Reaching Goals by a Deadline: Digital Options and Continuous-Time Active Portfolio Management
38 Pages Posted: 22 Jan 1997
Date Written: February 1996
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 that maximizes the probability of reaching a given wealth level by a given fixed 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 investment opportunity. We then use this to solve related problems for the following cases: (i) when the investor has an external source of income; (ii) when the investor faces external liabilities, as arises in pension fund management; and (iii) when the investor is interested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management and certain risk management applications. One of the benchmarks we consider for this last problem is that of the return of the optimal growth policy, for which the resulting controlled process is always a supermartingale. For the general case, we provide a thorough and complete analysis of the optimal strategy. For one special case, namely that of a single stock with constant coefficients, the optimal policy is independent of the underlying drift. We explain this by exhibiting a remarkable correspondence between the probability maximizing results and the pricing and hedging 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 probability of reaching a given value of wealth by a predetermined time is equivalent to simply buying a European digital option with a particular strike price and payoff. A similar result holds for the general case, but with the stock replaced by a particular (index) portfolio, namely the optimal growth or log-optimal portfolio.
JEL Classification: G11, G23, C73
Suggested Citation: Suggested Citation
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By Sid Browne
By Sid Browne