Xva Principles, Nested Monte Carlo Strategies, and Gpu Optimizations

Abstract

We present a nested Monte Carlo (NMC) approach implemented on graphics processing units (GPUs) to X-valuation adjustments (XVAs), where X ranges over C for credit, F for funding, M for margin, and K for capital. The overall XVA suite involves five compound layers of dependence. Higher layers are launched first, and trigger nested simulations on-the-fly whenever required in order to compute an item from a lower layer. If the user is only interested in some of the XVA components, then only the sub-tree corresponding to the most outer XVA needs be processed computationally. Inner layers only need a square root number of simulation with respect to the most outer layer. Some of the layers exhibit a smaller variance. As a result, with GPUs at least, error-controlled NMC XVA computations are doable. But, although NMC is naively suited to parallelization, a GPU implementation of NMC XVA computations requires various optimizations. This is illustrated on XVA computations involving equities, interest rate, and credit derivatives, for both bilateral and central clearing XVA metrics.

Stochastic approximation schemes for economic capital and risk margin computations

“…From a practical point of view, an incremental SA scheme uses a limited amount of memory but, being a loop, is less easy to parallelize than a simulation-and-sort algorithm, on which several processors can fruitfully be used (see [1,Appendix C]). On the other hand SA schemes can be efficiently combined with importance sampling as studied in [7,8], whereas [17] and [10] introduce respective jacknife and weighted regression acceleration procedures for the simulation-and-sort schemes.…”

“…Our case study is based on the setup of Armenti and Crépey [5], Section 4 (see also Section 4.4 in [1]), which we recall as a starting point. We consider a clearing house (or central counterparty, CCP for short) with a finite number (≥ 2) of clearing members labeled by i.…”

“…For our numerics we consider the CCP toy model of Section 4 in [5] and Section 4.4 in [1], where nine members are clearing (interest rate or foreign exchange) swaps on a Black-Scholes underlying rate process X, with all the numerical parameters used there. The default times of the clearing members are defined through the common shock model or dynamic Marshall-Olkin copula (DMO) model of [12], Chapt.…”

“…Regarding now the abstract assumptions there, we only make a general comment that they should all hold in our lognormal model for X combined with randomized sampling at the times of defaults of the counterparties, which are all times with an intensity, recalling the corresponding modeling assumptions related to Algorithms 0 (SA scheme for the basic case without liabilities), 1 (SA scheme with nested simulation of future liabilities) and Algorithm 2 (SA scheme with regression of future liabilities), respectively: Regarding the regression algorithm for CVA ccp t+1 that is required in the context of Algorithm 2, we apply to CVA ccp t+1 = E t+1 ψ(Z [t+1,T ] ) the approach that is used for computing the "CA process" in Section 4.4 of [1], using as a regression basis 1, X t+1 , X 2 t+1 (recall X is the underlying Black-Scholes rate) and the default indicator processes at time (t + 1) of the clearing members. In the present case of CVA ccp t+1 the situation is in fact a bit simpler as no time-stepping is required, i.e.…”

Sparse Grid Method for Highly Efficient Computation of Exposures for xVA

“…An alternative way to improving computational speed is significant investments in hardware, like multi-core CPUs/GPUs. [9,1].…”