Utility indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation

TL;DR

This paper develops a utility indifference option pricing model incorporating non-constant risk aversion and transaction costs, introduces a transformation method for solving complex PDEs, and provides a numerical scheme with computational examples.

Contribution

It introduces a novel transformation approach for solving nonlinear PDEs in utility-based option pricing with variable risk aversion and transaction costs.

Findings

Derived bounds on option prices using comparison principles.

Proposed a finite difference scheme for numerical approximation.

Provided computational examples demonstrating the method's effectiveness.

Abstract

Our goal is to analyze the system of Hamilton-Jacobi-Bellman equations arising in derivative securities pricing models. The European style of an option price is constructed as a difference of the certainty equivalents to the value functions solving the system of HJB equations. We introduce the transformation method for solving the penalized nonlinear partial differential equation. The transformed equation involves possibly non-constant the risk aversion function containing the negative ratio between the second and first derivatives of the utility function. Using comparison principles we derive useful bounds on the option price. We also propose a finite difference numerical discretization scheme with some computational examples.