Illiquidity Effects in Optimal Consumption-Investment Problems

TL;DR

This paper analyzes how liquidity freezes impact optimal consumption and investment strategies within a modified Black-Scholes-Merton framework, providing explicit formulas and utility loss estimates for different liquidity regimes.

Contribution

It introduces a model incorporating liquidity shocks governed by a Markov chain and derives explicit asymptotic formulas for optimal strategies under small shock probability and fast regime switching.

Findings

Explicit formulas for optimal consumption and investment under liquidity shocks.

Quantification of utility loss due to liquidity freezes.

Comparison with finite-horizon models in existing literature.

Abstract

We study the effect of liquidity freezes on an economic agent optimizing her utility of consumption in a perturbed Black-Scholes-Merton model. The single risky asset follows a geometric Brownian motion but is subject to liquidity shocks, during which no trading is possible and stock dynamics are modified. The liquidity regime is governed by a two-state Markov chain. We derive the asymptotic effect of such freezes on optimal consumption and investment schedules in the two cases of (i) small probability of liquidity shock; (ii) fast-scale liquidity regime switching. Explicit formulas are obtained for logarithmic and hyperbolic utility maximizers on infinite horizon. We also derive the corresponding loss in utility and compare with a recent related finite-horizon model of Diesinger, Kraft and Seifried (2009).