×

zbMATH — the first resource for mathematics

Portfolio selection in stochastic markets with exponential utility functions. (English) Zbl 1163.91374
Summary: We consider the optimal portfolio selection problem in a multiple period setting where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has an exponential structure and the market states change according to a Markov chain. 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. The problem is solved using the dynamic programming approach to obtain the optimal solution and an explicit characterization of the optimal policy. We also discuss the stochastic structure of the wealth process under the optimal policy and determine various quantities of interest including its Fourier transform. The exponential return-risk frontier of the terminal wealth is shown to have a linear form. Special cases of multivariate normal and exponential returns are disussed together with a numerical illustration.

MSC:
91G10 Portfolio theory
90C39 Dynamic programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arrow, K. J. (1965). Aspects of the theory of risk-bearing. Helsinki: Yrjö Hahnsson Foundation.
[2] Asmussen, S. (2000). Ruin probabilities. Singapore: World Scientific. · Zbl 0960.60003
[3] Bäuerle, N., & Rieder, U. (2004). Portfolio optimization with Markov-modulated stock prices and interest rates. IEEE Transactions on Automatic Control, 49, 442–447. · Zbl 1366.91135 · doi:10.1109/TAC.2004.824471
[4] Bertsekas, D. P. (2000). Dynamic programming and optimal control. Massachusetts: Athena Scientific.
[5] Bielecki, T. R., & Pliska, S. R. (1999). Risk sensitive dynamic asset management. Journal of Applied Mathematics and Optimization, 39, 337–360. · Zbl 0984.91047 · doi:10.1007/s002459900110
[6] Bielecki, T.R., Hernández-Hernández, D., & Pliska, S. R. (1999). Risk sensitive control of finite state Markov chains in discrete time, with applications to portfolio management. Mathematical Methods of Operations Research, 50, 167–188. · Zbl 0959.91029 · doi:10.1007/s001860050094
[7] Bodily, S. E., & White, C. C. (1982). Optimal consumption and portfolio strategies in a discrete time model with summary-dependent preferences. Journal of Financial and Quantitative Analysis, 17, 1–14. · doi:10.2307/2330925
[8] Çakmak, U., & Özekici, S. (2006). Portfolio optimization in stochastic markets. Mathematical Methods of Operations Research, 63, 151–168. · Zbl 1136.91409 · doi:10.1007/s00186-005-0020-x
[9] Çelikyurt, U., & Özekici, S. (2007). Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach. European Journal of Operational Research, 179, 186–202. · Zbl 1163.91375 · doi:10.1016/j.ejor.2005.02.079
[10] Chen, A. H., Jen, F. C., & Zionts, S. (1971). The optimal portfolio revision policy. Journal of Business, 44, 51–61. · doi:10.1086/295332
[11] Çınlar, E., & Özekici, S. (1987). Reliability of complex devices in random environments. Probability in the Engineering and Informational Sciences, 1, 97–115. · Zbl 1133.90321 · doi:10.1017/S0269964800000322
[12] Di Massi, G. B., & Stettner, L. (1999). Risk sensitive control of discrete time partially observed Markov processes with infinite horizon. Stochastics and Stochastics Reports, 67, 309–322. · Zbl 0942.93047
[13] Dumas, B., & Luciano, E. (1991). An exact solution to a dynamic portfolio choice problem under transaction costs. Journal of Finance, 46, 577–595. · doi:10.2307/2328837
[14] Ehrlich, I., & Hamlen, W. A. (1995). Optimal portfolio and consumption decisions in a stochastic environment with precommitment. Journal of Economic Dynamics and Control, 19, 457–480. · Zbl 0900.90034 · doi:10.1016/0165-1889(94)00790-O
[15] Elliott, R. J., & Mamon, R. S. (2003). A complete yield curve description of a Markov interest rate model. International Journal of Theoretical and Applied Finance, 6, 317–326. · Zbl 1079.91027 · doi:10.1142/S0219024903001852
[16] Elton, E. J., & Gruber, M. J. (1974). On the optimality of some multiperiod portfolio selection criteria. Journal of Business, 47, 231–243. · doi:10.1086/295633
[17] Fleming, W. H., & Hernández-Hernández, D. (2003). An optimal consumption model with stochastic volatility. Finance and Stochastics, 7, 245–262. · Zbl 1035.60028 · doi:10.1007/s007800200083
