  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.
