×

Optimal consumption and investment strategies with liquidity risk and lifetime uncertainty for Markov regime-switching jump diffusion models. (English) Zbl 1431.91361

Summary: In this paper, we consider the optimal consumption and investment strategies for households throughout their lifetime. Risks such as the illiquidity of assets, abrupt changes of market states, and lifetime uncertainty are considered. Taking the effects of heritage into account, investors are willing to limit their current consumption in exchange for greater wealth at their death, because they can take advantage of the higher expected returns of illiquid assets. Further, we model the liquidity risks in an illiquid market state by introducing frozen periods with uncertain lengths, during which investors cannot continuously rebalance their portfolios between different types of assets. In liquid market, investors can continuously remix their investment portfolios. In addition, a Markov regime-switching process is introduced to describe the changes in the market’s states. Jumps, classified as either moderate or severe, are jointly investigated with liquidity risks. Explicit forms of the optimal consumption and investment strategies are developed using the dynamic programming principle. Markov chain approximation methods are adopted to obtain the value function. Numerical examples demonstrate that the liquidity of assets and market states have significant effects on optimal consumption and investment strategies in various scenarios.

MSC:

91G10 Portfolio theory
49L20 Dynamic programming in optimal control and differential games
93E20 Optimal stochastic control
Full Text: DOI

References:

[1] Amihud, Y.; Mendelson, H., Liquidity, maturity, and the yields on U.S. treasury securities, Journal of Finance, 46, 4, 1411-1425 (1991)
[2] Ang, A.; Papanikolaou, D.; Westerfield, M., Portfolio choice with illiquid assets, Management Science, 60, 11, 2737-2761 (2014)
[3] Budhiraja, A.; Ross, K., Convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem, SIAM Journal on Control and Optimization, 45, 6, 2169-2206 (2007) · Zbl 1138.93062
[4] Constantinides, G. M., Capital market equilibrium with transaction costs, Journal of Political Economy, 94, 4, 842-862 (1986)
[5] Dai, M.; Li, P.; Liu, H.; Wang, Y., Portfolio choice with market closure and implications for liquidity premia, Management Science, 62, 2, 368-386 (2016)
[6] Davis, M. H.; Norman, A. R., Portfolio selection with transaction costs, Mathematics of Operations Research, 15, 4, 676-713 (1990) · Zbl 0717.90007
[7] Dufresne, D., Fitting combinations of exponentials to probability distributions, Applied Stochastic Models in Business and Industry, 23, 23-48 (2007) · Zbl 1142.60321
[8] Fleming, H.; Soner, M., Controlled Markov processes and viscosity solutions (2006), Springer: Springer New York · Zbl 1105.60005
[9] Framstad, C.; Oksendal, B.; Sulem, A., Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs, Journal of Mathematical Economics, 35, 2, 233-257 (2001) · Zbl 1013.91055
[10] French, R.; Roll, R., Stock return variances: the arrival of information and the reaction of traders, Journal of Financial Economics, 17, 1, 5-26 (1986)
[11] Hamilton, J., A new approach to the economic analysis of non-stationary time series, Econometrica, 57, 2, 357-384 (1989) · Zbl 0685.62092
[12] Jang, B.; Koo, H.; Liu, H.; Loewenstein, M., Liquidity premia and transaction costs, Journal of Finance, 62, 5, 2329-2366 (2007)
[13] Keykhaei, R., Portfolio selection in a regime switching market with a bankruptcy state and an uncertain exit-time: Multi-period mean-variance formulation, Operational Research, 1-24 (2018)
[14] Kushner, H. J.; Dupuis, P., Numerical methods for stochastic control problems in continuous time (2001), Springer: Springer New York · Zbl 0968.93005
[15] Levy, M.; Kaplanski, G., Portfolio selection in a two-regime world, European Journal of Operational Research, 242, 2, 514-524 (2015) · Zbl 1341.91126
[16] Liu, H., Optimal consumption and investment with transaction costs and multiple risky assets, Journal of Finance, 59, 1, 289-338 (2004)
[17] Liu, H.; Loewenstein, M., Market crashes, correlated illiquidity, and portfolio choice, Management Science, 59, 3, 715-732 (2013)
[18] Mei, X.; Nogales, F. J., Portfolio selection with proportional transaction costs and predictability, Journal of Banking and Finance, 94, 131-151 (2018)
[19] Merton, R. C., Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51, 3, 247-257 (1969)
[20] Song, Q. S.; Yin, G.; Zhang, Z., Numerical methods for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42, 7, 1147-1157 (2006) · Zbl 1117.93370
[21] Zheng, X.; Chau, K. W.; Hui, C. M., Liquidity risk and cross-sectional return in the housing market, Habitat International, 49, 426-434 (2015)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.