Optimal strategies in equity securities and derivatives

We consider an asset allocation problem of three asset classes: bond, equities and derivatives. The problem is to maximize the expected utility of the terminal wealth with the investor's preference described by an exponential utility function. The equity class consists of multiple stocks and the derivative class consists of European options with all possible strikes written on each stock. This problem, under a single-period is solved under the assumptions that the stock price processes follow Geometric Brownian motions with a budget constraint. Optimal payoff at the terminal date for the investor is also obtained and the corresponding replications of the asset positions for the derived payoff are explicitly worked out. Performances and usefulness of the obtained optimal strategies are illustrated through numerical examples, where the stock price movements are generated by Monte Carlo simulations. In testing the performance of the optimal strategies, a well-known portfolio performance measure, Sharpe ratio, is used and the portfolio created by the mean-variance method is also used as a benchmark.