Notes on Backward Stochastic Differential Equations for Computing XVA

TL;DR

This paper analyzes backward stochastic differential equations (BSDEs) with random horizons in the context of XVA, providing explicit pricing formulas and conditions to prevent arbitrage in derivative securities.

Contribution

It introduces explicit conditions and formulas for XVA computation using BSDEs with random horizons, advancing mathematical finance methods.

Findings

Explicit non-arbitrage conditions for XVA pricing

A new explicit formula for XVA as correction terms

Analysis of BSDE properties with random horizons

Abstract

The X-valuation adjustment (XVA) problem, which is a recent topic in mathematical finance, is considered and analyzed. First, the basic properties of backward stochastic differential equations (BSDEs) with a random horizon in a progressively enlarged filtration are reviewed. Next, the pricing/hedging problem for defaultable over-the-counter (OTC) derivative securities is described using such BSDEs. An explicit sufficient condition is given to ensure the non-existence of an arbitrage opportunity for both the seller and buyer of the derivative securities. Furthermore, an explicit pricing formula is presented in which XVA is interpreted as approximated correction terms of the theoretical fair price.