Interest rate products form a large segment of over-the-counter derivatives. When the interest rate became negative, for the first time, in July 2009, interest rate models needed to adjust. Where first a log-normal model, as the Brace Gatarek Musiela (BGM) model, might have seemed logical for interest-rate products, as they were
bounded by zero, now a normally distributed model, as the Hull-White model, could be considered more practical. To our knowledge, no comparison of the Hull-White model and the Brace Gatarek Musiela model has been made on the Expected Positive Exposure (EPE) (and thus Credit Valuation Adjustment (CVA)) of a swap portfolio. Therefore, this thesis compares the Hull-White model with the BGM model on the EPE of a swap portfolio. First, we show how both models can be simulated with Monte Carlo simulation and calibrated to caplets, after which we validate the used simulation. Finally, both models are compared on the convergence, computation time and EPE. It was found that the Hull-White model had a faster convergence and computation time than the BGM model for our implementation. Moreover, it was shown that the Hull-White model and BGM model have significantly different swap EPEs, except for far in-the-money (ITM) swaps and single payment swaps. Therefore, the models used for the EPE of a swap portfolio have a model risk.

Interbank-offered-rates play a critical role in the hedging processes of banks, hedge funds or institutional investors. However, the financial stability board recommended to replace these rates by alternative risk-free-rates at the end of 2021. The new rates will be backward-looking rates and therefore, the payoff definitions of interest rate derivatives will change and the currently used Libor Market model to price exotic interest rate derivatives is no longer feasible. This thesis examines a new type of model, the forward market model, which is able to generate both the new backward-looking rates as the current forward-looking rates under the same stochastic process. Besides, contrary to the Libor Market Model, the dynamics under the risk-neutral measure can obtained. Consequently, the new forward market model should always be chosen over the Libor market model. Two issues regarding the forward market model are also considered in this thesis. First of all, the forward market model cannot deal with negative interest rate, this is solved by implementing a shifted version of the log-normal model. Second, a log-normal model is unable to reproduce the implied volatility smile which is present in the market. We solve this issue by combining the forward market model together with the SABR model. Under a few assumptions we derive the shifted SABR forward market model which hasn't been derived in the literature. The model is validated by pricing a new type of caplet that will be present in the post-Libor world, where the payoff won't be known until the payment date. We find that the implementation of this new shifted SABR-FMM can accurately price zero-coupon bonds and caplets in the market. Therefore, we conclude that this new type of model is a possible solution to price exotic interest rate derivatives in the post-Libor world.

In this research, different models are used to construct volatility surfaces and these models are compared with each other in terms of accuracy. The models range from the SSVI to neural networks. Specifically, we look at the SSVI, the feedforward neural network and the gated neural network. Attention is also paid to the incorporation of financial conditions in the considered models. In addition, we propose a framework which uses two neural networks in combination with the weighted mean squared error as a loss function to construct a volatility surface belonging to one trading day. We found out that this approach appears to be very accurate and outperforms all the other approaches. The thesis is structured in such a way that it starts with the construction of a volatility surface for 1 trading day, later in the research for the sake of examining the robustness of the models we used data belonging to several trading days.

By sampling financial correlation matrices over sliding windows, it has been shown in recent work that the quantum majorization induced partial ordering on this space of correlation matrices known as the "quantum Lorenz ordering" (QLO) can be used to characterize systemic risk by clustering correlation matrices according to their degree centrality on the associated directed graph called the "quantum majorization graph" (QMG). In this work, clusterings of the QMG are used to construct an online Bayesian nonparametric alarm system for the prediction of stock market crashes via the so-called "reinforced urn process" (RUP). To test the efficacy of this modelling methodology we exploit extreme value theory to systematically define stock market crashes by studying the tail of an appropriately fitted generalized pareto distribution (GPD) for stock market drawdowns. This approach identified 13 extreme drawdowns between 1985-2020, for which the RUP was trained from 1986-2005 to predict the 8 extreme drawdowns from 2005-2020. Of the three correlation metrics used to test this approach, the QLO corresponding to the set of upper Tail-Dependence matrices was shown to outperform the others: Pearson's and the Gini correlation. Tail-Dependence was able to predict all 8 crashes with just 5 false alarms over a 12 month time horizon, all 8 with 7 false alarms over an 8 month time horizon, 7 out of 8 with 9 false alarms over a 4 time horizon, and 7 out of 8 with 17 false alarms over a 2 month time horizon. This approach was then tested against the usage of the Log-Periodic Power Law Singularity (LPPLS) model's confidence indicators with promising results. The quantum Lorenz ordering is meant to rank a set of correlation matrices by the amount of dispersion reflected in their spectra: a true heterogeneity. We consider this dispersion from the standpoint of measurement error as has been in the application of random matrix theory (RMT) to correlation matrices in portfolio risk theory. We provide analytical relations between quantum majorization and random matrix cleaning for a few RMT filtering schemes posing quantum majorization as a desirable condition for RMT filtering. The RUP is tested using these RMT cleaned correlation matrices as well.

