The classical optimal trading problem is the closure of an initial position in a financial asset over a fixed time interval; the trader tries to maximize an expected utility under the constraint that the position is fully closed by terminal time. Given that the asset price is stochastic, the liquidation constraint may be too restrictive; the trader may want to relax the full liquidation constraint or slow down/stop trading depending on price behavior.We consider two additional parameters that serve these purposes within the Almgren-Chriss liquidation 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 all based on a lower bound specified for the price process. The terminal cost of the stochastic optimal control problem is ∞ over S; this represents the liquidation constraint. The permanent price impact defines the negative part of the terminal cost over the complement of S. The I parameter enters the stochastic optimal control problem as a multiplier of the running cost. Except for quadratic liquidation costs the problem turns out to be non-convex. 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 −∞ backward in time and 2) the existence results on minimal supersolutions of BSDE with singular terminal values and monotone drivers are not directly applicable. To tackle these we introduce an assumption that balances the market volume process and the permanent price impact in the model over the trading horizon. In the quadratic case, assuming only that the noise driving the asset price is a martingale, we show that the minimal supersolution of the BSDE gives both the value function and the optimal control of the stochastic optimal control problem. For the non-quadratic case, we assume a Brownian motion driven stochastic volatility model and focus on choices of I and S that are either Markovian or can be broken into Markovian pieces. These assumptions allow us to represent the value functions as solutions of PDE or PDE systems. The PDE arguments are based on the smoothness of the value functions and do not require convexity. We quantify the financial performance of the resulting liquidation algorithms by the percentage difference between the initial stock price and the average price at which the position is (partially) closed in the time interval 0, T. We note that this difference can be divided into three pieces: one corresponding to permanent price impact (A1), one corresponding to random fluctuations in the price (A2) and one corresponding to transaction/bid-ask spread costs (A3). We provide a numerical study of the distribution of the closed portion under the assumption that the price process is Brownian for I = 1 and an S corresponding to a lower bound on terminal price.
Aksu et al. (Thu,) studied this question.