Analysis of the nonlinear option pricing model under variable transaction costs

TL;DR

This paper investigates a nonlinear Black--Scholes model incorporating variable transaction costs, providing theoretical analysis, existence proofs, bounds, and an efficient numerical scheme with computational examples.

Contribution

It introduces a transformation approach for analyzing nonlinear option pricing models with variable transaction costs, including existence proofs and a new numerical scheme.

Findings

Existence of smooth solutions for the nonlinear model

Derived bounds on option prices under variable transaction costs

Developed an efficient numerical discretization scheme

Abstract

In this paper we analyze a nonlinear Black--Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price $V$ is assumed to be a function of the underlying asset price and the Gamma of the option. We show that the generalizations of the classical Black--Scholes model can be analyzed by means of transformation of the fully nonlinear parabolic equation into a quasilinear parabolic equation for the second derivative of the option price. We show existence of a classical smooth solution and prove useful bounds on the option prices. Furthermore, we construct an effective numerical scheme for approximation of the solution. The solutions are obtained by means of the efficient numerical discretization scheme of the Gamma equation. Several computational examples are presented.