The optimal liquidation model under stochastic market impact

This paper is concerned with optimal liquidation of a large block of stocks. The stock price follows a geometric Brownian motion (GBM), and is affected by the rate of selling. The investor attempts to unwind large block of stocks, which moves the stock price downward for the liquidity reasons. The price impact function, assumed stochastic, is related to the activity or liquidity of the market which is characterized by a factor following another SDE. By means of dynamic programming principle, the HJB equation of this optimal control problem is derived. The method of viscosity solution is used to describe the value function. Furthermore, we obtain a comparison principle for the viscosity solutions. Finally, the convergence for the numerical scheme is verified and some numerical examples are given to illustrate the results.