no. 3
Partial differential equations/Mathematical economics
Mathematical analysis of a nonlinear PDE model for European options with counterparty risk
[Analyse mathématique d'un modèle d'EDP non linéaire pour les options européennes avec risque de contrepartie]

Présenté par : Olivier Pironneau

Arregui, Iñigo1, 2 ; Salvador, Beatriz1, 2 ; Ševčovič, Daniel3 ; Vázquez, Carlos1, 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, Tome 357 (2019) no. 3, pp. 252-257.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.03.001
Arregui, Iñigo 1, 2 ; Salvador, Beatriz 1, 2 ; Ševčovič, Daniel 3 ; Vázquez, Carlos 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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     title = {Mathematical analysis of a nonlinear {PDE} model for {European} options with counterparty risk},
     journal = {Comptes Rendus. Math\'ematique},
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Arregui, Iñigo; Salvador, Beatriz; Ševčovič, Daniel; Vázquez, Carlos. Mathematical analysis of a nonlinear PDE model for European options with counterparty risk. Comptes Rendus. Mathématique, Tome 357 (2019) no. 3, pp. 252-257. doi : 10.1016/j.crma.2019.03.001. http://www.numdam.org/articles/10.1016/j.crma.2019.03.001/

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

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

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

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

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