Efficient XVA computation using supervised learning algorithms
Published:
We will show how supervised learning algorithms can help to handle the main issues of the computations of XVAs.
This post will be related to the work I did during my internship as a Quantitative Research at Forvis Mazars during the period May-November 2024. To access to a PDF version of the presentation, click here.
Table of contents
A quick definition of XVAs
We will focus on computational aspects of $\text{XVAs}$. For a deep understanding of theses quantities, see this book.
$\text{XVAs}$ can be summarized as the following :
- $\text{CVA}$ and $\text{DVA}$ represent 2 important values which can be defined as follows :
- $\text{FVA}$ and $\text{MVA}$ represent
- $\text{KVA}$ represents
In the following, we will focus on the main quantity of interest for banks which is the $\text{CVA}$.
The $\text{CVA}$ equation is given by :
$\text{with the following notations :}$
- $\mathbb{1}_{t \leq \tau^C \leq T}$ represents the potential default time between t and $T$.
- $R^C$ refers to the recovery rate
- Test
Gaussian Processes Regressions for CVA computation
To be done asap
Neural Networks for MVA computation
Description of the method
This method is based on a simple but fundamental representation of the conditional expectation for $L^2$ random variables. Assuming $X$ and $Y$ 2 random variables such that $E[Y|X] \in L^2(X)$. Therefore, we know that :
\(\begin{align} E[Y|X] = \underset{f \in L^2(X)}{argmin} \quad \mathbb{E}[|Y-f(X)|^2] \tag{2} \end{align}\) The natural idea is therefore with neural networks to look for a good approximation of the function $f$ by a neural network parametrization $f^{\theta}$ where $\theta$ is defined on a suitable space such that instead of solving $(2)$, we look for :
\(\begin{align} E[Y|X] = \underset{\theta }{argmin} \quad \mathbb{E}[|Y-f^{\theta}(X)|^2] \tag{3} \end{align}\) In a practical way, we will considerer $i.i.d$ sampling of random variables $(X_i,Y_i)_{i \in \mathbb{N}^*}$ and we are going to optimize over $\theta$ the quantity
\[\begin{align} \frac{1}{N} \sum_{i=1}^{N} (Y_i -f^{\theta}(X_i))^2 \tag{4} \end{align}\]