Optimal Liquidation with Conditions on Minimum Price

Mervan, Aksu
The classical optimal trading problem is the closure of an initial position q_0 in a financial asset over a time interval [0 T]; the trader tries to maximize an expected utility under the constraint q_T = 0, which is the liquidation constraint. Given that the trading takes place in a stochastic environment, the constraint q_T=0 may be too restrictive; the trader may want to relax this constraint or slow down/stop trading depending on price behavior. The goal of this thesis is the formulation and a study of these types of modified liquidation orders. We introduce two new parameters to the stochastic optimal control formulation of this problem: a process I taking values in \{0,1\} and a measurable set { S}. The set { S} prescribes when full liquidation is required and I prescribes when trading takes place. We give four examples for { S} and I which are all based on a lower bound specified for the price process. We show that the minimal supersolution of a related backward stochastic differential equation (BSDE) with a singular terminal value and with a convex driver term gives both the value function and the optimal control of the modified stochastic optimal control problem. The novelties of the BSDE arising from the modified control problem are as follows: the relaxation of the constraint q_T = 0 implies that the terminal value of the BSDE can take negative values; this and the convexity of the driver imply that the driver is no longer monotone and results from the currently available literature giving the minimal supersolution of this type of BSDE are not directly applicable. The same aspects of the problem imply that the BSDE can explode to -infty backward in time. 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. The BSDEs reduce to PDE for Markovian price processes; we also present an analysis of these PDE for a Markovian price process involving stochastic volatility. We quantify the financial performance of our models by the percantage 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 (A_1), one corresponding to random fluctuations in the price (A_2) and one corresponding to transaction/bid-ask spread costs (A_3). A_1 turns out to be a linear function of 1-q_T/q_0, the portion of the portfolio that is closed; therefore, its distribution is fully determined by that of q_T/q_0. We provide a numerical study of the distribution of q_T/q_0 and the conditional distributions of A_2 and A_3 given q_T/q_0 under the assumption that the price process is Brownian for a range of choices of I and { S}.


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Citation Formats
A. Mervan, “Optimal Liquidation with Conditions on Minimum Price,” Ph.D. - Doctoral Program, Middle East Technical University, 2022.