Static use of options in dynamic portfolio optimization under transaction costs and solvency constraints

We study a dynamic portfolio optimization problem where it is possible to invest in a risk-free bond, in a risky stock modeled by a lognormal diffusion and in call options written on the stock. The use of the options is limited to static strategies at the beginning of the investment period. The investor faces transaction costs with a fixed component and solvency constraints and the objective is to maximize the expected utility of the final wealth. We characterize the value function as a constrained viscosity solution of the associated quasi- variational inequality and we prove the local uniform convergence of a Markov chain approximation scheme to compute numerically the optimal solution. Because of transaction costs and solvency constraints the options cannot be pefectly replicated and despite the restriction to static policies our numerical results show that in most cases the investor will keep a significant part of his portfolio invested in options.