Monotone iteration scheme for nonlinear PDEs in risk models

TL;DR

This paper develops a monotone iteration scheme with semi-explicit solutions for nonlinear PDEs in risk models, simplifying calculations especially for non-negative payoffs, and demonstrates its effectiveness on various financial derivatives.

Contribution

It introduces a novel monotone iteration method with explicit formulas for nonlinear Black-Scholes-type PDEs in risk modeling, particularly for contracts with non-negative payoffs.

Findings

The scheme simplifies solving nonlinear PDEs in risk models.

Effective for options with non-negative payoffs.

Validated on call, forward, and gap options.

Abstract

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.