Numerical approximations of McKean Anticipative Backward Stochastic Differential Equations arising in Initial Margin requirements

TL;DR

This paper introduces a new class of anticipative McKean-type backward stochastic differential equations (MKABSDE) relevant for initial margin calculations in derivative pricing, providing existence, uniqueness, and numerical approximation methods.

Contribution

The paper defines MKABSDE, proves their well-posedness under general conditions, and develops numerical schemes for their approximation in the context of initial margin requirements.

Findings

Established existence and uniqueness of MKABSDE solutions.

Developed linear and non-linear numerical approximation methods.

Applied the framework to initial margin calculations using CVaR.

Abstract

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.