In this report an Agent-Based Model created by Bulleit and Drewek used for analysing terrorist attacks will be implemented and compared with reality. First the implementation of the model is explained based on different articles written by Bulleit and Drewek. After that the data of the simulations are analysed statistically. In this analysis the model is compared with the Global Terrorism Database for consistency with historical data and the model is compared for consistency in distributions of the data based on research reports. Finally, a few extensions will be implemented and again analysed statistically. An Agent-Based Model for terrorist activity can give us insights in the behaviour of terrorists. It can especially be used for security policies or when we want to investigate what happens with an implementation of a new type of security.

This thesis is a research into the relationship between performance and sales of new financial contracts of financial products providers in the employer’s market. This thesis is written in collaboration with IG&H Consulting. Combining the performance scores given by advisors on financial providers and the production of new contracts over the past two years, a logistic regression is fitted with a selected group of performance variables. The results show a significant positive effect of the performance on the production of new financial contracts. The model predicts a potential growth in the number of contracts when there is an improvement in the performance of providers in the eyes of advisors. The performance of the model is not fully satisfying but it is a good starting point. It can be ameliorated in the future with the collection of contract specific information.

This work is on the extension of the SWIFT method to option pricing problems where the sum of lognormals occurs. The SWIFT method (ShannonWavelet Inverse Fourier Technique) is extended to the valuation of geometric Asian options and arithmetic Asian options with a Lévy process as underlying price process and the valuation of European options under SABR dynamics. In both applications a sum of lognormals (or sum of increments) occurs. The main result in this thesis is the SWIFT-SIA method (SWIFT sinc integral approximation), which is applied to the valuation of arithmetic Asian options as well as to the valuation of European options under the SABRmodel. Within the SWIFT-SIA method the recovery of the probability density function is obtained by an approximation, instead of a numerical integration method, which results in a very fast method compared to an alternative method based on cosine expansions as well as high accuracy in the option values.

Three interest rate models are researched: Displaced Exponential-Vasicek, Hull-White one factor and Hull-White two factors with time-dependent volatility parameters. The motivation for this is two-fold: firstly, we would like to understand how the capital calculations would be impacted when yield curves are modelled under the three different models. This is done by looking at both magnitude and stability of the risk profiles and scalar risk-measures for three counterparties, which are highly representative for the bank. Secondly, we investigate the benefits and drawbacks of using one model and its corresponding calibration method over the others, with a special attention to the impact on yields correlations. The first model, calibrated to historical data, is used as a nine-factors model for forward rates and is currently being used within the bank for PFE profiles and CVA regulatory capital. Historical backtest has proven the current model to perform reasonably well on real data and therefore it is used as a benchmark against which the other two models are tested. The two Hull-White models, used as short rate models, are calibrated to the risk-neutral measure (namely, to European swap- tions). Precisely, a two-steps calibration procedure suited for piece-wise constant volatility functions is implemented for both. The stability analysis reveals that the variation of Exposure at Defaults is significant, which might be undesired. On the other side, the two short rate models retain the correlation structure of interest rates better than the current model. This in turn translates into higher capital impact.

In the Netherlands Dutch pension funds perform the feasibility test. This test is originally designed for DB pension schemes. Nowadays DC pension schemes are getting a more prominent role in the Dutch pension system. Therefore an improved design of the feasibility test for DC pension schemes would be beneficial. In this research we search for a new definition for the pension result in DC pension schemes. The pension result is an important part of the feasibility test. We prefer to stay close to the concept of pension result in DB schemes and we conclude that a pension result based on indexed pension entitlements is an appropriate definition for DC pension schemes since it measures the maintenance of purchasing power similar to how it is done in the current feasibility test. We investigate how robust this new definition of the pension result is by considering an alternative premium policy, an alternative investment strategy, an alternative pension payment policy and an extension of the financial market model. All factors have an influence on the pension result. The pension result is calculated using economic scenarios which are based on the financial market model from Koijen, Nijman and Werker (KNW model). The pension result is highly sensitive to the interest rate and inflation rate development during the pension accrual period and the retirement period. Historical data give a motivation for the use of a jump diffusion model for both the interest rate and inflation rate. These variables are modeled as diffusion only in the KNW model. An extension of the KNW model is proposed in which jumps are added to the interest rate process and the inflation rate process. The addition of jumps influences the pension result in DC schemes.

