XVA Valuation under Market Illiquidity

TL;DR

This paper develops a framework for valuing XVA of derivatives considering market crises, using a regime-switching model and BSDEs, with numerical validation via deep learning algorithms.

Contribution

It introduces a state-dependent model incorporating market crises into XVA valuation using an alternating renewal process and BSDEs, extending prior models that ignored crises.

Findings

The model captures the impact of crises on XVA valuation.

Deep learning algorithms effectively solve the BSDEs in this context.

Parameter variations significantly influence XVA estimates.

Abstract

Before the 2008 financial crisis, most research in financial mathematics focused on pricing options without considering the effects of counterparties' defaults, illiquidity problems, and the role of the sale and repurchase agreement (Repo) market. Recently, models were proposed to address this by computing a total valuation adjustment (XVA) of derivatives; however without considering a potential crisis in the market. In this article, we include a possible crisis by using an alternating renewal process to describe the switching between a normal financial regime and a financial crisis. We develop a framework to price the XVA of a European claim in this state-dependent situation. The price is characterized as a solution to a backward stochastic differential equation (BSDE), and we prove the existence and uniqueness of this solution. In a numerical study based on a deep learning algorithm for BSDEs, we compare the effect of different parameters on the valuation of the XVA.