counterparty risk. We transform the nonlinear PDE into an equivalent one, involving a sectorial operator, and prove the existence and uniqueness of a solution.

Dans ce travail, nous analysons un modèle d'équations aux dérivées partielles (EDP) non linéaires pour l'ajustement XVA d'options européennes en présence d'un risque de contrepartie. Nous transformons l'EDP non linéaire en une équation équivalente, impliquant un opérateur sectoriel, et prouvons l'existence et l'unicité de la solution.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.03.001

Iñigo Arregui 1, 2; Beatriz Salvador 1, 2; Daniel Ševčovič 3; Carlos Vázquez 1, 2, 4

1 Department of Mathematics, University of A Coruña, Campus de Elviña, 15071 A Coruña, Spain
2 CITIC, Campus de Elviña, 15071 A Coruña, Spain
3 Department of Applied Mathematics and Statistics, Comenius University, Mlynska Dolina, 84248 Bratislava, Slovakia
4 ITMATI, Campus Vida, 15782 Santiago de Compostela, Spain 
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     author = {I\~nigo Arregui and Beatriz Salvador and Daniel \v{S}ev\v{c}ovi\v{c} and Carlos V\'azquez},
     title = {Mathematical analysis of a nonlinear {PDE} model for {European} options with counterparty risk},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {252--257},
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Iñigo Arregui; Beatriz Salvador; Daniel Ševčovič; Carlos Vázquez. Mathematical analysis of a nonlinear PDE model for European options with counterparty risk. Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 252-257. doi : 10.1016/j.crma.2019.03.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.03.001/

[1] I. Arregui; B. Salvador; C. Vázquez PDE models and numerical methods for total value adjustment in European and American options with counterparty risk, Appl. Math. Comput., Volume 308 (2017), pp. 31-53

[2] I. Arregui; B. Salvador; D. Ševčovič; C. Vázquez Total value adjustment for European options with two stochastic factors. Mathematical model, analysis and numerical simulation, Comput. Math. Appl., Volume 76 (2018), pp. 725-740

[3] C. Burgard; M. Kjaer PDE representations of derivatives with bilateral counterparty risk and funding costs, J. Credit Risk, Volume 7 (2011), pp. 1-19

[4] D. Henry Geometric Theory of Semilinear Parabolic Equations, Springer, 1981

[5] B. Salvador Modelling, mathematical analysis and numerical simulation of problems related to counterparty risk and CVA, Universidade da Coruña, Spain, 2018 (Ph.D. Thesis)

[6] P. Wilmott; S. Howison; J. Dewynne The Mathematics of Financial Derivatives. A Students Introduction, Cambridge University Press, Cambridge, UK, 1996

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