[18] Hernández-Hernández, D., & Marcus, S. I. (1999). Existence of risk sensitive optimal stationary policies for controlled Markov processes. Applied Mathematics and Optimization, 40, 273–285. · Zbl 0937.90115 · doi:10.1007/s002459900126
[19] Hu, Y., Imkeller, P., & Müller, M. (2005). Utility maximization in incomplete markets. Annals of Applied Probability, 15, 1691–1712. · Zbl 1083.60048 · doi:10.1214/105051605000000188
[20] Korn, R., & Kraft, H. (2001). A stochastic control approach to portfolio problems with stochastic interest rates. SIAM Journal on Control and Optimization, 40, 1250–1269. · Zbl 1020.93029 · doi:10.1137/S0363012900377791
[21] Li, D., & Ng, W. L. (2000). Optimal dynamic portfolio selection: multiperiod mean-variance formulation. Mathematical Finance, 10, 387–406. · Zbl 0997.91027 · doi:10.1111/1467-9965.00100
[22] Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7, 77–91. · doi:10.2307/2975974
[23] Marshall, A. W., & Olkin, I. (1967). A multivariate exponential distribution. Journal of American Statistical Association, 62, 30–44. · Zbl 0147.38106 · doi:10.2307/2282907
[24] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous-time case. The Review of Economics and Statistics, 51, 247–257. · doi:10.2307/1926560
[25] Mossin, J. (1968). Optimal multiperiod portfolio policies. Journal of Business, 41, 215–229. · doi:10.1086/295078
[26] Nagai, H., & Peng, S. (2002). Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Annals of Applied Probability, 12, 173–195. · Zbl 1042.91048 · doi:10.1214/aoap/1015961160
[27] Norberg, R. (1995). A continuous-time Markov chain interest model with applications to insurance. Journal of Applied Stochastic Models and Data Analysis, 11, 245–256. · Zbl 1067.91509 · doi:10.1002/asm.3150110306
[28] Özekici, S. (1996). Complex systems in random environments. In S. Özekici (Ed.), NATO ASI series : Vol. F154. Reliability and maintenance of complex systems (pp. 137–157). Berlin: Springer. · Zbl 0868.90035
[29] Prabhu, N. U., & Zhu, Y. (1989). Markov-modulated queueing systems. Queueing Systems, 5, 215–246. · Zbl 0694.60087 · doi:10.1007/BF01149193
[30] Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica, 32, 122–136. · Zbl 0132.13906 · doi:10.2307/1913738
[31] Pye, G. (1966). A Markov model of the term structure. The Quarterly Journal of Economics, 80, 60–72. · doi:10.2307/1880579
[32] Rolski, T., Schmidli, H., Schmidt, V., & Teugels, J. (1999). Stochastic processes for insurance and finance. Wiley: Chichester. · Zbl 0940.60005
[33] Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. The Review of Economics and Statistics, 51, 239–246. · doi:10.2307/1926559
[34] Song, J.-S., & Zipkin, P. (1993). Inventory control in a fluctuating demand environment. Operations Research, 41, 351–370. · Zbl 0798.90035 · doi:10.1287/opre.41.2.351
[35] Steinbach, M. C. (2001). Markowitz revisited: mean-variance models in financial portfolio analysis. Society for Industrial and Applied Mathematics Review, 43, 31–85. · Zbl 1049.91086
[36] Stettner, L. (1999). Risk sensitive portfolio optimization. Mathematical Methods of Operations Research, 50, 463–474. · Zbl 0949.93077 · doi:10.1007/s001860050081
[37] Stettner, L. (2004). Risk-sensitive portfolio optimization with completely and partially observed factors. IEEE Transactions on Automatic Control, 49, 457–464. · Zbl 1366.91146 · doi:10.1109/TAC.2004.824476
[38] Tehranchi, M. (2004). Explicit solutions of some utility maximization problems in incomplete markets. Stochastic Processes and Their Applications, 114, 109–125. · Zbl 1152.91555 · doi:10.1016/j.spa.2004.05.007
[39] Yin, G., & Zhou, X. Y. (2004). Markowitz’s mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits. IEEE Transactions on Automatic Control, 49, 349–360. · Zbl 1366.91148 · doi:10.1109/TAC.2004.824479
[40] Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance and Stochastics, 5, 61–82. · Zbl 0977.93081 · doi:10.1007/PL00000040
[41] Zhang, Q. (2001). Stock trading: an optimal selling rule. SIAM Journal on Control and Optimization, 40, 64–87. · Zbl 0990.91014 · doi:10.1137/S0363012999356325
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.