A new closed-form solution for optimal portfolio selection with liquidity risk
DOI: https://doi.org/10.3846/mma.2026.24351Abstract
In the literature on optimal portfolio selection problems, it is rare that closed-form solutions are found. It is even more so when liquidity risk needs to be taken into consideration. In this paper, we present a closed-form solution for the optimal weights of a portfolio that consists of a risky and riskless asset under a new key assumption that the liquidity risk is directly proportional to the wealth of the portfolio invested in the risky asset. The solution found is for the Constant Relative Risk Aversion (CRRA) utility function, after successfully solving the associated HJB (HamiltonJacobi-Bellman) equation exactly. Due to the presence of liquidity risk, the research findings reveal that the optimal weights are consistently lower than those found by [19]. Finally, the quantitative impact of the proposed solution is discussed.
Keywords:
portfolio choice problem, log-return assumption, liquidity cost, closed-form solutionHow to Cite
Share
License
Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
R. Aaberge, K. Liu and Y. Zhu. Political uncertainty and household savings. Journal of Comparative Economics, 45(1):154–170, 2017. https://doi.org/10.1016/j.jce.2015.12.011
V.V. Acharya and L.H. Pedersen. Asset pricing with liquidity risk. Journal of Financial Economics, 77(2):375–410, 2005. https://doi.org/10.1016/j.jfineco.2004.06.007
Y. Amihud, A. Hameed, W. Kang and H. Zhang. The illiquidity premium: International evidence. Journal of Financial Economics, 117(2):350–368, 2015. https://doi.org/10.1016/j.jfineco.2015.04.005
Y. Amihud, H. Mendelson and L.H. Pedersen. Liquidity and asset prices. MPRA Paper 24768, University Library of Munich, Germany, 2005.
U. Ҫetin, R. Jarrow, P. Protter and M. Warachka. Option pricing with liquidity risk. 2004. Preprint, Cornell University.
M.H.A. Davis and A.R. Norman. Portfolio selection with transaction costs. Mathematics of Operations Research, 15(4):676–713, 1990. https://doi.org/10.1287/moor.15.4.676
B. Dumas and E. Luciano. An exact solution to a dynamic portfolio choice problem under transactions costs. The Journal of Finance, 46(2):577–595, 1991. https://doi.org/10.1111/j.1540-6261.1991.tb02675.x
I. Erauskin. The impact of financial openness on the size of utility-enhancing government. Economics - The Open-Access, Open-Assessment E-Journal (20072020), 7:1–56, 2013. https://doi.org/10.5018/economics-ejournal.ja.2013-38
N. Gârleanu and L.H. Pedersen. Dynamic portfolio choice with frictions. Journal of Economic Theory, 165:487–516, 2016. https://doi.org/10.1016/j.jet.2016.06.001
P. Guasoni and M.H. Weber. Nonlinear price impact and portfolio choice. Mathematical Finance, 30(2):341–376, 2020. https://doi.org/10.1111/mafi.12234
C.W. Holden, S. Jacobsen and A. Subrahmanyam. The empirical analysis of liquidity. Foundations and Trends(R) in Finance, 8(4):263–365, 2014. https://doi.org/10.1561/0500000044
R.A. Jarrow and P. Protter. Chapter 17 liquidity risk and option pricing theory. In John R. Birge and Vadim Linetsky(Eds.), Financial Engineering, volume 15 of Handbooks in Operations Research and Management Science, pp. 727–762. Elsevier, 2007. https://doi.org/10.1016/S0927-0507(07)15017-9
H. Kraft. Optimal portfolios and Heston’s stochastic volatility model: An explicit solution for power utility. Quantitative Finance, 5:303–313, 2005. https://doi.org/10.1080/14697680500149503
H. Li, R. Novy-Marx and M. Velikov. Liquidity risk and asset pricing. Critical Finance Review, 8(1-2):223–255, 2019. https://doi.org/10.1561/104.00000076
Mr.T. Lybek and Mr.A. Sarr. Measuring liquidity in financial markets. IMF Working Papers 2002/232, International Monetary Fund, 2002.
G. Ma, C.C. Siu and S.-P. Zhu. Optimal investment and consumption with return predictability and execution costs. Economic Modelling, 88(C):408–419, 2020. https://doi.org/10.1016/j.econmod.2019.09.051
G. Ma, C.C. Siu and S.-P. Zhu. Portfolio choice with return predictability and small trading frictions. Economic Modelling, 111(C), 2022. https://doi.org/10.1016/j.econmod.2022.105823
G. Ma and S.-P. Zhu. Revisiting the Merton problem: from HARA to CARA utility. Computational Economics, 59:651–686, 2021. https://doi.org/10.1007/s10614-021-10102-z
R.C. Merton. Lifetime portfolio selection under uncertainty: The continuoustime case. The Review of Economics and Statistics, 51(3):247–257, 1969. https://doi.org/10.2307/1926560
R. Musneh, M.R.A. Karim and C. Geetha A/P A. Baburaw. Liquidity risk and stock returns: empirical evidence from industrial products and services sector in Bursa Malaysia. Future Business Journal, 7(1):1–10, 2021. https://doi.org/10.1186/s43093-021-00106-4
S.E. Shreve and H.M. Soner. Optimal investment and consumption with transaction costs. The Annals of Applied Probability, 4(3):609–692, 1994. https://doi.org/10.1214/aoap/1177004966
M. Vaihekoski. Pricing of liquidity risk: empirical evidence from Finland. Applied Financial Economics, 19(19):1547–1557, 2009. https://doi.org/10.1080/09603100802599548
X. Zeng and M. Taksar. A stochastic volatility model and optimal portfolio selection. Quantitative Finance, 13(10):1547–1558, 2013. https://doi.org/10.1080/14697688.2012.740568
R. Zhang, N. Langrené, Y. Tian, Z. Zhu, F. Klebaner and K. Hamza. Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach. Quantitative Finance, 19(3):519–532, 2019. https://doi.org/10.1080/14697688.2018.1524155
View article in other formats
CrossMark check
Published
Issue
Section
Copyright
Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
License
This work is licensed under a Creative Commons Attribution 4.0 International License.