×

zbMATH — the first resource for mathematics

Portfolio selection with imperfect information: a hidden Markov model. (English) Zbl 1276.91091
Summary: We consider a utility-based portfolio selection problem, where the parameters change according to a Markovian market that cannot be observed perfectly. The market consists of a riskless and many risky assets whose returns depend on the state of the unobserved market process. The states of the market describe the prevailing economic, financial, social, political or other conditions that affect the deterministic and probabilistic parameters of the model. However, investment decisions are based on the information obtained by the investors. This constitutes our observation process. Therefore, there is a Markovian market process whose states are unobserved, and a separate observation process whose states are observed by the investors who use this information to determine their portfolios. There is, of course, a probabilistic relation between the two processes. The market process is a hidden Markov chain and we use sufficient statistics to represent the state of our financial system. The problem is solved using the dynamic programming approach to obtain an explicit characterization of the optimal policy and the value function. In particular, the return-risk frontiers of the terminal wealth are shown to have linear forms.

MSC:
91G10 Portfolio theory
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
90C39 Dynamic programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Çanakoğlu, Portfolio selection in stochastic markets with exponential utility functions, Annals of Operations Research 166 pp 281– (2009) · Zbl 1163.91374 · doi:10.1007/s10479-008-0406-2
[2] Çanakoğlu, Portfolio selection in stochastic markets with HARA utility functions, European Journal of Operational Research 201 pp 520– (2010) · Zbl 1180.91252 · doi:10.1016/j.ejor.2009.03.017
[3] Pye, A Markov model of the term structure, The Quarterly Journal of Economics 80 pp 60– (1966) · doi:10.2307/1880579
[4] Piger, Encyclopedia of Complexity and System Science (2009)
[5] Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 pp 357– (1989) · Zbl 0685.62092 · doi:10.2307/1912559
[6] Gray, Modeling the conditional distribution of interest rates as a regime-switching process, Journal of Financial Economics 42 pp 27– (1996) · doi:10.1016/0304-405X(96)00875-6
[7] Costa, A generalized multi-period mean-variance portfolio optimization with Markov switching parameters, Automatica 44 pp 2487– (2008) · Zbl 1157.91356 · doi:10.1016/j.automatica.2008.02.014
[8] Schaller, Regime switching in stock market returns, Applied Financial Economics 7 pp 177– (1997) · doi:10.1080/096031097333745
[9] Elliott, Hidden Markov Models: Estimation and Control (1994)
[10] Elliott, Portfolio optimization, hidden Markov models, and technical analysis of point and figure charts, International Journal of Theoretical and Applied Finance 5 pp 385– (2002) · Zbl 1107.91331 · doi:10.1142/S0219024902001493
[11] Sass, Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain, Finance and Stochastics 8 pp 553– (2004) · Zbl 1063.91040 · doi:10.1007/s00780-004-0132-9
[12] Rieder, Portfolio optimization with unobservable Markov-modulated drift process, Journal of Applied Probability 42 pp 362– (2005) · Zbl 1138.93428 · doi:10.1239/jap/1118777176
[13] Corsi, Handbook of Numerical Analysis, in: Mathematical Modelling and Numerical Methods in Finance, vol XV (2008)
[14] Stiglitz, The contributions of the economics of information to twentieth century economics, Quarterly Journal of Economics 115 pp 1441– (2000) · Zbl 0970.91515 · doi:10.1162/003355300555015
[15] Stiglitz, Credit rationing in markets with imperfect information, American Economic Review 71 pp 393– (1981)
[16] Stiglitz, Equlibrium in product markets with imperfect information, American Economic Review 69 pp 339– (1979)
[17] Smallwood, Optimal control of partially observable processes over finite horizon, Operations Research 21 pp 1071– (1973) · Zbl 0275.93059 · doi:10.1287/opre.21.5.1071
[18] Monahan, A survey of partially observable Markov decision processes: Theory, models, and algorithms, Management Science 28 pp 1– (1982) · Zbl 0486.90084 · doi:10.1287/mnsc.28.1.1
[19] Bell, Improved algorithms for inventory and replacement stock problems, SIAM Journal on Applied Mathematics 18 pp 558– (1970) · Zbl 0205.48202 · doi:10.1137/0118048
[20] Pratt, Risk aversion in the small and in the large, Econometrica 32 pp 122– (1964) · Zbl 0132.13906 · doi:10.2307/1913738
[21] Arrow, Aspects of the Theory of Risk-bearing (1965)
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.