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

**Pedro Pólvora and Daniel Ševˇcoviˇc \***

Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava, Slovakia; pedro.polvora@gmail.com

**\*** Correspondence: sevcovic@fmph.uniba.sk

**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.

**Keywords:** option pricing; utility indifference pricing; transaction costs; Hamilton-Jacobi-Bellman equation; penalty methods; finite difference approximation
