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

Part of: Stochastic processes Stochastic analysis Stochastic systems and control Markov processes

Published online by Cambridge University Press:  01 July 2016

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

Keywords

Optimal gamblingstochastic controlportfolio theorymartingalesoption pricinghedging strategiesdigital options

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 93E20: Optimal stochastic control 60G40: Stopping times; optimal stopping problems; gambling theory 60J60: Diffusion processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 551 - 577
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

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

References

Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Polit. Econom. 81, 637–659.CrossRefGoogle Scholar
Breiman, L. (1961). Optimal gambling systems for favorable games. Fourth Berkeley Symp. Math. Stat. and Prob., 1, 65–78.Google Scholar
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, 937–958.Google Scholar
Browne, S. (1997). Survival and growth with a liability: optimal portfolios in continuous time. Math. Oper. Res. 22, 468–493.Google Scholar
Cox, J. C. and Huang, C. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econ. Theory 49, 33–83.Google Scholar
Dubins, L. E. and Savage, L. J. (1976). How to Gamble If You Must: Inequalities for Stochastic Processes. Dover, New York.Google Scholar
Duffie, D. (1996). Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press, Princeton, NJ.Google Scholar
Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, New York.Google Scholar
Hakansson, N. H. (1970). Optimal investment and consumption strategies under risk for a class of utility functions. Econometrica 38, 587–607.Google Scholar
Heath, D. (1993). A continuous time version of Kulldorff's Result. Unpublished manuscript.Google Scholar
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, 195–205.CrossRefGoogle Scholar
Hull, J. C. (1993). Options, Futures, and Other Derivative Securities, 2nd edn. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Kulldorff, M. (1993). Optimal control of favorable games with a time limit. SIAM J. Contr. and Opt. 31, 52–69.CrossRefGoogle Scholar
Majumdar, M. and Radner, R. (1991). Linear models of economic survival under production uncertainty. Econ. Theory 1, 13–30.Google Scholar
Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3, 373–413.Google Scholar
Merton, R. (1990). Continuous Time Finance. Blackwell, Oxford.Google Scholar
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, 599–611.CrossRefGoogle Scholar
Pestien, V. C. and Sudderth, W. D. (1988). Continuous-time casino problems. Math. Oper. Res. 13, 364–376.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York.Google Scholar
Sharpe, W. F., Alexander, G. F. and Bailey, J. V. (1995). Investments, 5th edn. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Wilmott, P., Dewynne, J. and Howison, S. (1993). Option Pricing: Mathematical Models and Computation. Oxford Financial Press, London.Google Scholar