This thesis is devoted to option pricing on backward-looking rates. For the last decades, interest rate products were often linked to IBOR rates. IBORs are short-term borrowing rates charged between global banks in the unsecured interbank market. The purpose of this thesis is to compare the Hull-White model to the Black-Karasinski model for the pricing of caps/floors on compounded rates. Both models are so-called short-rate models, which are widely used for interest rate modelling. Due to the IBOR reform, new products are expected to appear in the market. One type of these products is caps/floors linked to the new Risk-Free Rate (RFR). The new RFRs will be in-arrears backward-looking rates and, as a consequence, have an impact on the choice of pricing models.
This thesis considers caps/floors on the new compounded RFR rates. For both models various pricing techniques for caps/floors on compounded rates are investigated. For the Hull-White model, the pricing kernel approach and a Monte Carlo simulation are explored. The pricing kernel approach yields an analytic formula for caps/floors on compounded rates. This formula is also used for the comparison. For the Black-Karasinski model, the pricing kernel approach, the trinomial tree method and the Monte Carlo simulation are considered. The pricing kernel approach yields a semi-analytic formula for caps/floors on compounded rates. However, in practice the computation time of this semi-analytic formula turned out to be substantial. Further, despite the fast computation time of the trinomial tree for LIBOR caps/floors, the trinomial tree method is rather slow for the caps/floors on compounded rates. As a result, the Monte Carlo simulation is the most suitable pricing technique, along the three
explored methods, for caps/floors on compounded rates under the Black-Karasinski model. Therefore, the Monte Carlo simulation is used for pricing caps on compounded rates in the model comparison.
Since there exists no liquid market yet for caps/floors linked to the new RFR, the models are calibrated to a proxy market. First, the Black-Karasinski model is calibrated to the proxy market using a heuristic approach. This heuristic approach is chosen as a compromise between computation time and accuracy. Then, the Hull-White model is calibrated to the Black-Karasinski model using the stripping method. Having both models calibrated, the price of caplets/floorlets on compounded rates are calculated. Thereafter, these prices are inverted to Bachelier implied volatilities for a uniform comparison. From the comparison of the two models, a difference in Bachelier implied volatility is observed in a range of 4 to 4 bps. This is of one order less than the volatility itself.

In the fight against money laundering, demand for data-driven Anti-Money Laundering (AML) solutions is growing. Particularly anomaly detection algorithms have proven effective in the detection of suspicious customer behaviour, as well as observing patterns otherwise hidden in customer transaction data. In this thesis, the Isolation Forest anomaly detection algorithm is studied in combination with the model-specific local explanation method, Multiple Indicator Local Depth-based Isolation Forest Feature Importance (MI-Local-DIFFI). To expand Isolation Forest to mixed-attribute data sets, the incorporation of nominal features is explored in more detail. This analysis resulted in the introduction of Isolation Forest with Categorical Sampling (iForestCS ), a methodology that directly incorporates nominal attributes into an isolation tree without the need of encoding it onto a numerical scale. This method is tested against different encoding strategies and Isolation Forest Conditional Anomaly Detection (iForestCAD) using different synthetic data sets. The method shows improved performance to the utilization of encoding strategies for different parameters of the underlying synthetic data. Furthermore, this thesis explores the potential of ternary Isolation Forest, in which the branching strategy of an isolation tree is expanded to produce three child nodes. It is demonstrated using synthetic data, that particularly the performance of MI-Local-DIFFI reduces when applied to a ternary Isolation Forest. Finally, the research considers a practical use-case. Using customer transaction data from Triodos Bank, the locally explainable Isolation Forest is applied to mixed-attribute customer transaction data. This has provided useful insight and resulted in the detection of suspicious customer behaviour and the introduction of new rules into business practices. Although the most interesting customer behaviour did not directly emanate from the nominal attributes, the method of incorporating nominal features resulted in differences when considering the anomalies with the highest anomaly scores.

