Electronic trading algorithms are at the centre of every buy-side equity trading desk. These algorithms rely often on market impact models, which are stochastic models for the stock prices that account for the feedback effects of trading. Propagator models are central tools for describing the evolution of market impact during and after a trade. This thesis extends the linear propagator model by proposing a new variant that incorporates time-varying liquidity and general decay kernels. To bridge the gap between theory and practice, we use Robeco's proprietary order data base to calibrate the model and validate its performance. The main findings reveal a two-stage decay pattern of market impact, the absence of a single best admissible decay kernel, and a model performance which is in line with our expectations based on the low-signal to noise ratio of the data. The main application of the model in this research is its use in the optimal execution problem, in both a intraday and multiday setting. In an intraday setting we formulate the problem for a risk aware trader and incorporate short-term alpha signals. The discrete analogs of these problems are solved analytically and we highlight significant cost reduction compared to industry benchmarks. In the multiday framework, we quantify the expected cost of trading two adjacent orders and use this to find optimal multiday execution strategies. In a final simulation study we quantify the expected cost of rebalancing two similar investment accounts on consecutive days with a varying number of overlapping stocks. The simulation study accentuate a significant additional cost for the account trading on the second day, which stresses the importance of multiday cost management in rebalancing investment accounts.

This thesis is concerned with finding the asymptotic distributions of linear spectral statistics of the nonlinear shrinkage estimator for large covariance matrices derived by Ledoit and Wolf (2012). It provides some new inferential procedures for large-dimensional data and shed some light on the power of the new statistical tests on the structure of the large covariance matrices. After a short review of the relevant theory, two linear spectral statistics are proposed which are deduced from the nonlinear shrinkage estimator where for one of these linear spectral statistics its limiting distribution is derived. This results in a ready to use sphericity test statistic in the large-dimensional framework and is one of the main results of this thesis. The sphericity test corresponding to this new test statistic is called the nonlinear shrinkage test (NLS-ε). The theoretical results are illustrated by means of a simulation study where the new nonlinear shrinkage test is compared with al- ready existing tests, in particular the commonly used corrected likelihood ratio test and the corrected John’s test. It is demonstrated that the new nonlinear shrinkage test is most powerful under a non homogeneous variance alternative where it outperforms well known sphericity tests. Moreover, it is observed that the new nonlinear shrinkage test encounters some problems when different alternatives are combined.