Optimal Liquidation under Stochastic Liquidity

TL;DR

This paper explicitly solves a stochastic control problem for optimal asset liquidation in a market with stochastic liquidity, deriving the optimal strategy through advanced probabilistic and variational techniques.

Contribution

It provides an explicit solution to a complex stochastic control problem involving stochastic liquidity and transient price impact, using novel probabilistic and calculus of variations methods.

Findings

Explicit optimal liquidation strategy derived.

Solution involves local time of reflected diffusion.

Addresses stochastic liquidity effects in market impact.

Abstract

We solve explicitly a two-dimensional singular control problem of finite fuel type for infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determines the inter-temporal resilience of the market in spirit of Predoiu, Shaikhet and Shreve (2011), is taken to be stochastic, being driven by own random noise. The optimal control is obtained as the local time of a diffusion process reflected at a non-constant free boundary. To solve the HJB variational inequality and prove optimality, we need a combination of probabilistic arguments and calculus of variations methods, involving Laplace transforms of inverse local times for diffusions reflected at elastic boundaries.