X-Valuation adjustments computations by nested simulation on graphics processing units.

Summary This thesis deals with the computation of X-value adjustments, where X includes C for credit, F for financing, M for margin and K for capital. We study different approaches based on nested simulation and implemented on graphics processing units (GPUs). We first consider the problem, for an insurance company or a bank, of numerically computing its economic capital in the form of a value-at-risk or an expected shortfall over a given time horizon. Using a stochastic approximation approach on the value-at-risk or the expected shortfall we establish the convergence of the resulting patterns of the economic capital simulation. Then, we present a nested Monte Carlo (NMC) approach for the computation of XVA. We show that the overall computation of XVAs involves five layers of dependence. The highest layers are run first and trigger nested simulations on the fly if needed to compute an element from a lower layer. Finally, we present a single-layer nested Monte Carlo (1NMC) based algorithm to simulate the U-functions of a Markov process X. The main originality of the proposed method comes from the fact that it provides a recipe for simulating U_{t>=s} conditionally on X_s. The generality, the stability and the iterative character of this algorithm, even in high dimension, make its strength.