Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function

TL;DR

This paper develops a numerical method to price American call options using a nonlinear Black-Scholes model where volatility depends on the asset price and Gamma, incorporating variable transaction costs.

Contribution

It introduces a transformation of the nonlinear free boundary problem into a Gamma variational inequality and applies a modified relaxation method for numerical solutions.

Findings

Effective numerical scheme for nonlinear Black-Scholes with variable volatility

Successful computational examples demonstrating the method

Inclusion of variable transaction costs in option pricing

Abstract

In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a numerical method for pricing American style call options by means of transformation of the free boundary problem for a nonlinear Black-Scholes equation into the so-called Gamma variational inequality with the new variable depending on the Gamma of the option. We apply a modified projective successive over relaxation method in order to construct an effective numerical scheme for discretization of the Gamma variational inequality. Finally, we present several computational examples for the nonlinear Black-Scholes equation for pricing American style call option under presence of variable transaction costs.