The digitalisation of financial markets has led to an increase in the share of trades being executed by algorithmic trading systems. A key application of these algorithms arises when a trader needs to buy or sell a large volume of shares within a fixed time window. The trader must decide between placing reliable but costly market orders or cheaper limit orders, at a risk of not filling her target amount within the desired time frame. This is known as the Optimal Placement Problem (OPP) and this thesis aims to provide a general solution to this problem. Our research builds forth upon two models. The first, proposed by Cont and Kukanov [2017] [15], formulates the problem as a stochastic convex optimization problem. This model provides a static solution to the OPP, where the trader can only place orders at the initial time. We extend their model beyond the best quote price-level to consider any number of price-levels. This generalization still allows us to derive analytical expressions for the optimal placement strategy, including a closed-form solution under the assumption that market and limit order arrivals follow exponential distributions. However, large deviations in the calibrated parameters or distributions can lead to substantially higher costs. A dynamic strategy that allows the trader to adjust her placement strategy at intermediate times offers increased flexibility and resilience against such unexpected order flow deviations. The second model, proposed by Cartea and Jaimungal [2015] [5], provides such a dynamic solution and frames the problem as a stochastic optimal stopping and control problem. We extend this model to include mid-price drift and a general running inventory penalty, and formulate a recursive expression for the general optimal strategy for an inventory of any size. The static solution gives an accurate, efficient and robust indication of the optimal allocation for larger inventories and longer trading windows. The dynamic solution, on the other hand, is more effective at liquidating (or acquiring) smaller inventories in shorter time frames at an optimal price, by providing a precise, adaptive strategy in volatile market conditions. Overall, this thesis improves our understanding of the Optimal Placement Problem by analytically deriving its solution under two distinct model setups: the static model of [15] and the dynamic model of [5]. In the static approach, we find the optimal allocation switches from limit orders at one price-level to the next at distinct critical points. This provides an intuitive and broadly applicable rule of thumb for practical implementation, leading to lower expected costs across the board. The optimal strategy, derived from the dynamic model, leads to higher average earnings per share than benchmark models like the Time Weighted Average Price, the solution only considering limit orders and the solution found in [5], by taking into account market orders and possible mid-price drift. Future extensions, such as testing on different arrival distributions, depth discretisation, parameter calibration and using heuristic methods to improve computational speed, could further assess the practical viability and performance trade-offs of these strategies in real-world trading systems.

Random walks have been applied in a many different fields for a long time. More recently, classical random walks are being used in wide variety of computer algorithms used to solve complex computational problems like 2-SAT, 3-SAT and the estimation of the volume of complex bodies. In this thesis we look into the quantum version of the familiar random walk, distinguishing between the discrete- and continuous-time quantum randomwalk. Some general properties of random walks are studied throughout this thesis. First we look at the behaviour of both the classical and the random walk on a simple 1-dimensional lattice. Then, to investigate the speed and efficiency of algorithms that use quantumwalks, we analyse quantumrandom walks in graphs, as graphs do a good job of representing the decision trees of algorithms. The graph we look at specifically is the ndimensional hypercube. The criterion we use to see how fast a quantumwalk can traverse certain types of graph is the hitting time. The hitting time is the expected amount of time a random walk takes to reach a certain vertex in a graph for the first time. Literature has shown that classical random walks have a hitting time that scales exponentially with the dimension n of the hypercube. We find that quantum random walks offer a significant speed-up to its classical counterpart. Generally they have a polynomial hitting time, that depends on the Hamming distance between the start and sink vertex. Furthermore we find that quantum walks can have interesting properties such as a non-unitary hitting probability, meaning the quantum walker will never visit the sink vertex. Finally we see that, in some simple, symmetric cases, the hitting time of the quantum walk even is sub-linear, scaling almost like a square root, rather than a polynomial.