Liquidation of a large block of stock

In the financial engineering literature, stock-selling rules are mainly concerned with liquidation of the security within a short period of time. This is practically feasible only when a relative smaller number of shares of a stock is involved. Selling a large position in a market place normally depresses the market if sold in a short period of time, which would result in poor filling prices. In this paper, we consider the liquidation strategy for selling a large block of stock by selling much smaller number of shares over a longer period of time. In particular, we treat the selling rule problem by using a fluid model in the sense that the number of shares are treated as fluid (continuous) and the corresponding liquidation is dictated by the rate of selling over time. The objective is to maximize the expected overall return. The underlying problem may be formulated as a stochastic control problem with state constraints. Method of constrained viscosity solution is used to characterize the dynamics governing the optimal reward function and the associated boundary conditions. Numerical examples are reported to illustrate the results.