Backward stochastic differential equations (BSDE) are known to be a powerful tool in mathematical modeling due to their inherent connection with second-order parabolic partial differential equations (PDE) established by the non-linear Feynman-Kac relations. The fundamental power of BSDEs lies in the fact that with them one does not merely obtain the solution of the corresponding PDE but also its spatial gradient through the control process Z. Classical numerical methods tackling the system face the so-called curse of dimensionality and cannot be used to solve high-dimensional problems. In recent years, multiple approaches have been developed to overcome this computational burden, building on deep learning and showing remarkable empirical success even beyond 10 dimensions. However, such Deep BSDE methods struggle with giving accurate approximations for the Z-process throughout the whole time horizon. In this thesis we propose a novel approach aimed to give better estimations for the control problem, exploiting the natural dynamics of the Z-process given by Malliavin calculus. The proposed methods use deep learning parametrizations taking advantage of the universal approximation capability of neural networks. The Malliavin derivatives are estimated through the Malliavin chain rule. Two discrete numerical methods are developed which are called One-Step Malliavin (OSM) and Multi-Step Malliavin (MSM) schemes respectively. An error analysis is carried out proving the consistency of the algorithms and showing first-order convergence under certain assumptions. Numerical experiments are presented to demonstrate the efficiency of the Malliavin formulation compared to other Deep BSDE solvers.

In this thesis we build a goal based planning framework that takes into account goal priorities. Goal based planning is a type of personal wealth planning, with a focus on the feasibility of an investor's goals. Currently, to take goal priorities into account a dynamic asset allocation is created. We focus on incorporating goal priorities by ensuring goal cash flows reflect investors' preferences, no matter the asset allocation. No money should be spent on less important goals if it jeopardises achieving more important goals. With such a goal cash flow strategy, an investor gains valuable insight into the feasibility of her goals. We build such a framework in two ways. First, we use utility theory. With utility theory, we can quantify an investor's satisfaction with the goal cash flows. For that purpose, we create a total utility function that takes into account goal priorities. Then, we optimise over the goal cash flows such that it results in the highest expected total utility. Second, we use conditional expectations. With conditional expectations, we determine the relationship between earlier wealth and future goal achievability. Therefore, we create a framework with a spending rule based on the conditional expectation. We implement conditional goal cash flows. With conditional cash flows, we create a goal based planning framework that takes into account goal priorities for multi-goal investors.

Machine learning methods like outlier detection are becoming increasingly more popular as tools in the fight against money laundering. In this thesis, we analyse the Isolation Forest outlier detection algorithm in detail and introduce a new local explanation method for Isolation Forest, the MI-Local-DIFFI (Multiple Indicator Local-DIFFI) method. The method uses the structure of the isolation trees and the traversal of individual outliers down the trees to determine an importance weight for each of the features. These weights are then combined into feature importance scores that are used to explain why a specific outlier is identified as such. In anti-money laundering (AML), such explanations are very valuable when determining whether an outlying customer is suspicious or not. MI-Local-DIFFI is based on a global explanation method called DIFFI and while we were conducting our research, another local version, Local-DIFFI, was also introduced. In the thesis, we use a synthetic data set to compare the performance of four different explanation methods including our MI-Local-DIFFI, Local-DIFFI and the state-of-the-art TreeSHAP method. Our MI-Local-DIFFI shows excellent results in terms of performance and runtime. Furthermore, we use a data set from Triodos bank to apply the explainable outlier detection methodology consisting of the combination of Isolation Forest and MI-Local-DIFFI. This resulted in interesting findings like the revealing of data quality issues in the current system and references to EDRs and SARs. However, after further inspection, no SARs were filed but some customers were put to higher risk classes. This procedure will be performed on a monthly basis with the goal of continuing to improve the AML processes of the bank.

