VWAP OPTIMAL EXECUTION Deterministic and stochastic approaches

Understanding market impact and optimizing trading strategies to minimize market impact has long been an important goal for investors who wishes to execute large orders. Nowadays, among all strategies they can choose, VWAP orders correspond to implementation strategies where traders act with market volume in the attempt to achieve an average execution price equal to the VWAP (Volume Weighted Average Price) benchmark price. In a framework inspired by Robert Almgren and Neill Chriss' original market model, VWAP strategies are analysed in the presence of permanent and temporary impact in order to provide a closed-form solution of the problem. Firstly, following Oliver Guéant's studies in his "The Financial Mathematics of Market Liquidity", the problem boils down to find an optimal deterministic control minimizing a functional, with mathematical tools from variational calculus. Secondly, going beyond the deterministic case, thanks to the results of Alvaro Cartea, Sebastian Jaimungal and José Penalva in "Algorithmic and High-Frequency Trading", two explicit closed-form optimal execution strategies to target VWAP are provided, under general assumptions about the stochastic process followed by the volume traded. So, in short, our main goal consists in trying to check similarities and/or differences between the results we compute along this dissertation.