Optimal Liquidation with Conditions on Minimum Price

The classical optimal trading problem is the closure of a position in an asset over a time interval; the trader maximizes an expected utility under the constraint that the position be fully closed by terminal time. Since the asset price is stochastic, the liquidation constraint may be too restrictive; the trader may want to relax it or slow down/stop trading depending on price behavior. We consider two additional parameters that serve these purposes within the Almgren-Chriss framework: a binary valued process $I$ that prescribes when trading takes place and a measurable set $S$ that prescribes when full liquidation is required. We give four examples for $S$ and $I$ which are defined in terms of a lower bound for the price process. The terminal cost of the control problem is $\infty$ over $S$ representing the liquidation constraint. The permanent price impact parameter enters the problem as the negative part of the terminal cost over $S^c$. $I$ modifies the running cost. A terminal cost that can take negative values implies 1) the backward stochastic differential equation (BSDE) associated with the value function of the control problem can explode to $-\infty$ backward in time and 2) existence results on minimal supersolutions of BSDE with singular terminal values and monotone drivers are not directly applicable. A key part of the solution is an assumption that balances market volume and the permanent price impact parameter and a lower bound on the BSDE based on this assumption. When liquidation costs are quadratic, the problem is convex and, under a general filtration, the minimal supersolution of the BSDE gives the value function and the optimal control. For the non-quadratic case, we assume a stochastic volatility model and focus on choices of $I$ and $S$ that are Markovian or can be broken into Markovian pieces. These give PDE/PDE-system representations for the value functions.