In this thesis we introduce valuation techniques to price electricity storage contracts, where the electricity prices follow a structural model based on polynomial processes. In particular we focus on a Fourier-based pricing method known as the COS method, which performs impressively to price the contracts accurately. We provide details on how to formalize an electricity storage contract, taking into account the physical limitations of an electricity storage and the operational constraints of the electricity grid. In addition to the electricity storage contract, other well-known options are being considered, such as the European option, Bermudan option and Bermudan option with multiple early-exercise rights, where the same asset price model is used based on polynomial processes. We propose an approximation of the characteristic function, so that the Fast Fourier Transform (FFT) can be applied to significantly reduce the computational complexity of the COS method, which is especially suitable for pricing Bermudan options and Bermudan options with multiple early-exercise rights. With the FFT-based algorithm, the computation time of the valuation of the discussed Bermudan-type options with the COS method is reduced from seconds to milliseconds. Furthermore, the Least Squares Monte Carlo (LSMC) method is presented to value the discussed financial derivatives and used to validate the results obtained with the COS method.

In this study, two interest rate models are analysed in context of counterparty credit risk. The goal of the study is to find a model that performs well on historical simulation for the PFE and EPE. The two models analysed are the Dynamic Nelson-Siegel model and the Displaced Diffusion model.
In the Dynamic Nelson-Siegel model, a Nelson-Siegel curve is fitted against the historical yield curves. The fit gives an historical series of the parameter values of the Nelson-Siegel curve, which are modelled via a stochastic process to obtain future yield curve predictions. In historical backtesting, the classic model using AR(1) processes for the parameters performs inadequate. Analysis on the underlying assumptions of the model show that the mean-reverting behaviour that is modeled is the cause. In addition the data is likely to feature heteroskedastic behaviour, which is not incorporated by the model. An adjusted model in which one parameter is modeled with a random walk with drift performs well on longer maturity rates, however shorter maturity rates are not modeled satisfactory.
The Displaced Diffusion model uses a lognormal diffusion process that is shifted to model Libor rates. As it is a Libor market model, all libor rates are modelled seperately using correlated Brownian motions. The shift parameter allows negative rates to be modeled, and is initially assumed constant. The backtesting results are mixed; some observed libor rates are modeled inadequately and some cannot be rejected to come from the Displaced Diffusion model and thus are modelled correctly. When backtesting the PFE, the results are good at the short term. At the 2-year window, PFE estimates are not always conservative but the number of excesses are of medium severity when compared to the probabilities used in the green-orange-red system dictated by the Basel committee for VaR backtesting.

The bond market is affected by the shortage of liquidity problem, which means that many bonds are not frequently traded. This implies that market data for these bonds are missing. This lack of data represent a problem for financial risk measures such as Value at Risk (VaR). This research provides the framework for the construction of a proxy which replaces the missing data with artificial data such that VaR can be calculated. The data used for the VaR calculation are bond z-spreads, which is a credit spread measure. This research represents an improvement of the current proxy methodologies under different aspects. A major improvement is provided by the usage of machine learning algorithms such as Random Forest, Support Vector Regression and CatBoost which significantly increased the predictive accuracy of the proxy. Another main difference from the current proxy methods relies in the
prediction of z-spreads daily changes (shifts), instead of z-spread levels. This modification required a shift types assessment and it has been beneficial both for the proxy performance and for the VaR calculation. The main result of this thesis from a financial and statistical perspective is the theoretical and empirical convergence of the VaR obtained through the proxy with the VaR calculated with real market data.

This thesis showcases a rather contemporary method of solving a generalized system of stochastic differential equations (SDE's) comparable to the SABR model. The solution is derived from a stochastic-local volatility (SLV) model in which the local volatility (LV) component is kept general. This generality is maintained throughout all derivations, eventually yielding a model containing an undefined LV function. This function can then be specified however the user of the model deems suitable, as long as minor constraints are satisfied. Obviously, this is a very valuable quality of the model as it is highly customizable. The solution consists of a set of pricing functions that seemingly possess all the aforementioned desirable properties, i.e. fast in evaluation, computational tractability, flexibility etc., with little disadvantages. The generalized SLV model that is used is typically denoted in the form of two SDE's, though in the majority of this thesis an atypical three SDE form is used. This extended system is used to isolate the LV component, in turn enabling for appropriate application of an SDE transformation called the Lamperti transform, which will provide the key to solving the entire system. The Lamperti transform is a highly versatile method for transforming SDE's into new equations typically more suitable for simulation and parameter estimation procedures, and its inner-workings and various applications will be the main focus of this thesis.

