Abstract
We study a Merton type optimization problem under a reallocation constraint. Under this restriction, the stock holdings can not be liquidated faster than a certain rate. This is a common restriction in certain type of investment firms. Our main objective is to study the large time optimal growth rate of the expected value of the utility from wealth. We also consider a discounted infinite horizon problem as a step towards understanding the first problem. A numerical study is done by solving the dynamic programming equations. Under the assumption of a power utility function, an appropriate dimension reduction argument is used to reduce the original problem to a two dimensional one in a bounded domain with convenient boundary conditions. Computation of the optimal growth rate introduces additional numerical difficulties as the straightforward approach is unstable. In this direction, new analytical results characterizing the growth rate as the limit of a sequence of finite horizon problems with continuously derived utility are proved.
Similar content being viewed by others
References
Bakshi GS, Chen ZW (1996) The spirit of capitalism and stock-market prices. Am Econ Rev 86(1): 133–157
Basak S, Shapiro A, Tepla L (2006) Risk management with benchmarking. Manage Sci 52(4): 542–557
Ben-Tahar I, Soner HM, Touzi N (2008) The dynamic programming equation for the problem of optimal investment under capital gains taxes. SIAM J Control Optim 46(5): 1779–1801
Ben-Tahar I, Soner HM, Touzi N (2010) Merton problem with taxes: characterization, computation and approximation. SIAM J Financ Math
Çetin U, Jarrow R, Protter P (2004) Liquidity risk and arbitrage pricing theory. Financ Stoch 8: 311–341
Çetin U, Rogers LCG (2007) Modelling liquidity effects in discrete time. Math Financ 17: 15–29
Çetin U, Soner HM, Touzi N (2010) Option hedging for small investors under liquidity costs. Financ and Stoch 14
Constantinides GM (1979) Multiperiod consumption and investment behavior with convex transaction costs. Manag Sci 25: 1127–1137
Constantinides GM (1983) Capital market equilibrium with personal tax. Econometrica 51: 611–637
Cox J, Huang C (1989) Optimal consumption and portfolio policies when asset prices follow a diffusion process. J Econ Theory 49: 33–83
Cvitanic J, Karatzas I (1995) On portfolio optimization under drawdown constraints. IMA Vol Math Appl 65: 35–46
Davis MHA, Norman A (1990) Portfolio selection with transaction costs. Math Oper Res 15: 676–713
Duffie D, Fleming WH, Soner HM, Zariphopoulou T (1997) Hedging in incomplete markets with HARA utility. J Econ Dyn Control 21: 753–782
Elie R, Touzi N (2008) Optimal lifetime consumption and investment under a drawdown constraint. Financ Stoch 12(3): 299–330
Fleming WH, Soner HM (1993) Controlled markov processes and viscosity solutions. Springer, Berlin
Grossman SJ, Vila J-L (1992) Optimal dynamic trading with leverage constraints. J Financ Quant Anal 27(2): 151–168
Grossman SJ, Zhou Z (1993) Optimal investment strategies for controlling drawdowns. Math Financ 3(3): 241–276
Hamilton JH (1987) Taxation, savings, and portfolio choice in a continuous-time model. Financ Publiques/Public Financ 42: 264–282
He H (1989) Essays in dynamic portfolio optimization and diffusion estimations. Ph.D. dissertation, A.P. Sloan School of Management, Massachusetts Institute of Technology, Cambridge
He H, Pearson ND (1991) Consumption and portfolio policies with incomplete markets and short-sale constraints: the infinite dimensional case. J Econ Theory 54: 259–304
Heath DC, Jarrow RA (1987) Arbitrage, continuous trading, and margin requirements. J Financ 42: 1129–1142
Karatzas I, Lehoczky JP, Sethi SP, Shreve SE (1986) Explicit solution of a general consumption/investment problem. Math Oper Res 11: 261–294
Karatzas I, Lehoczky JP, Shreve SE (1987) Optimal portfolio and consumption decisions for a small investor on a finite horizon. SIAM J Control Optim 25: 1557–1586
Karatzas I, Lehoczky JP, Shreve SE, Xu GL (1991) Martingale and duality methods for utility maximization in an incomplete market. SIAM J Control Optim 29: 702–730
Leizarowitz A (1989) Optimal trajectories of infinite-horizon deterministic control systems. Appl Math Optim 19(1): 11–32
Magill MJP, Constantinides GM (1976) Portfolio selection with transaction costs. J Econ Theory 13: 245–263
Merton RC (1969) Lifetime portfolio selection under uncertainty: the continuous-time case. Rev Econ Stat 51: 247–257
Merton RC (1971) Optimum consumption and portfolio rules in a continuous time model. J Econ Theory 3: 373–413
Roche H (2006) Optimal Consumption and Investment Strategies under Wealth Ratcheting. (Preprint)
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Rogers LCG (1998) The relaxed investor, liquidity, and parameter uncertainty. (Preprint)
Rogers LCG (2001) The relaxed investor and parameter uncertainty. Financ Stoch 5(2): 131–154
Shreve SE, Soner HM (1994) Optimal investment and consumption with transaction costs. Ann Appl Probab 4(3): 609–692
Smith WH (2001) How does the spirit of capitalism affect stock market prices?. Rev Financ Stud 14: 1215–1232
Sundaresam SM (1989) Intertemporally dependent preferences and the volatility of consumption and wealth. Rev Financ Stud 2: 73–89
Tourin A, Zariphopoulou T (1997) Viscosity solutions and numerical schemes for investment/consumption models with transaction costs. In: Numerical methods in finance. Publ. Newton Inst., Cambridge Univ. Press, Cambridge, pp 245–269
Xu GL, Shreve SE (1992) A duality method for optimal consumption and investment under short-selling prohibition. I. General market coefficients. Ann Appl Probab 2(1): 87–112
Xu GL, Shreve SE (1992) A duality method for optimal consumption and investment under short-selling prohibition. II. Constant market coefficients. Ann Appl Probab 2(2): 314–328
Zariphopoulou T (1994) Consumption investment models with constraints. SIAM J Control Optim 52: 59–84
Zariphopoulou T (2002) Stochastic control methods in asset pricing. Handbook of stochastic analysis and applications. In: Statist Textbooks Monogr, vol. 163. Dekker, New York, pp 679–753
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Egriboyun, F., Soner, H.M. Optimal investment strategies with a reallocation constraint. Math Meth Oper Res 71, 551–585 (2010). https://doi.org/10.1007/s00186-010-0306-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-010-0306-5