Viscosity solution of a Delta Greek nonlinear Black-Scholes equation

TL;DR

This paper investigates a nonlinear Black-Scholes equation involving transaction costs, focusing on the Delta Greek, and proves the existence of viscosity solutions using the vanishing viscosity method, enabling robust numerical approaches.

Contribution

It introduces a viscosity solution framework for a nonlinear PDE of the Delta Greek in option pricing with transaction costs, employing the vanishing viscosity method.

Findings

Existence of viscosity solutions proved.

Regularized equations have good regularity properties.

Approximations converge to the viscosity solution.

Abstract

In this paper, a class of nonlinear option pricing models involving transaction costs is considered. The diffusion coefficient of the nonlinear parabolic equation for the price $V$ is assumed to be a linear function of the option's underlying asset price and the Gamma Greek $V_{xx}$. The main aim of this work is to study the governing PDE of the Delta Greek. The existence of viscosity solutions is proved using the vanishing viscosity method. Regularizing the equation by adding a small perturbation to the initial problem, a sequence of approximate solutions $u^{\varepsilon}$ is constructed and then the method of weak limits is applied to prove the convergence of the sequence to the viscosity solution of the Delta equation. The approximate problems constructed are shown to have good regularity, which allows the use of efficient and robust numerical methods.