This thesis is about pricing European options using a Fourier-based numerical method called the COS method under the rough Heston model. Besides examining the efficiency and accuracy of the COS method for pricing options under the rough Heston model, it is also investigated if the rough Heston model produces the advantages of the so-called rough volatility models. To do so, the characteristic function of the rough Heston model is derived, and the COS method for the rough Heston model and also a Monte Carlo simulation scheme is introduced. Throughout the thesis, the theoretical background of the rough Heston model, the numerical techniques and some numerical experiments on European option prices and implied volatility behaviors are presented. Also, a calibration of the rough Heston model is performed using Artificial Neural Networks. As a result of this thesis, pricing of European options using COS method is succeeded. Moreover, it is shown that the rough Heston model produces the rough volatility behaviors as expected.

In financial and egineering problems, we are often faced with solving Partial-Integro Differential Equations (PIDEs). Rarely we can find an analytic solution in a closed form expression for these PIDEs, hence we turn to numerical schemes to accurately approximate the solution instead. Classically these methods are based on finite difference methods, however, we can turn certain kinds of PIDEs into a probabilistic representation, called Forward Backward Stochastic Differential Equations with Jumps (FBSDEJs). Solving the PIDE can now be done alternatively by solving a FBSDEJ. In this thesis we will first investigate the stochastic framework behind FBSDEJs and we will look into the uniqueness and existence of their solutions. Furthermore we propose a new numerical method which can efficiently solve FBSDEJs. The semi-discretisation is based on the classical Backward Differentiation Formula (BDF) methods, for the computation of the conditional expectations we use the COS method which makes use of Fourier cosine expansions, exploiting the knowledge we have about characteristic functions. Finally we implement the new method and we investigate it extensively both numerically and theoretically. We show that the BDFn schemes are highly stable and efficient for computing FBSDEJs, the initial steps still have to be investigated in greater detail so that we can make use of the high-order BDFn schemes.

This thesis explores existing and proposes new methods for assessing concentration risk in default-only credit risk models. Within the existing methods, the analytic Granularity Adjustment is studied in the single factor Gaussian threshold and in the CreditRisk+ framework. These adjustments are tested on a sample portfolio in the presence of recovery risk, and we show that the CreditRisk+ adjustment is more conservative than the Gaussian threshold adjustment. Furthermore, we show that in the presence of recovery risk, the accuracy of the adjustment on exposure level deteriorates. Additionally, the Granularity Adjustment is extended to an independent single factor \textit{t}-threshold model to account for heavier tailed asset returns. Based on the independent single factor \textit{t}-threshold model, we suggest an ASRF equivalent that could serve as an alternative to the current IRB framework. Although much existing literature is focusing on developing analytical methods for measuring concentration risk, recent advances in computational speed make Monte Carlo methods an interesting substitute for measuring concentration risk. To this extend, we propose a split between Monte Carlo based Economic and Regulatory Concentration Risk and show that these measures do not coincide for a given portfolio. This method involves a novel way of assessing idiosyncratic risk in multi factor frameworks. In order to assess sector concentration risk, this thesis proposes a Diversification Factor and a Capital Diversification Index as risk management tools. Finally, this thesis provides a clear account of the effects of concentration, diversification and recovery risk on the portfolio loss distribution for both Gaussian and \textit{t}-threshold models. This thesis was carried out in close cooperation with ING Bank.

Currently, quantitative asset pricing models are often not equipped to deal with merger and acquisition events. In such cases, portfolio managers make the assumption that the model is not working and they override its decisions for an entire year. This thesis studies the performance of quantitative models after these events and provides research based guidelines for future decisions on model overrides.A selection of investment factors representing the model will be used to explain the post-event abnormal returns. In general, the data consists of monthly observations on one cross-section of firms over a time period of several years. Consequently, it may contain unobserved firm or time effects. Ignoring such effects leads to inefficient estimates and biased results. Hence, robust inference is conducted by adjusting the standard errors of a pooled regression, and through modelling the effects in a panel regression.Multiple approaches show that the separate factors are not sufficiently able to explain the abnormal returns after an event, if compared to the model's performance in regular market circumstances. Consistent results for the post-event performance are particularly hard to find. Equally weighting all factors in a single regressor, leads to model performance being equivalent to that in non-event times after 9 months, which indicates that a override period of equal length is sufficient. Therefore, the current assumption that the model is not working properly after a merger or acquisition, is correct. The robust pooled model is favoured over the panel model in this research, due to its low complexity and its straightforward results. 

