OPTIMIZATION OF A TIME CONTINUOUS PORTFOLIO OF ASSETS AND DERIVATIVES

Abstract

We consider an incomplete market and suggest a self-financing time continuous investment strategy consisting of a risk-free asset (bond), a risky asset, and a financial derivative whose value moves inversely to that of the risky asset. We optimize the wealth process by introducing a parametric convex utility function that simultaneously maximizes wealth and minimizes the mean square of it. Using the HJB equation, we compute precisely the optimal portfolio process, where, notably, a range of processes can optimize the problem. This advantage enables investors to gain fringe benefits while maintaining their overall investment strategy by adjusting their portfolios accordingly. As an application of the results, we optimize a portfolio process with a European put as the derivative and compute the corresponding optimal wealth numerically. Additionally, we will outline a method to calculate the market return rate and the martingale parameter, which are necessary for optimization.