Geodesic
Numerical approximations of McKean anticipative backward stochastic differential equations arising in initial margin requirements
A. Agarwal1 ; S. De Marco2 ; E. Gobet3 ; J. G. López-Salas4 ; F. Noubiagain5 ; A. Zhou6
1 Adam Smith Business School, University of Glasgow, University Avenue, G12 8QQ Glasgow, United Kingdom
2 Centre de Mathématiques Appliqués (CMAP), Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
3 CMAP, Ecole Polytechnique
4 Department of Mathematics, Faculty of Informatics, Universidade da Coruña, Campus de Elviña s/n, 15071 - A Coruña, Spain
5 This author research was conducted while at Laboratoire Manceau de Mathématiques, Le Mans Université, France
6 This author research was conducted while at Université Paris-Est, CERMICS (ENPC), 77455, Marne-la-Vallé, France
ESAIM. Proceedings, Tome 65 (2019), pp. 1-26.

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We introduce a new class of anticipative backward stochastic differential equations with a dependence of McKean type on the law of the solution, that we name MKABSDE. We provide existence and uniqueness results in a general framework with relatively general regularity assumptions on the coefficients. We show how such stochastic equations arise within the modern paradigm of derivative pricing where a central counterparty (CCP) requires the members to deposit variation and initial margins to cover their exposure. In the case when the initial margin is proportional to the Conditional Value-at-Risk (CVaR) of the contract price, we apply our general result to define the price as a solution of a MKABSDE. We provide several linear and non-linear simpler approximations, which we solve using different numerical (deterministic and Monte-Carlo) methods.
DOI : 10.1051/proc/201965001

A. Agarwal 1 ; S. De Marco 2 ; E. Gobet 3 ; J. G. López-Salas 4 ; F. Noubiagain 5 ; A. Zhou 6

1 Adam Smith Business School, University of Glasgow, University Avenue, G12 8QQ Glasgow, United Kingdom
2 Centre de Mathématiques Appliqués (CMAP), Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
3 CMAP, Ecole Polytechnique
4 Department of Mathematics, Faculty of Informatics, Universidade da Coruña, Campus de Elviña s/n, 15071 - A Coruña, Spain
5 This author research was conducted while at Laboratoire Manceau de Mathématiques, Le Mans Université, France
6 This author research was conducted while at Université Paris-Est, CERMICS (ENPC), 77455, Marne-la-Vallé, France 
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     author = {A. Agarwal and S. De Marco and E. Gobet and J. G. L\'opez-Salas and F. Noubiagain and A. Zhou},
     title = {Numerical approximations of {McKean} anticipative backward stochastic differential equations arising in initial margin requirements},
     journal = {ESAIM. Proceedings},
     pages = {1--26},
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     volume = {65},
     year = {2019},
     doi = {10.1051/proc/201965001},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201965001/}
}
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A. Agarwal; S. De Marco; E. Gobet; J. G. López-Salas; F. Noubiagain; A. Zhou. Numerical approximations of McKean anticipative backward stochastic differential equations arising in initial margin requirements. ESAIM. Proceedings, Tome 65 (2019), pp. 1-26. doi : 10.1051/proc/201965001. http://geodesic.mathdoc.fr/articles/10.1051/proc/201965001/

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