In this work, we developed a Python demonstrator for pricing total valuation adjustment (XVA) based on the stochastic grid bundling method (SGBM). XVA is an advanced risk management concept which became relevant after the recent financial crisis. This work is a follow-up work on Chau and Oosterlee in (Int J Comput Math 96(11):2272–2301, 2019), in which we extended SGBM to numerically solving backward stochastic differential equations (BSDEs). The motivation for this work is basically two-fold. On the application side, by focusing on a particular financial application of BSDEs, we can show the potential of using SGBM on a real-world risk management problem. On the implementation side, we explore the potential of developing a simple yet highly efficient code with SGBM by incorporating CUDA Python into our program.

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The main aims of this research are to study various numerical schemes in the approximation of the occurring expectations and their applications in numerically solving BSDEs. We focus on numerical expectation/finite measure integration since the majority of the BSDE solvers consists of two parts, conditional expectations computations, and deterministic functions to map these expectations to target approximations. By simply changing the approximation for conditional expectations, we can effectively generate various schemes for BSDEs that can suit different requirements. Furthermore, our results carry implications in numerical integration too. In this thesis, we focus on the mathematical properties of these approximations. We will discuss the fundamental assumptions for them, give complete descriptions, derive error bounds and conduct numerical experiments. The main goal is to analyze these approximations. We will also touch upon the financial applications of BSDEs.@en

In this work, we apply the Stochastic Grid Bundling Method (SGBM) to numerically solve backward stochastic differential equations (BSDEs). The SGBM algorithm is based on conditional expectations approximation by means of bundling of Monte Carlo sample paths and a local regress-later regression within each bundle. The basic algorithm for solving the backward stochastic differential equations will be introduced and an upper error bound is established for the local regression. A full error analysis is also conducted for the explicit version of our algorithm and numerical experiments are performed to demonstrate various properties of our algorithm.@en

We propose a numerical algorithm for backward stochastic differential equations based on time discretization and trigonometric wavelets. This method combines the effectiveness of Fourier-based methods and the simplicity of a wavelet-based formula, resulting in an algorithm that is both accurate and easy to implement. Furthermore, we mitigate the problem of errors near the computation boundaries by means of an antireflective boundary technique, giving an improved approximation.We test our algorithm with different numerical experiments.

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