HARA frontiers of optimal portfolios in stochastic markets.

*(English)*Zbl 1253.91164Summary: We consider the optimal portfolio selection problem in continuous-time settings where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has the structure of the HARA family and the market states change according to a Markov process. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. This includes the distributions of the random asset returns as well as the utility function. We analyzed Black-Scholes type continuous-time models where the market parameters are driven by Markov processes. The Markov process that affects the state of the market is independent of the underlying Brownian motion that drives the stock prices. The problem of maximizing the expected utility of the terminal wealth is investigated and solved by stochastic optimal control methods for exponential, logarithmic and power utility functions. We found explicit solutions for optimal policy and the associated value functions. We also constructed the optimal wealth process explicitly and discussed some of its properties. In particular, it is shown that the optimal policy provides linear frontiers.

##### MSC:

91G10 | Portfolio theory |

91G80 | Financial applications of other theories |

93E20 | Optimal stochastic control |

##### Keywords:

Markov processes; dynamic programming; portfolio optimization; optimal control; HARA utility functions; HARA frontiers
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\textit{E. Çanakoğlu} and \textit{S. Özekici}, Eur. J. Oper. Res. 221, No. 1, 129--137 (2012; Zbl 1253.91164)

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##### References:

[1] | Arrow, K., Aspects of the theory of risk-bearing, (1965), Yrjö Hahnsson Foundation Helsinki |

[2] | Bäuerle, N.; Rieder, U., Portfolio optimization with Markov-modulated stock prices and interest rates, IEEE transactions on automatic control, 49, 442-447, (2004) · Zbl 1366.91135 |

[3] | Benth, F.; Karlsen, K.; Reikvam, K., Merton’s portfolio optimization problem in a black and Scholes market with non-Gaussian stochastic volatility of ornstein – uhlenbeck type, Mathematical finance, 13, 215-244, (2003) · Zbl 1049.91060 |

[4] | Bielecki, T.; Pliska, S., Risk sensitive dynamic asset management, Journal of applied mathematics and optimization, 39, 337-360, (1999) · Zbl 0984.91047 |

[5] | Bollen, N., Real options and product life cycles, Management science, 670-684, (1999) · Zbl 1231.91288 |

[6] | Çanakog˜lu, E., 2009. Portfolio selection in stochastic markets: Utility based approach. Ph.D. thesis, Koç University. |

[7] | Çanakog˜lu, E.; Özekici, S., Portfolio selection in stochastic markets with exponential utility functions, Annals of operations research, 166, 281-297, (2009) · Zbl 1163.91374 |

[8] | Çanakog˜lu, E.; Özekici, S., Portfolio selection in stochastic markets with HARA utility functions, European journal of operational research, 201, 520-536, (2010) · Zbl 1180.91252 |

[9] | Çanakog˜lu, E.; Özekici, S., Portfolio selection with imperfect information: A hidden Markov model, Applied stochastic models in business and industry, 27, 95-114, (2011) · Zbl 1276.91091 |

[10] | Costa, O.; Araujo, M., A generalized multi-period mean – variance portfolio optimization with Markov switching parameters, Automatica, 44, 2487-2497, (2008) · Zbl 1157.91356 |

[11] | Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Mathematische annalen, 300, 463-520, (1994) · Zbl 0865.90014 |

[12] | Detemple, J.; Rindisbacher, M., Closed-form solutions for optimal portfolio selection with stochastic interest rate and investment constraints, Mathematical finance, 15, 539-568, (2005) · Zbl 1138.91432 |

[13] | Dybvig, P.; Huang, C., Nonnegative wealth, absence of arbitrage, and feasible consumption plans, The review of financial studies, 1, 377-401, (1988) |

[14] | Fleming, W.; Hernández-Hernández, D., An optimal consumption model with stochastic volatility, Finance and stochastics, 7, 245-262, (2003) · Zbl 1035.60028 |

[15] | Fleming, W.; Sheu, S.J., Optimal long term growth rate of expected utility of wealth, Annals of applied probability, 9, 871-903, (1999) · Zbl 0962.91036 |

[16] | Gray, S., Modeling the conditional distribution of interest rates as a regime-switching process, Journal of financial economics, 42, 27-62, (1996) |

[17] | Hamilton, J., A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, 357-384, (1989) · Zbl 0685.62092 |

[18] | Honda, T., Optimal portfolio choice for unobservable and regime-switching Mean returns, Journal of economic dynamics and control, 28, 45-78, (2003) · Zbl 1179.91233 |

[19] | Karatzas, I.; Shreve, S., Methods of mathematical finance, (1998), Springer · Zbl 0941.91032 |

[20] | Kreps, D., Three essays on capital markets, (1979), Institute for Mathematical Studies in the Social Sciences, Stanford University |

[21] | Lindberg, C., Portfolio optimization and a factor model in a stochastic volatility market, Stochastics, 78, 259-279, (2006) · Zbl 1280.91152 |

[22] | Mamon, R.; Rodrigo, M., Explicit solutions to European options in a regime-switching economy, Operations research letters, 33, 6, 581-586, (2005) · Zbl 1116.91047 |

[23] | Merton, R., Optimum consumption and portfolio rules in a continuous-time model, Journal of economic theory, 3, 373-413, (1971) · Zbl 1011.91502 |

[24] | Moler, C.; van Loan, C., Nineteen dubious ways to compute the exponential of a matrix, SIAM review, 20, 801-836, (1978) · Zbl 0395.65012 |

[25] | Nagai, H.; Runggaldier, W., PDE approach to utility maximization for market models with hidden Markov factors, Progress in probability, 59, 493-506, (2008) · Zbl 1140.93049 |

[26] | Pham, H.; Quenez, M., Optimal portfolio in partially observed stochastic volatility models, Annals of applied probability, 11, 210-238, (2001) · Zbl 1043.91032 |

[27] | Piger, J., 2009. Econometrics: Models of regime changes. In: Encyclopedia of Complexity and System Science. Springer, New York, Ch. Econometrics: Models of Regime Changes. |

[28] | Pratt, J., Risk aversion in the small and in the large, Econometrica, 32, 122-136, (1964) · Zbl 0132.13906 |

[29] | Pye, G., A Markov model of the term structure, The quarterly journal of economics, 80, 60-72, (1966) |

[30] | Sass, J.; Haussmann, U., Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain, Finance and stochastics, 8, 553-577, (2004) · Zbl 1063.91040 |

[31] | Schaller, H.; van Norden, S., Regime switching in stock market returns, Applied financial economics, 7, 177-191, (1997) |

[32] | Sotomayor, L.; Cadenillas, A., Explicit solutions of consumption – investment problems in financial markets with regime switching, Mathematical finance, 19, 215-236, (2009) |

[33] | Yao, D.; Zhang, Q.; Zhou, X., A regime-switching model for European options, Stochastic processes, optimization, and control theory: applications in financial engineering, queueing networks, and manufacturing systems, 281-300, (2006) · Zbl 1136.91015 |

[34] | Zariphopoulou, T., A solution approach to valuation with unhedgeable risks, Finance and stochastics, 5, 61-82, (2001) · Zbl 0977.93081 |

[35] | Zhou, X.; Li, D., Continuous-time mean – variance portfolio selection: A stochastic LQ framework, Applied mathematics and optimization, 42, 19-33, (2000) · Zbl 0998.91023 |

[36] | Zhou, X.; Yin, G., Markowitz’s mean – variance portfolio selection with regime switching: A continuous-time model, SIAM journal on control and optimization, 42, 4, 1466-1482, (2003) · Zbl 1175.91169 |

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