Liquidity risk in portfolio selection: A stochastic control approach

In the literature, on optimal portfolio selection, closed-form solutions are rare, specially when market friction is present. This becomes even more challenging if market friction comes in the form of liquidity risk. This paper bridges this gap by formulating a portfolio selection problem with liquidity risk as a stochastic control problem with Constant Relative Risk Aversion (CRRA) utility function. Assuming that liquidity costs are generated by rebalancing the portfolio over finite intervals, rather than by instantaneous position changes, we derive a closed-form expression for the optimal portfolio weights. This formulation offers two notable advantages. First, it explicitly takes liquidity costs into the wealth dynamics and provides an exact analytical solution to the problem. Second, this formulation has the feature to trace back to the classical [Merton, RC (1969). Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51(3), 247–257] model in the case of perfect asset liquidity. In high-liquidity regimes, our numerical results show that new optimal portfolio weights initially exhibit variation but ultimately converge to the frictionless solution proposed by Merton [(1969). Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51(3), 247–257] as the terminal time T approaches. Finally, the quantitative impact of this new solution is thoroughly analyzed, providing valuable insights into portfolio optimization under liquidity constraints.