# Efficient Computation of Various Valuation Adjustments Under Local L\'evy Models

## TL;DR

This paper introduces a Fourier-based numerical method to efficiently price Bermudan derivatives with XVAs under local Le9vy models, accommodating local volatility and jump measures despite the lack of characteristic functions.

## Contribution

It develops an innovative Fourier-based approach for solving FBSDEs in local Le9vy models, enabling accurate valuation of derivatives with XVAs.

## Key findings

- Effective pricing of Bermudan options with XVAs under complex models
- Approximation method compensates for unavailable characteristic functions
- Framework accommodates local volatility and jump dynamics

## Abstract

Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FBSDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local L\'evy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.