This thesis investigates efficient Monte Carlo methods for estimating the 99.9% Value-at-Risk of
concentrated credit portfolios modelled through a normal copula framework. Crude Monte Carlo
simulation is inefficient when estimating extreme loss levels. To address this inefficiency, variance
reduction techniques are applied, including importance sampling and its adaptive variant. This thesis
provides empirical evidence that these techniques make extreme quantile estimation computationally
feasible in a realistic portfolio credit risk setting.
Multiple methods from the literature are reviewed to approach an appropriate importance sampling
proposal. Bernoulli tilting conditional on the common factors is an effective method when obligors
are weakly correlated. However, when large losses are mainly driven by common factors in the credit
portfolio, shifting their means turns out to be more effective. This has to be done with care, since an
improper shift may increase the variance. To this end, a deterministic method that approaches the mode
of the zero-variance distribution is discussed. Furthermore, adaptive importance sampling is studied,
which aims to minimize the variance directly by iteratively updating the shift based on past samples.
This thesis introduces a method that uses the generalized Poisson–Binomial distribution, together with
an inverse Fourier transform, to approximate the zero-variance distribution nearly exactly for simple
portfolio cases.
A deterministic mean-shifted importance sampling (M-IS) and an adaptive importance sampling
(AIS) approach are applied on stylized portfolios. These portfolios are constructed such that they mimic
real-world conditions. The M-IS method achieves an estimated variance reduction factor of 175 on a
large portfolio with 20 000 obligors. Performance on the actual Triodos Bank portfolio delivered similar
performance, but detailed results are withheld for reasons of confidentiality. For a moderately sized
portfolio consisting of 1 000 obligors, M-IS is no more effective than crude Monte Carlo because of the
higher impact of idiosyncratic concentrations. In this case, AIS finds a better balance between the
various kinds of concentrations. With conservative parameters, AIS achieves an estimated variance
reduction factor of 5 on the moderately sized portfolio. Bernoulli tilting could further increase the
variance reduction in these portfolio scenarios. Moreover, as the number of samples increases, AIS further
reduces variance. Finally, this thesis investigates how to construct asymptotic confidence intervals for
the extreme quantile estimates of these stylized portfolios and demonstrates via experiments that they
exhibit asymptotic normal behaviour despite the discreteness of the loss distribution.

The Weibull distribution is one of the most widely used distributions in reliability analysis. The ability to accurately estimate the parameters of Weibull distributed data can be very useful, and particularly important when dealing with small data sets and high degrees of censoring. This project aims to compare the performance of the maximum likelihood (ML) method and a generalised least squares (GLS) method in estimating the parameters of small, censored Weibull distributed data. The study involves a simulation of type II censored Weibull data to estimate the log-Weibull parameters and predict the quantiles of the next failure. Simulations were performed across a broad range of sample sizes and number of observed failures. We evaluated the efficiency and accuracy of both estimation methods. The simulation results demonstrated that the unbiased versions of the ML estimators consistently outperform the GLS estimators in terms of efficiency. The root mean squared error (RMSE) of the ML estimator were lower, indicating a higher accuracy in parameter estimation.

In this thesis, we define a new concept of duration for FX Swaps and more broadly for sovereign bonds. The con-cept of duration already exists for bonds and more specifically coupon bonds, where it is also called ”Macauley Duration”. We aim to define a concept for FX Swaps with similar financial properties and derive mathematical properties from the new definition. A major result of this thesis is the Duration Equivalence Theorem, which states that FX Swap portfolios with the same duration have roughly the same payoff and by extension similar risk exposures. This theorem can be used as a tool for portfolio management and hedging purposes. We also derive a bound of the remainder of the approximation in the theorem so that the applicability of the theorem can be assessed. Based on this first major result, the remainder of the thesis is focused on FX Swap portfolio optimization.

