Optimal Trade Execution and Absence of Price Manipulations in Limit Order Book Models

We continue the analysis of optimal execution strategies in the model for a limit order book with nonlinear price impact and exponential resilience that was considered in Alfonsi, Schied, and Fruth (2009). We now allow for non-homogeneous resilience rates and arbitrary trading dates and consider the extended problem of optimizing jointly over trading dates and sizes. Our main results show that, under general conditions on the shape function of the limit order book, placing the deterministic trade sizes identified in Alfonsi, Schied, and Fruth (2009) at trading dates that are homogeneously spaced is optimal also within the large class of adaptive strategies with arbitrary trading dates. Perhaps even more importantly, our analysis yields as a corollary that our model does not admit price manipulation strategies in the sense of Huberman and Stanzl (2004). This latter result contrasts the recent findings of Gatheral (2008), where, in a related but different model, exponential resilience was found to give rise to price manipulation strategies when price impact is nonlinear