Financial markets continue to see an increase in the share of trades executed by algorithmic trading systems. A key component of an efficient algorithmic trading system is its ability to accurately estimate the probability an order will be executed: the fill probability. This thesis aims to determine whether the dynamics of the order book can be captured accurately by a stochastic model, which subsequently leads to the following research question: Can we accurately compute fill probabilities of limit orders, conditional on the state of the order book? Our research builds upon the stochastic model proposed by Cont, Stoikov and Talreja, which assumes orders and cancellations have unit size and arrive according to independent Poisson processes [12]. There are two main advantages of the model proposed by Cont et al., which include: • Straightforward parameter estimation based on observable order book data, • The possibility to compute various probabilities of interest semi-analytically. These probabilities include the fill probability of limit orders and the direction of the next mid-price move. In this model, the number of orders at each price level in the order book is modelled as a birth-death process, in which births represent an increase in the number of orders and conversely, deaths represent a decrease. The first-passage time is the time it takes for such a process to reach a certain state for the first time. Since the order arrivals follow Poisson processes, we can find an expression for the Laplace transforms of the density functions of the first-passage times. By numerically inverting these Laplace transforms we can calculate the probabilities of interest. The parameters of the Poisson processes that determine the arrivals of orders are calibrated using observable order book data of the Euro-US Dollar currency pair of the foreign exchange (FX) spot market. We extend the model of Cont et al. by integrating spread-dependent arrival rates of orders and cancellations, meaning that the arrival rates change for different sizes of the spread. This extension is based on the order book data, which clearly shows a difference in arrival rates for various spread sizes. Additionally, we provide an expression to compute fill probabilities of orders posted not only at the best bid price, but also for one price level below. This expression can be generalised to obtain an expression for the fill probabilities of orders posted even deeper in the order book. Finally, by employing this stochastic model, we also show that this model has the potential to be applied to asset classes other than equities, which in our case is the FX market. To evaluate this model, we compare the computed probabilities with the empirical probabilities based on the FX data. Our findings indicate that the model has the ability to effectively capture the dynamics at each price level and the short-term movements of the mid-price. We are able to show that the fill probability for orders posted at the best quote can be estimated with reasonable accuracy when the order has the highest priority of execution. For orders with lower priority, the model is not able to capture all the dynamics, resulting in most cases in an overestimation of the fill probability. Additionally, we evaluate the method for computing the fill probability at one price level lower than the best quote. We also compare the results of this method to empirical probabilities. However, it is challenging to evaluate the performance of the model due to limited data availability and the small magnitude of the computed probabilities, which are typically less than 1.0%, resulting in large relative errors. Overall, this research contributes to better understanding order book dynamics and provides insights on the precision of the computation of fill probabilities via an extension of [12]. Further improvements and refinement of the model, such as a more realistic order flow or allowing multiple order sizes, could lead to more accurate estimations and a better comprehension of the market dynamics and fill probabilities.

In dit verslag beschrijven we met behulp van een wiskundig model hoe een malaria infectie zich ontwikkelt in een persoon. We gebruiken hiervoor een stelsel van gewone differentiaalvergelijkingen, wat beschreven wordt in een artikel van Mary Bushman. Hierbij volgen we veranderingen in verschillende aantallen die een rol spelen bij een infectie, zoals het aantal geïnfecteerde en ongeínfecteerde rode bloedcellen, het aantal parasieten en de immuniteit, die wordt opgedeeld in een aangeboren deel en een adaptief deel. Hierbij kijken we naar de invloed van medicijngebruik en de infectiehistorie op een infectie. We gebruiken een numerieke rekenmethode om het stelsel op te lossen. We gebruiken de functie ’ode45’ in het programma MATLAB om tot een oplossing te komen. Bij een infectie door zowel vatbare parasieten sterven de vatbare parasieten eerder uit. Dit heeft te maken met de ontwikkeling van de adaptieve immuniteit tegen een bepaald soort parasiet. Het toedienen van medicijn vroeg in de infectie heeft meer effect dan later. Als laatste hebben we gekeken naar de invloed van eerdere infecties op de ontwikkeling van een nieuwe infectie. Doordat het immuunsysteem een soort parasiet al eerder is tegengekomen, geeft dit een nadeel voor de parasiet. Dit zorgt ervoor dat de parasieten sneller uit zullen sterven.