Reaching goals by a deadline: Digital options and continuous-time active portfolio management

We study a variety of optimal investment problems for objectives related to attaining goals by a fixed terminal time. We start by finding the policy t hat maximizes the probability of reaching a given wealth level by a given f ired terminal time, for the case where an investor can allocate his wealth at any time between n + 1 investment opportunities: n risky stocks, as well as a risk-free asset that has a positive return. This generalizes results recently obtained by Kulldorff and Heath for the case of a single investmen t opportunity. We then use this to solve related problems for cases where t he investor has an external source of income, and where the investor is int erested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management. One of the benchmark s we consider for this last problem is that of the return of the optimal gr owth policy, for which the resulting controlled process is a supermartingal e. Nevertheless, we still find an optimal strategy. For the general case, w e provide a thorough analysis of the optimal strategy, and obtain new insig hts into the behavior of the optimal policy. For one special case, namely t hat of a single stock with constant coefficients, the optimal policy is ind ependent of the underlying drift. We explain this by exhibiting a correspon dence between the probability maximizing results and the pricing and hedgin g of a particular derivative security, known as a digital or binary option. In fact, we show that for this case, the optimal policy to maximize the pr obability of reaching a given value of wealth by a predetermined time is eq uivalent to simply buying a European digital option with a particular strik e price and payoff. A similar result holds for the general case, but with t he stock replaced by a particular tinder) portfolio, namely the optimal gro wth or log-optimal portfolio.