In financial mathematics, stochastic processes are regularly used to describe observed financial indicators such as stocks, options, futures or interest rates. Identifying the underlying dynamics of observed financial time series is crucial in risk management, as it greatly affects pricing and hedging strategies. The large number of available stochastic processes make selecting the most suitable stochastic process a non-trivial problem. Additionally, realisations of stochastic processes are elements of the path space which is infinite-dimensional and non-locally compact. Given these observations, we find that model selection methods from classical statistics, such as distribution metrics, are inadequate.
In this thesis a signature-based model recognition method is proposed. The goal of this model is to select the most suitable stochastic process to describe an observed financial time series. Signatures are transformations from a path to an infinite-length sequence of properties of that path, which makes signatures a highly interesting approach to construct input features which can be used by a machine learning model. In our evaluation, we use the aforementioned methodology to distinguishing between various classification settings of Arithmetic Brownian Motion, Geometric Brownian Motion and the exponential jump diffusion process, both in a binary and multi-class classification setting. This evaluation shows that the proposed method can adequately distinguish between the same model with different parameters, models with and without jumps and models with different jump sizes and jump intensities.

The Competitive Investor Game from Bell & Cover (1980) and the 푘-Player Ranking Game from Alpern & Howard (2017) are analysed in thesis. Optimal strategies have been derived and the related proofs have been given a new look. The Symmetric Multiplayer Ranking Game is considered as the general interpretation of financial competition among hedge funds. This study focused on the hedge funds that manage a Long/Short U.S. Equity strategy. Some minor evidence has been found to support the hypothesis that the studied hedge funds manage a strategy that has the objective to beat the competition in terms of annual performance in order to achieve the highest ranking.

We study the impact of wrong-way risk (WWR) on the credit valuation adjustment (CVA) of a portfolio of interest rate swaps (IRSs), using an intensity-based reduced form model. To model WWR in IRSs we create a dependence between he underlying market risk factor of the IRS and the survival probability of the
counterparty. The focus lies on modeling the credit part of the CVA, choosing the most suitable default model. Using a Monte Carlo (MC) framework, we correlate the Brownian increments of the stochastic processes that describe the interest rate and the hazard rate. Assuming the correlation parameter is chosen based on historical data, we solve the problem of leaking correlation. Two
stochastic processes for the credit part are chosen, the CIR++ and the JCIR++, such that the influence of the model choice can be separated from the impact of WWR. With a case study, we vary the correlation parameter to quantify the impact of WWR on CVA of the portfolio. We show that the introduced dependence has a significant impact on the CVA, that depends on the calibration, the portfolio and the chosen correlation. Solving the leaking correlation problem can increase the impact of WWR on the CVA, especially using JCIR++ dynamics.

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.

This thesis captures the calibration of a FX hybrid model: The FX Black-Scholes Hull-White model. The main focus is on the calibration of the parameters in the Hull-White process: The mean reversion and the volatility parameter. The latter is commonly calibrated as a time-dependent parameter, whilst the mean reversion parameter is not. This thesis covers the calibration of the mean reversion as a time-dependent parameter. A known method from the Literature is researched, where we calibrate the mean reversion independently from the volatility parameter to the ratio of two swaptions with the same expiry but different tenor. In our research this method is extended to the negative interest rate environment by assuming that the swap rate follows a shifted lognormal distribution instead of a lognormal distribution. We show that a specific set of swaptions can be chosen, so that the calibration problem is simplified. This choice leads to sequential calibration of convex optimization problems. Numerical results of calibration to artificial and market data are presented, where we compare our method to picking the mean reversion parameter arbitrarily. The findings suggest that the choice of mean reversion parameter affects the calibration procedure. Therefore, we argue that a calibration method for the mean reversion would be appropriate. Besides the focus on the Hull-White process, the calibration of the volatility parameter in the FX Black-Scholes process of the hybrid model is investigated. For the calibration of the FX volatility parameter ATM options on FX rates are used. Numerical results of calibration to artificial and market data are shown. A typical problem of calibration in the industry is discussed: Precision for late maturities. The results in this thesis suggest that this problem could be solved. For research on homogeneity constraints on the time-dependent parameters, the performance of delta-hedging is investigated. The delta-hedging is performed in a simple Black-Scholes world with time-dependent volatility. We show that given a fixed amount hedges beforehand and interest rate zero, there exists an optimal distribution of time points that will lead to equal variance on the profit and loss for every volatility function that has equal implied volatility from t0 to maturity T. Using this strategy, the homogeneity of the volatility parameter has no impact on the performance of delta-hedging. Therefore, in this thesis no homogeneity constraints are used for the calibration of the time-dependent parameters.

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.

Efficiently managing hedging portfolios on behalf of pension funds is key in achieving the target hedging strategy, which can significantly impact coverage ratios. A new optimization approach to fixed income portfolio management for pension funds is proposed that finds interest rate risk hedging strategies while incorporating additional requirements. These are relevant requirements for pension funds such as country allocations, low transaction costs and reasonable investment costs. In doing so, pension fund regulations and common practices are investigated in a rigorous mathematical framework. The hedging strategies are shown to perform well when back-testing. In addition, simulation of the interest rate and cash flows in a Defined Benefits pension scheme displays the good performance of the strategies. These strategies are further tailored to specific pension funds by considering the trade-off between yield and risk, which could contribute to increasing a pension fund’s coverage ratio. Alternatively, a procedure is also proposed to generate more diversified albeit less optimal hedging portfolios using the optimization approach.

In financial applications it is often necessary to determine conditional expectations in Monte Carlo type of simulations. The industry standard at the moment relies on linear regression, which is characterized by the inconvenient problem of having to choose the type and number of basis functions used to build the model, task which is made harder by the frequent impossibility to use an alternative numerical method to evaluate the "ground truth". In this thesis Gaussian Process Regression is investigated as potential substitute for linear regression, as it is a flexible Bayesian non-parametric regression model, which requires little tuning to be used. Its downfall is the computational complexity related to its "training" phase, namely cubic, which requires the use of algorithmic approximations. The most prominent approximations are reviewed and tested in different scenarios requiring the approximation
of conditional expectation by regression, among which the Longstaff-Schwartz algorithm for the pricing of Bermudan options. This thesis was carried out in cooperation with ABN-AMRO bank.

When dealing with datasets where the observations are obtained from the same cross-sectional units at multiple time points, most of the times, heterogeneity arises across he cross-sectional units. If one ignores this heterogeneity, assuming that the data are pooled, the parameters estimations run the risk of being inconsistent. This thesis studies the difference between panel data and pooled data models with regard to their construction procedure and their predictive performance. An application is discussed per credit risk modelling for a mortgage portfolio. Therein, different models were constructed, covering pooled and panel linear models and pooled and panel logistic models. By model performance and testing comparison, we found that by adding the heterogeneity effect in the regression model the discriminatory power is improved. At the same time, however, it provides lower predicted losses than the observed ones. We have also noted that, most of the times, the pooled model fails to estimate accurate predictions. This thesis has been carried out jointly with TU Delft / Department of Applied Mathematics and the Central Risk Management / Model Validation department of ABN AMRO Bank.

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.

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.

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.

In this thesis, we find optimal strategies for both players in the generalized tally game, mainly using Glicksberg's theorem and minimax theorem. Analytical and numerical solutions provided for the tally game with one draws and with two draws.