Generative adversarial networks (GANs) have shown promising results when applied on partial differential equations and financial time series generation. This thesis investigates if GANs can be used to provide a strong approximation to the solution of stochastic differential equations (SDEs) of the Ito type. Standard GANs are only able to approximate processes in conditional distribution, yielding a weak approximation to the SDE.
A novel GAN architecture is proposed that enables strong approximation, called the constrained GAN. The discriminator of this GAN is informed with the random sample that corresponds to the Brownian motion increment between two time steps. This way, the constrained GAN does not only learn the conditional distribution, but the unique map from a random increment to the next asset value along the path, conditional on the previous value. The architecture was tested on geometric Brownian motion (GBM) and the Cox Ingersoll Ross (CIR) process in one dimension, where it was conditioned on a range of time steps and previous values of the asset process. The constrained GAN was shown to outperform discrete-time schemes in strong error on a discretisation with large time steps. It also outperformed the standard conditional GAN when approximating the conditional distribution. A method is proposed to extend the constrained GAN to general one-dimensional Ito SDEs, beyond the SDEs tested in this work. In future work, the constrained GAN should be conditioned on the SDE parameters as well, allowing it to learn an entire family of solutions at once. Furthermore, the architecture could be extended to higher dimensions, including systems of SDEs, such as the Heston model.

Demand deposits modeling is of top importance for banking institutions and usually represents a large part of a bank portfolio. Even though these products seem rather simple at first glance, demand deposits are without a fixed maturity, generating uncertainties in the model. A significant amount of academic literature on this subject is available. However, we are experiencing negative interest rates for a few years now, that may have affected customer behavior and have led to an excess in deposits-taking for a large majority of bank. In addition to the low-rate environment, this small European bank has been growing rapidly in recent years and new customers have different characteristics from the old ones. Demand deposits modeling is therefore a major challenge for the bank. It can be divided in three steps, respectively the market rates, the deposit volumes and the deposit rates, however, only the market rates and the deposit volumes will be considered in this thesis. The dynamics of market rate follow a single factor Hull-White model. The mean-reverting parameter and the volatility are calibrated on historical data of the one-month Euribor rates, and then simulated using an exact Monte-Carlo approach. The deposit volume model is based on an Ornstein-Uhlenbeck process, where the constant drift term μ is replaced by a trend that depends on the rates' level. The trend has a different slope whether we are in a normal-rate environment or in a low-rate environment, and the probability of being in either state depends on the market rates. We create age and wealth categories for customer segmentation purpose. We then perform a cluster analysis, either using a k-means method or a hierarchical clustering algorithm, that will be included in the deposit volume model. The clustering forms similar groups of customers, reflecting better customers' diversity. Lastly, we compare two output variables, the average life and the optionality, for different simulations, one without the clustering, and two with the clustering. The best-case scenario for any bank being a high average life associated with a low optionality with regards to demand deposits modeling, the clustering integration in the model leads to more optimal results.

The yield curve represents market supply and demand implied expectations of future interest rates and is calibrated from the most liquidly traded interest rate derivatives like cash deposits, forward rate agreeents, swaps and futures. Due to the daily margining mechanism of futures contracts, interest rate futures require the substraction of a convexity adjustment in order for them to be used in curve calibration. It is common practice to use externally computed convexity adjustments, which treats the convexity adjustment as a black-box parameter. We will argue the inherent relationship between the convexity adjustment and cap/floor volatility smiles and derive a nested calibration algorithm for the simultaneous calibration of the yield curve to futures and the convexity adjustment to cap/floor volatility surfaces. This introduces dependencies of the yield curve to option volatilities and we will argue that for simple interest rate derivatives the implied vegas are negligible. .