Monotone methods in counterparty risk models with non-linear Black-Scholes-type equations

TL;DR

This paper investigates non-linear Black-Scholes equations in counterparty risk models, introducing monotone iterative schemes and discussing efficient numerical methods like Monte Carlo and finite differences for solving the associated linear diffusion equations.

Contribution

It presents a transformation of non-linear Black-Scholes equations into a form suitable for monotone iterative solutions and explores computational strategies for efficient numerical approximation.

Findings

Monotone iterative schemes converge monotonically to the true solution.

Transformation simplifies the non-linear problem into a linear diffusion equation.

Proposes solution methods including Monte Carlo and finite difference techniques.

Abstract

A non-linear Black-Scholes-type equation is studied within counterparty risk models. The classical hypothesis on the uniform Lipschitz-continuity of the non-linear reaction function allows for an equivalent transformation of the semi-linear Black-Scholes equation into a standard parabolic problem with a monotone non-linear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most computational power for computing an approximate solution to the inhomogeneous linear diffusion equation, we discuss also this question and suggest several solution methods, including those based on Monte Carlo and finite differences/elements.