  In this work we study a finite horizon optimal liquidation problem with
multiplicative price impact in algorithmic trading, using market orders. We
analyze the case when an agent is trading on a market with two financial
assets, whose difference of log-prices is modelled with a mean-reverting
process. The agent's task is to liquidate an initial position of shares of one
of the two financial assets, without having the possibility of trading the
other stock. The criterion to be optimized consists in maximising the expected
final value of the agent, with a running inventory penalty. The main result of
this paper consists in finding a classical solution of the
Hamilton-Jacobi-Bellman (HJB) equation associated to this problem, which is
proved to not coincide with the value function. However, we find the value
function as a solution to the forward-backward stochastic differential equation
(FBSDE) associated to the problem. We provide numerical tests showing that the
HJB and FBSDE solutions are close to each other and analysing performance of
the described model. We also prove a verification theorem and a comparison
principle for the viscosity solution to the HJB equation.
