Dynamic mean–variance problem with frictions

• Published in: Finance and stochastics Vol. 26; no. 2; pp. 267 - 300
• Main Authors: Bensoussan, Alain, Ma, Guiyuan, Siu, Chi Chung, Yam, Sheung Chi Phillip
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2022; Springer Nature B.V
• Subjects: Control theory; Cost analysis; Differential equations; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Equilibrium; Finance; Insurance; Liquidity; Management; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Quantitative Finance; Statistics for Business; Trading; Dynamic mean–variance problem; 91G60; Mean-field type control problems; 91G10; G11; G21; Time-inconsistency; G24; Asymptotics; Price impact; 91G80
• ISSN: 0949-2984; 1432-1122
• DOI: 10.1007/s00780-022-00474-x

We study a dynamic mean–variance portfolio selection problem with return predictability and trading frictions from price impact. Applying mean-field type control theory, we provide a characterisation of an equilibrium trading strategy for an investor facing stochastic investment opportunities. An explicit equilibrium strategy is derived in terms of the solution to a generalised matrix Riccati differential equation, and a sufficient condition is also provided to ensure the latter’s well-posedness. Our solution indicates that the investor should trade gradually towards a target portfolio which accounts for return predictability, price impact and time-consistency. Moreover, an asymptotic analysis around small liquidity costs shows that the investor’s target portfolio is an equilibrium portfolio without price impact in the first-order sense, and that her first-order approximated value function does not deteriorate significantly for sufficiently small liquidity costs. Finally, our numerical results demonstrate that the target portfolio is more conservative than an equilibrium portfolio without price impact.