Insurers are required to have buffers to be able to meet financial obligations that result from their portfolios, which are determined using a cash flow model. The input of such a cash flow model consists among of things, of two mortality tables and the portfolio of an insurer. Mortality rates are simulated using the Lee-Carter model. These simulated rates are in turn used to simulate the cash flow corresponding to a portfolio. This results in one possibility of incoming and outgoing money over a period of time. Lots of simulations are required to get a reliable estimate for the future cash flow which is (depending on the number of simulations) computationally heavy and therefore time consuming. The calculation time is decreased by applying an extreme gradient boosting (XGBoost) machine learning method in which cash flows are considered target variables and the mortality tables are considered features of the model. The trained XGBoost model can predict the cash flows based on the mortality tables. The standard XGBoost model is extended to a multi-target regression model which is able to predict multiple target variables at once. This XGBoost model reduces the computation time and ensures that 99.5% of the predictions deviates within either 1% or 0.5% of the observed values. XGBoost gives a good method of determining a reliable estimate of the future cash flow.

In this thesis, we evaluate the statistical procedure based on the similarity index proposed by Ypma and Ross in 'Determing the similarity between expected and observed ageing behavior'. The function for the similarity index is defined based on the inner product of two probability density functions, assumed to be related to the Weibull distribution. The evaluated statistical test aims to verify whether an observation of censored lifetime data is compliant with a given reference distribution. The test is compared to both the likelihood ratio test and a test using a variation of the similarity index. The comparison between these tests is based on their power function. First, a clear explanation of the complete statistical procedure using the similarity index is provided. Then the observations for which the test can be used are explained further. Its application and small limitations are shown using an example. Then, the two alternative tests are introduced. This will be followed by a theoretical overview of the power comparison method. Finally, the simulation for the power comparison is conducted. The power function is estimated for multiple relevant base cases along a few alternative parameter lines. It can then be concluded that the likelihood ratio test has consistently higher power than the similarity index test. However, the variation of the similarity index demonstrates a varying power, with instances of both higher and lower values than the original similarity index test.

This thesis investigates the estimation of option-implied probability density functions for inflation using inflation options, focusing not only on the expected value but the whole distribution. The aim is to identify the most effective method for measuring the market expectation of future inflation. The research explores both parametric and non-parametric approaches for deriving these density functions from inflation option prices. Methodologies include parametric models such as expansion, generalised distribution, and mixture methods, alongside non-parametric techniques using Breeden and Litzenberger’s result, such as curve-fitting and kernel methods. Implementing these methods involved analysing inflation option data sourced from the BVOL Bloomberg database, specifically for Harmonised Index of Consumer Prices excluding Tobacco (HICPxT) options from January 1, 2013. The study employed Shimko’s method, various spline methods, the Delta method, and Kernel method, assessing their effectiveness and challenges. Results reveal diverse implications for each method. Visual comparisons showcase the varying outcomes of the implemented methods, Likelihood-based assessments present a more numerical approach benefiting the Delta and Kernel methods due to higher scores and fewer negative likelihoods. Conclusions suggest that while multiple methods offer insights into inflation prediction, the Kernel method shows promise in its reliability while the Delta method scores highest in the numerical methods. However, challenges in accurately modelling extreme values and tail behaviours persist across methodologies. Recommendations for further research involve addressing these limitations and exploring enhancements to refine inflation prediction models.F

In this report, our goal is to find a way to get some information such as the mean out of high dimensional densities. If we want to calculate the mean we need to calculate integrals, which are difficult to do for high dimensional densities. We cannot use the analytical or classical (deterministic) numerical rules for high dimensional problems for which we want to calculate the mean. These methods take a lot of computational time. To solve this problem we introduced the Markov Chain Monte Carlo (MCMC) method, the method samples from the distribution, and with these samples, we can approximate the mean. Then we explain the theory behind these methods and how we can use it. Then we introduced one MCMC method, in particular, the Metropolis-Hastings Algorithm. We explain how this method works and the theory behind it. From this, we see that the method is very easy to implement and can be used to approximate the mean. Then we approximate the mean for some examples using this method.

