2023/06/29 by Falko Baustian, Baustian, Falko, Jan Pospíšil +3
Economics, Econometrics and Finance · Engineering · Mathematics · #35A16 #35K58 #91G20 #91G40 #Analysis of PDEs (math.AP) #Applied mathematics #Capital Investment and Risk Analysis #Counterparty #Credit risk #Derivative (finance) #Economics #FOS: Mathematics #Finance #Fixed point #Fixed-point iteration #Mathematical analysis #Mathematical optimization #Mathematics #Monotone polygon #Nonlinear system #Partial derivative #Partial differential equation #Reservoir Engineering and Simulation Methods #Scheme (mathematics) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2306.17320
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2023/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study nonlinear partial differential equations (PDEs) that are used to model different value adjustments denoted generally as xVA. These adjustments are nowadays commonly added to the risk-free financial derivative values and the PDE approach allows their easy incorporation. The aim of this paper is to apply the method of monotone iterations with sub- and supersolutions to the nonlinear Black-Scholes-type equation that occurs especially in the counterparty risk models. We introduce a monotone iteration scheme with semi-explicit solution formulas for each iteration step. Moreover, we show that the problem greatly simplifies for contracts with non-negative payoffs. To show the viability of the approach we apply our method to the call option, the forward, and the gap option.