Total value adjustment of Bermudan option valuation under pure jump L\'evy fluctuations

TL;DR

This paper develops a numerical method combining Monte Carlo and finite difference techniques to price Bermudan options under pure jump Lévy processes, incorporating total value adjustments like XVA.

Contribution

It introduces a novel MC-FF method for fractional PDEs in jump models and proves its convergence, enhancing accuracy in derivative pricing with counterparty risk.

Findings

The MC-FF method accurately approximates derivative exposure.

XVA calculations under pure jump Lévy processes are feasible with the proposed approach.

The convergence of the numerical scheme is theoretically established.

Abstract

During the COVID-19 pandemic, many institutions have announced that their counterparties are struggling to fulfill contracts.Therefore, it is necessary to consider the counterparty default risk when pricing options. After the 2008 financial crisis, a variety of value adjustments have been emphasized in the financial industry. The total value adjustment (XVA) is the sum of multiple value adjustments, which is also investigated in many stochastic models such as Heston and Bates models. In this work, a widely used pure jump L\'evy process, the CGMY process has been considered for pricing a Bermudan option with various value adjustments. Under a pure jump L\'evy process, the value of derivatives satisfies a fractional partial differential equation(FPDE). Therefore, we construct a method which combines Monte Carlo with finite difference of FPDE (MC-FF) to find the numerical approximation of exposure, and compare it with the benchmark Monte Carlo-COS (MC-COS) method. We use the discrete energy estimate method, which is different with the existing works, to derive the convergence of the numerical scheme.Based on the numerical results, the XVA is computed by the financial