A Pre-Trade Algorithmic Trading Model under Given Volume Measures and Generic Price Dynamics (GVM-GPD)

We make several improvements to the mean-variance framework for optimal pre-trade algorithmic execution, by working with volume measures and generic price dynamics. Volume measures are the continuum analogies for discrete volume profiles commonly implemented in the execution industry. Execution then becomes an absolutely continuous measure over such a measure space, and its Radon-Nikodym derivative is commonly known as the Participation of Volume (PoV) function. The four impact cost components are all consistently built upon the PoV function. Some novel efforts are made for these linear impact models by having market signals more properly expressed. For the opportunistic cost, we are able to go beyond the conventional Brownian-type motions. By working directly with the auto-covariances of the price dynamics, we remove the Markovian restriction associated with Brownians and thus allow potential memory effects in the price dynamics. In combination, the final execution model becomes a constrained quadratic programming problem in infinite-dimensional Hilbert spaces. Important linear constraints such as participation capping are all permissible. Uniqueness and existence of optimal solutions are established via the theory of positive compact operators in Hilbert spaces. Several typical numerical examples explain both the behavior and versatility of the model