Optimal execution strategy with linear price impact functions

When fund managers or traders in the financial institutions trade a large volume of a stock, the trading volume might impact the stock price. This paper discusses optimal execution strategies with linear price impact functions for trading a large volume of a stock. At first, we verify the fact that an optimal solution derived by dynamic programming algorithm can be satisfied with the optimality condition via mathematical programming formulation if a random variable in a price impact function is independently and identically distributed. We formulate the mathematical programming model with non-negativity constraints. The type of the problem can be formulated as a quadratic programming, but it is not always convex. In this paper, we decompose the matrix derived from the linear price impact function, and we calculate a closed-form condition that the matrix is positive definite. Similarly, we propose a model using matrix decomposition to solve the problem fast. We examine the model using a linear impact function of Huberman and Stanzl with numerical examples. We analyze the sensitivity of various parameters for seven kinds of the coefficients of linear price impact.