The aim of this thesis is to provide a formula for the value of a correlation swap. To get to this formula, a model from an article by Bossu is inspected and its resulting expression for fair the fair value of a correlation swap is simulated. The Jacobi process will be defined and two discretization schemes will be compared for it. Methods are discussed to make simulations of the Jacobi process as accurate as possible, making sure it crosses its boundaries -1 and 1 as little as possible. It will be shown that a correlation swap can be hedged by dynamically trading variance dispersion trades. The main result of the thesis is a partial differential equation for the fair value of a correlation swap. It will be shown that the expression for the value of a correlation swap obtained by Bossu's model satisfies this partial differential equation.

In this thesis we statistically analyze violent conflicts. The main focus lies on the risk of occurrence of large wars. We collected data that provides the total amount of casualties, for every known war, in the time span 768CE - 2019. The distribution of the data suggests a presence of a long right tail. We have used different graphical tools to determine that the tail is Paretian and to locate the threshold value u for which the tail starts. Fitting the Generalized Pareto Distribution we have found that this tail starts from the 70% quantile which corresponds to a total number of casualties of 70000. We have found through the method of maximum likelihood a shape parameter of ξ = 1.3 and a scale parameter β = 193397. The threshold and these estimates provide us enough material to determine the tail risk using the survival function. We have researched the inter-arrival times and we support the idea from earlier studies that the occurrence of wars follow a homogeneous Poisson process and that therefore no particular trend can be stated.

Wrong-way risk (WWR), which is the dependence between the probability of default (PD) and the exposure at default of a counterparty, is an aspect of credit risk that can lead to high losses. This thesis aims firstly to quantify WWR in interest rate swaps (IRSs) using a copula model, where a copula is used to couple the PD implied by credit default swap (CDS) spreads and the 10 year swap rates. Specific attention is given to choosing and fitting a copula. For datasets obtained from periods of regular economic conditions and periods of financial crisis, the t copula is found to be the best fit. In general, negative dependence between the log-returns of the implied PDs and the log-returns of the swap rates is observed, especially in the tails of high PD log-returns and low swap rate log-returns during regular periods. This means that the general demand for loans is lower during a worsening economy. During crisis periods however, the dependence is weaker than in regular periods; implying that loans are in higher demand in these times, e.g. because companies need loans to survive. The second purpose of this thesis is the introduction of a novel product aimed at hedging WWR in IRSs. This product, named an IRS partial insurance contract (IPI), introduces an upper bound to the credit losses on an IRS. If this bound is exceeded, the amount by which the losses exceed the bound is paid out by the IPI seller to the IPI buyer. An analytical expression for the no-arbitrage price of an IPI is obtained, using a copula model for WWR. This expression shows that an IPI can be replicated by a portfolio of swaptions and CDSs.

Clients with a mortgage loan may prepay a part of their loan before the contractual date. This is called prepayment. In the case of a prepayment, the bank who issued the loan earns less interest than ini- tially agreed. It is therefore essential to build accurate models for predicting prepayment behavior. In this thesis, machine learning models called artificial neural networks are used to predict conditional prepayment rates. In particular, the possibility to quantify the model uncertainty is studied. It is important to acknowledge that the accuracy of a model is not constant over its domain. Most ma- chine learning methods do not provide information about the uncertainty regarding a prediction and, unfortunately, methods that do so are often computationally expensive. This thesis studies uncertainty estimates generated by a neural network with dropout applied. Dropout is a method that randomly drops out neurons in each layer and has been suggested to prevent overfitting. We will see that this method can also be used to extract a measure of uncertainty regarding the prediction of a neural net- work efficiently. This method is used first to evaluate uncertainty for three simple functions under different distributions of the train data. Then the uncertainty estimates are evaluated for a neural network that predicts conditional prepayment rates. It will be illustrated that the uncertainty estimates provide an accurate indication of regions of the domain where predictions are inaccurate. Moreover, using a different model in the regions identified as uncertain can result in higher model performance.

