Optimal execution with log-linear price impact and transaction costs

We consider the so-called “optimal execution problem” in algorithmic trading, which is the problem faced by an investor who has a large number of stock shares to sell over a given time horizon and whose actions have impact on the stock price. In particular, we develop and study a price model that presents the stochastic dynamics of a geometric Brownian motion and incorporates a log-linear effect of the investor’s transactions. We then formulate the optimal execution problem as a two-dimensional degenerate singular stochastic control problem. Using both analytic and probabilistic techniques, we establish a simple sufficient condition for the market to allow for no arbitrage opportunities in a finite time horizon and we develop a detailed characterisation of the value function and the optimal strategy. In particular, we derive an explicit solution to the problem if the time horizon is infinite. Interesting features of the problem’s solution include the facts that (a) the value function may be discontinuous as a function of the time horizon and (b) an optimal strategy may not exist even when the value function is finite.