Strategic Execution Trajectories

We obtain the optimal execution strategy for two sequential trades in the presence of a transient price impact. We first present a novel and general solution method for the case of a single trade (a metaorder) that is executed as a sequence of sub-trades (child orders). We then analyze the case of two sequential metaorders, including the case where the size and direction of the second metaorder are uncertain at the time the first metaorder is initiated. We obtain the optimal execution strategy under two different cost functions. First, we minimize the total cost when each metaorder is benchmarked to the price at its initiation, the total separate costs approach widely used by practitioners. Although simple, we show that optimizing total separate costs can lead to a significant understatement of the real costs of trading whilst also adversely impacting order scheduling. We overcome these issues by introducing a new cost function that splits the second metaorder into two parts, one that is predictable when the first metaorder is initiated and a residual that is not. The predictable and residual parts of the second metaorder are benchmarked using the initiation prices of the first and second metaorders, respectively. We prove existence of an optimal execution trajectory for linear instantaneous price impact and positive definite decay, and derive the explicit form of the minimizer in the special case of exponentially decaying impact, however uniqueness in general remains unproven. Various numerical examples are included for illustration.