This thesis develops a continuous time framework to value deferred taxes using Black and Scholes (1973) type option pricing techniques. The valuation renders a market consistent pricing procedure, which avoids the necessity of subjective accounting principles. Our framework is flexible enough to value deferred taxes like carry forward, carry back or liabilities arising from temporary differences relying solely on quantities observed in the market. A simulation study over multiple time horizons shows that carry forward value is negatively influenced by leverage, whereas carry back and tax liability values increase. Two empirical applications serve to illustrate the practical use of our model: the loss absorbing capacity of deferred taxes for European insurers and an estimate of BP's loss of deferred taxes following the U.S. tax overhaul.

This thesis is on the subject of modelling the probability of default in a low default portfolio. In these portfolios there is a high risk of underestimating the true probability of default. Two models are considered, a Gaussian one factor model and a Poisson model with Gamma mixture. Classical estimation methods as the maximum likelihood are shown to fail to produce conservative results, and therefore the Bayesian approach is considered. New on the subject is the consideration of multivariate prior distributions, which are shown to be an improvement of the univariate case.

This thesis adds to quantitative literature on terrorism by examining the relationship between various annual country statistics and the number of terrorist attacks. In addition, it assesses the potential of forecasting terrorism. Combining an extensive review of literature from social science, with data analysis of the Global Terrorism Database, results in a specific selection of attacks on country-level, along with various factors that allegedly affect terrorism. It is shown that, under certain conditions, the countries can be fit by a nonhomogeneous Poisson process model with a Weibull baseline intensity and a piecewise constant covariates component. For three out of five countries, the in-sample results are satisfactory, providing a tentative answer to the general debate whether terrorism can be explained by root causes. Out-of-sample results for Afghanistan and Somalia are promising for future research and implementation in policy making.

In this research, the returns of four cryptocurrencies (Bitcoin, Litecoin, Ripple and Ethereum) were analyzed in order to answer the following research question: “How do the returns of Bitcoin and other altcoins behave over time, and what can we say about extreme values for losses and profits?” With respect to volatility, cryptocurrencies can still be considered extremely volatile. For Bitcoin, the least volatile of the four, we found an annual volatility of approximately 70% based on daily exchange rates. For Ethereum, the most volatile of all four, this percentage was closer to 130%. Also, several distributions were fitted on the returns. It is shown that the Generalized Hyperbolic Distribution is the best fit for all four cryptocurrencies, apart from the tails in some cases. The tails were investigated seperately by using Extreme Value Analysis and by looking into both empirical and theoretical risk quantities (the Value at Risk and Expected Shortfall). Bitcoin appears to be the least risky of all four cryptocurrencies, but also the least profitable, whereas Ripple appears to be the most risky and also the most profitable. Compared to previous research, Bitcoin has also become less risky, showing a less fat tail for the losses than before. For Litecoin and Ripple, the reverse is true, as they appear to have become riskier. For Ethereum, no comparisons could be made, as this is a relatively new cryptocurrency that has not been investigated much yet. When tested for Paretianity, the left tails of Litecoin and Ripple appear to Pareto distributed: the losses seem to exhibit heavy tail behavior. For the profits, the tails turned out to be even heavier and can therefore also be considered Paretian. These results were confirmed by Maximum to Sum ratio plots, indicating infinite third and fourth moments for the losses and profits of Litecoin and Ripple, but not for Bitcoin and Ethereum. The results have implications for investment and risk management purposes.

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 extit{t}-threshold model to account for heavier tailed asset returns. Based on the independent single factor extit{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 extit{t}-threshold models. This thesis was carried out in close cooperation with ING Bank.

This thesis is about pricing European options and forward start options under the Heston LSV model. The impact of conditionally calibrating the Heston parameters on the satisfaction of the Feller condition and thereafter correcting with a local volatility surface is investigated here. The results show that this approach is computationally time efficient and accurate. Efficient numerical approaches for this LSV model, such as the multilevel Monte Carlo method, are also investigated. Furthermore, a comparison of several discretizations schemes for the SV part have been conducted. For the calibration of the local volatility surface, the efficiency of the Particle method and the Bin method are compared. An alternative numerical approach to this problem which builds on these two methods is developed and tested.

A medium size Dutch insurance company with third-party car insurance products initiated questions on whether the premium can be based on a statistical analysis where the expected future liabilities are taken into account. These questions are as follows: • Which statistical models can be used to base the premiums on expected future liabilities? • Are there enough data available to predict future liabilities accurately enough? • How can the ’best’ model be chosen? • How can the models be implemented? • What are the results when using these models for the third-party car products? After a practical introduction about insurances in society, the thesis starts with theory that can be used to answer the research questions. This analysis showed that generalized linear models are very useful models for the pricing of non-life insurance products. However, there are some disadvantages to these models which could be avoided by other models, such as hierarchical generalized linear models. We will explore several methods to determine if enough data is available to obtain credible enough estimates. One of these methods can be applied before implementing a generalized linear model. Choosing the ’best’ model is a non-trivial subject. Several statistical tests to choose which risk factors should be included in the model and how they should be included are discussed. These include tests for adding risk factors as random or fixed effects, but also which definition of an risk factor should best be used. This includes whether they should be added as a variate, as a factor or added dynamically. In addition, several statistical methods to choose the distribution that has the ’best’ fit for the observations for both the number of claims and the losses are discussed. These include graphical comparison methods, but also hypothesis testing. To answer the question how the models can be implemented, we will use the statistical programming language R. Algorithms that are used by some packages to calculate the estimates of the models are discussed, as well as several features of these algorithms. Codes are provided in the supplementary section of the thesis. Next, a statistical analysis is performed for the third-party car products of the insurance company. The performed theoretical analysis is applied in practice on the available data, and unknowns were calculated. Then an analysis is performed to determine which distribution ‘best’ fits the number of claims and which distribution ‘best’ fits the losses. The data was subdivided into several risk factors, such as age and region, and was analyzed again. A generalized linear model with a Poisson and log-link assumption was implemented for the number of claims, and a generalized linear model with a Gamma and log-link assumption was implemented for the losses. How and if the risk factors should be added was evaluated using a bottom-up approach. Initially, the models were applied without allowing interaction between risk factors, and subsequently the models were applied again, this time allowing interaction between risk factors to determine if this improved the models. Other models that may lead to a better fit for the data are also implemented. These include generalized linear mixed models, which do not assume that the observations are independent and assume a Normal distribution for the risk factors that are added as random effects. Also, a pure premium model in which the Tweedie family is used was applied. The study showed that the preferred models to calculate the pure premium are a generalized linear model with Negative Binomial and log-link assumption for the number of claims and a generalized linear model with Gamma and log-link assumption for the losses. Due to overdispersion of the observations for the number of claims, the Negative Binomial proved to be a better choice of distribution leading to a better fit of the model for the number of claims. The Normal and Pareto distribution were too symmetric and too right-skewed for the observations, respectively. The pure premium model showed a worse fit, when compared to the model for the number of claims. Furthermore, the effect of the risk factors on the risk profile of a risk group were very clear when a two-stage regression approach was used. The hierarchical models were not better models, because the estimates were less accurate. The results for the different models were then compared with the currently used pricing system of the company and the expected outcomes of the data analysis. This leads to recommendations for the insurance company, including recommendations for pricing in general but also specific recommendations for the pricing system of the third-party insurance product. The full thesis contains confidential information, therefore, a public version was provided in which the insurance company is anonymous. The full thesis was made available to the thesis committee.

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.