In dit onderzoek is een Monte Carlo algoritme beschreven voor het schatten van de dominante eigenwaarde en bijbehorende eigenvector van een niet-negatieve, irreducibele matrix. Naast de werking van het Monte Carlo algoritme is ook de convergentiesnelheid onderzocht. Tevens zijn een aantal simulaties in de praktijk uitgevoerd voor zowel het Monte Carlo algoritme als de power methode en aan de hand hiervan bekeken of het Monte Carlo algoritme een goede concurrent is van de power methode.

In tegenstelling tot een Europese optie, is de prijs van een Amerikaanse optie vaak niet te berekenen met behulp van standaard analysemethoden. Om toch een optieprijs te kunnen bepalen, wordt er gebruik gemaakt van simulatiemethoden. In stochastische modellen, gebaseerd op zogenoemde arbitragevrije prijsbepalingen, is de optieprijs gelijk aan de verdisconteerde verwachtingswaarde van de maximale opbrengst over alle mogelijke uitoefenmomentenm van de optie onder de risico-neutrale maat. Het uitoefenmoment waarvoor de verdisconteerde verwachte opbrengst maximaal is, wordt gevonden door een optimale uitoefenstrategie te hanteren. Deze uitoefenstrategie kan worden bepaald met behulp van de continueringswaarden die horen bij de optie. Het bepalen van de prijs van een Amerikaanse optie draait om het vinden van de continueringswaarden. Longstaff & Schwartz [2001] veronderstellen dat de continueringswaarden kunnen worden beschreven door een functie. Het algoritme dat zij beschrijven, maakt gebruik van regressietechnieken voor het bepalen van een functiebenadering van de continueringswaardefunctie. Met behulp van deze functiebenadering kan een ondergrens voor de optieprijs benaderd worden. Omdat het evalueren van de optieprijs relatief veel rekentijd kost, willen we dat de gevonden ondergrens ook relatief nauwkeurig is. Dit motiveert het gebruik van variantie reductie methoden. Bolia et al. [2004] beschrijven in hun artikel een methode die gebruik maakt van importance sampling als techniek voor de reductie van de variantie. In een iteratief proces proberen zij een zogenoemde variantie-nul kansmaat te benaderen. Wanneer de optieprijs onder deze kansmaat wordt geëvalueerd, kan de optieprijs met variantie nul worden bepaald. Net als Longstaff & Schwartz [2001] veronderstellen Bolia, Glasserman en Juneja dat zowel de optiewaarde als de continueringswaarden kunnen worden benaderd met een functie. Net als in het algoritme van Longstaff & Schwartz [2001], worden deze functies benaderd aan de hand van regressiemethoden. De optiewaardefunctie kan vervolgens gebruikt worden bij het bepalen van een benadering van de eerder genoemde variantie-nul maat. Met de gevonden benaderde kansmaat kan een nieuwe optiewaardeen continueringswaardefunctie worden benaderd. Intuïtief volgt dat elke iteratie een nauwkeurigere benadering geeft van de continueringswaardefunctie. Wanneer er voldoende iteraties zijn uitgevoerd kan de optieprijs worden geëvalueerd onder de verbeterde strategie. Dit geeft een betere benadering voor de ondergrens van de optieprijs. In het verslag wordt het algoritme van Bolia et al. [2004] behandeld. Het project kent twee voornamelijke doelen. In de eerste plaats is het project gericht op het doorgronden van het algoritme. Hierbij is het de bedoeling dat het algoritme wordt beschreven voor derdejaars bachelor wiskundestudenten. Vanwege de hoeveelheid en ingewikkelde wiskunde achter het algoritme, is er gekozen om alleen de delen wiskunde te behandelen die noodzakelijk zijn voor het begrip van het algoritme. Het tweede doel binnen het project is het algoritme toepassen. Het verslag wordt dan ook afgesloten met een voorbeeld ter illustratie van het algoritme. In de gevonden resultaten treden discrepanties op met eerder gedaan onderzoek. Uit verschillende testen kan echter geen duidelijke fout in de implementatie van het algoritme worden aangewezen.