Skip to main content Accessibility help
×
Home

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

  • Sid Browne (a1)

Abstract

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 cases where the investor has an external source of income, and where the investor is interested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management. 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 a supermartingale. Nevertheless, we still find an optimal strategy. For the general case, we provide a thorough analysis of the optimal strategy, and obtain new insights into the behavior of the optimal policy. 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 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.

Copyright

Corresponding author

Postal address: 402 Uris Hall, Graduate School of Business, Columbia University, New York, NY 10027, USA. Email address: sb30@columbia.edu

Footnotes

Hide All

An earlier version of this paper was presented at the Workshop on the Mathematics of Finance, Montreal, May 1996.

Footnotes

References

Hide All
[1] Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Polit. Econom. 81, 637659.
[2] Breiman, L. (1961). Optimal gambling systems for favorable games. Fourth Berkeley Symp. Math. Stat. and Prob., 1, 6578.
[3] Browne, S. (1995). Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Math. Operat. Res. 20, 937958.
[4] Browne, S. (1997). Survival and growth with a liability: optimal portfolios in continuous time. Math. Oper. Res. 22, 468493.
[5] Cox, J. C. and Huang, C. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econ. Theory 49, 3383.
[6] Dubins, L. E. and Savage, L. J. (1976). How to Gamble If You Must: Inequalities for Stochastic Processes. Dover, New York.
[7] Duffie, D. (1996). Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press, Princeton, NJ.
[8] Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, New York.
[9] Hakansson, N. H. (1970). Optimal investment and consumption strategies under risk for a class of utility functions. Econometrica 38, 587607.
[10] Heath, D. (1993). A continuous time version of Kulldorff's Result. Unpublished manuscript.
[11] Heath, D., Orey, S., Pestien, V. and Sudderth, W. (1987). Minimizing or maximizing the expected time to reach zero. SIAM J. Contr. and Opt. 25, 195205.
[12] Hull, J. C. (1993). Options, Futures, and Other Derivative Securities, 2nd edn. Prentice Hall, Englewood Cliffs, NJ.
[13] Kulldorff, M. (1993). Optimal control of favorable games with a time limit. SIAM J. Contr. and Opt. 31, 5269.
[14] Majumdar, M. and Radner, R. (1991). Linear models of economic survival under production uncertainty. Econ. Theory 1, 1330.
[15] Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3, 373413.
[16] Merton, R. (1990). Continuous Time Finance. Blackwell, Oxford.
[17] Pestien, V. C. and Sudderth, W. D. (1985). Continuous-time red and black: how to control a diffusion to a goal. Math. Oper. Res. 10, 599611.
[18] Pestien, V. C. and Sudderth, W. D. (1988). Continuous-time casino problems. Math. Oper. Res. 13, 364376.
[19] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York.
[20] Sharpe, W. F., Alexander, G. F. and Bailey, J. V. (1995). Investments, 5th edn. Prentice Hall, Englewood Cliffs, NJ.
[21] Wilmott, P., Dewynne, J. and Howison, S. (1993). Option Pricing: Mathematical Models and Computation. Oxford Financial Press, London.

Keywords

MSC classification

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

  • Sid Browne (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed