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This thesis is a quantitative study of the notion of 'market impact' and develops a new quantitative framework to understand the main determinants of execution costs. In the first part, we develop a market-microstructure model in which the market impact of an order is a latent variable that depends on the aggregate activity of the order book, as summarised by the order flow imbalance. The model stems from the observation that at each point in time, the liquidity in the limit order book available for immediate execution is very small, and most of the liquidity remains hidden. We empirically test this model using the Kalman filter and show that it delivers good out-of-sample forecasts. The second part is devoted to the identification of the main determinants of execution costs. Using a large dataset of institutional investors' transactions in the US equity market, we find that, for orderly trade executions, the main determinant of execution costs is not the 'market impact' of trades but price volatility. Our causal analysis shows that price changes during execution are mainly driven by the aggregate order flow imbalance, rather then by the characteristics of individual trades, and that causal claims that directly relate individual trade characteristics with execution costs may be spurious. In the third part, we investigate the notion of cross-impact. We show that the observed correlation between the returns of an asset and the order flow of other assets is due to the presence of common components in order flow across stocks, which may naturally arise from multi-asset trading strategies. We provide empirical evidence from order flow and returns of NASDAQ-100 stocks to supports this explanation. This leads to a parsimonious approach for causal modelling of multi-asset impact, which does not require introducing any concept of cross impact.

Strongly coupled systems are elusive and not suitable to be described by conventional perturbative approaches. However, they are ubiquitous in nature, especially in particle physics. The lattice formulation of quantum field theories provided a unique framework in which the physical content of these systems could be precisely determined. Combined with numerical techniques, the lattice formalism allowed to precisely determined physical quantities describing the thermodynamics, as well as the spectroscopy of strongly interacting theories. In this work, the lattice formulation has been employed to probe the effectiveness of a recently proposed method, which aims at determining the renormalized energy-momentum tensor in non perturbative regimes. The latter plays a fundamental role to quantitatively describe the thermodynamics and fluid-dynamics of hot, dense systems, or to characterize theories that enlarge the actual standard model. In all these aspects, only a non perturbative approach provides physically reliable results: hence a non perturbative determination of the energy momentum tensor is fundamental. The new method consists in defining suitable lattice Ward identities probed by observables built with the gradient flow. The new set of identities exhibits many interesting qualities, arising from the UV finiteness of such probes, and allows to define a numerical strategy for estimating the renormalization constants of the lattice energy-momentum tensor. In this work the method has been tested within two different quantum theories, with the purpose of understanding its effectiveness and reliability.