Selling a Large Stock Position: a Stochastic Control Approach with State Constraints

A common practice for stock-selling decision making is often concerned with liquidation of the security in a short duration. This is feasible when a relative smaller number of shares of a stock is treated. Selling a large position during a short period of time in the market frequently depresses the market, resulting in poor filling prices. In this work, liquidation strategies are considered for selling much smaller number of shares over a longer period of time. By using a fluid model in which the number of shares are treated as fluid, and the corresponding liquidation is dictated by the rate of selling over time. Our objective is to maximize the expected overall return. The problem is formulated as a stochastic control problem with state constraints. Using the method of constrained viscosity solutions, we characterize the dynamics governing the value function and the associated boundary conditions. Numerical algorithms are also provided along with an illustrative example for demonstration purposes.