Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

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 the Heston [B. Salvador and C. W. Oosterlee, Appl. Math. Comput. 391, 125489 (2020)] and Bates [L. Goudenège et al., Comput. Manag. Sci. 17, 163–178 (2020)] models. In this work, a widely used pure jump Lévy process, the Carr–Geman–Madan–Yor process has been considered for pricing a Bermudan option with various value adjustments. Under a pure jump Lévy process, the value of derivatives satisfies a fractional partial differential equation (FPDE). Therefore, we construct a method that combines Monte Carlo with a finite difference of FPDE to find the numerical approximation of exposure and compare it with the benchmark Monte Carlo simulation and Fourier-cosine series method. We use the discrete energy estimate method, which is different from the existing works, to derive the convergence of the numerical scheme. Based on the numerical results, the XVA is computed by the financial exposure of the derivative value.

Binomial Method in Bermudan Option

“…Thus, a numerical approach needs to be applied [9][10][11]. Other several methods for determining the price of Bermudan option are stochastic grid [12], low-discrepancy mesh [13], Lévy Process Models [14], recombining quadratures method [15], jump-diffusion processes [16], Merton jump-diffusion [17], least-squares Monte Carlo [15], neural network regression [18], regression trees/random forests [19], and a pure jump Lévy process method [20].…”

An Optimal Control Method to Compute the Most Likely Transition Path for Stochastic Dynamical Systems with Jumps

“…However, this kind of noise induced transition phenomena will not possibly happen in deterministic systems. In recent years, Lévy noise has been playing an increasingly important role in models of climate [7,8,9], thermal dynamics [10,11], biology [12,13,14], financial markets [15,16] and engineering [17]. Such a growing amount of research shows that non-Gaussian Lévy process is more appropriate to be utilized as the intrinsic noise in various complex systems.…”