Mathematical models and numerical methods for a capital valuation adjustment (KVA) problem

TL;DR

This paper develops rigorous mathematical models and numerical methods to compute the capital valuation adjustment (KVA) within the broader context of XVAs, using PDEs and market theory.

Contribution

It introduces a formal PDE-based framework for KVA calculation, establishing existence, uniqueness, and regularity of solutions, along with numerical methods for practical computation.

Findings

Numerical results for European options demonstrate the model's applicability.

The PDE approach accurately captures KVA in various market scenarios.

Theoretical results ensure well-posedness of the valuation models.

Abstract

In this work we rigorously establish mathematical models to obtain the capital valuation adjustment (KVA) as part of the total valuation adjustments (XVAs). For this purpose, we use a semi-replication strategy based on market theory. We formulate single-factor models in terms of expectations and PDEs. For PDEs formulation, we rigorously obtain the existence and uniqueness of the solution, as well as some regularity and qualitative properties of the solution. Moreover, appropriate numerical methods are proposed for solving the corresponding PDEs. Finally, some examples show the numerical results for call and put European options and the corresponding XVA that includes the KVA.