Volume 65, 2019CEMRACS 2017 - Numerical methods for stochastic models: control, uncertainty quantification, mean-field
|Page(s)||1 - 26|
|Published online||02 April 2019|
Numerical approximations of McKean anticipative backward stochastic differential equations arising in initial margin requirements*
Adam Smith Business School, University of Glasgow, University Avenue, G12 8QQ Glasgow, United Kingdom
e-mail: firstname.lastname@example.org. This author research was conducted while at CMAP, Ecole Polytechnique, and is part of the Chaire Risque Financiers
2 Centre de Mathématiques Appliqués (CMAP), Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
3 CMAP, Ecole Polytechnique
e-mail: email@example.com. This author research is part of the Chaire Risque Financiers of the Fondation du Risque and of the Finance for Energy Market Research Centre
4 Department of Mathematics, Faculty of Informatics, Universidade da Coruña, Campus de Elviña s/n, 15071 - A Coruña, Spain
e-mail: firstname.lastname@example.org. This author research was conducted while at CMAP, Ecole Polytechnique, and is part of the Chaire Risque Financiers
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
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.
Mathematics Subject Classification: 60H30 / 65C05 / 65C30
Key words: non-linear pricing / CVaR initial margins / anticipative BSDE / weak non-linearity
© EDP Sciences, SMAI 2019
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