Mathematical Foundations of Regression Methods for Approximating the Forward Initial Margin

In recent years, forward initial margin has attracted the attention of practitioners and academics. From a computational point of view, it poses a challenging problem as it requires the implementation of a nested Monte Carlo simulation. Abundant literature has been published on approximation methods aiming to reduce the dimensionality of the problem, the most popular ones being the family of regression methods.This paper describes the mathematical foundations, on which these regression approximation methods lie. Mathematical rigor is introduced to show that, in essence, all methods are just different variations of approximating the conditional expectation function. In short, all methods are performing orthogonal projections on Hilbert spaces, while simply choosing a different functional form to numerically estimate the conditional expectation.The most popular methods in the literature so far are covered here. These are polynomial approximations, Kernel regressions and Neural Networks