This master's thesis provides information on the subject of option pricing theory. Moreover, this topic is linked with sustainable Finance, which is essential in the battle against climate change. Green financing is a growing phenomenon, and a green bond is only a relatively new example of a green financial derivative. A three-dimensional PDE is at hand to determine the price of a green bond (also called a coupon value), but this PDE can only be solved numerically. This master thesis aims to determine the stability of certain numerical schemes that can be used to find a solution to the PDE. We will use both forward and backward differences to derive the numerical schemes. After the derivation process, a Von Neumann analysis is performed to conclude whether or not the schemes are stable. The research reveals that most of the numerical schemes are not stable. Amongst others, this is caused by the positive value of the interest rate $r$, the value of the carbon price, and the absence of damping factors. In some numerical schemes, the amplification factor is larger than one, but not by much. In other schemes, we can reduce the amplification factor to increase the stability. This means that the numerical schemes can still be used to find a reasonable value for a green bond.

Understanding multiphase flows is critical in nuclear engineering, particularly for processes such as coolant dynamics in nuclear reactors and safety scenario analyses involving different fluid phases. Numerical simulations are a valuable tool for studying these phenomena, especially when experimental approaches are impractical due to cost or safety concerns. While direct numerical simulations (DNS) offer detailed insights, their computational expense makes them impractical for turbulent flows, necessitating the use of turbulence models for efficiency.

This thesis introduces a novel machine learning framework designed to improve Reynolds-averaged Navier-Stokes models in turbulent stratified gas-liquid flows while employing the Boussinesq approximation. The framework encompasses two methods for turbulent viscosity field inversion and introduces correction terms in the turbulence model equations to ensure an accurate prediction of the turbulent viscosity field. Through sparse symbolic regression, the framework consistently discovers models that improve the accuracy of the baseline RANS model, even in untrained flow scenarios, though further testing is needed for varied flow regimes.

Key findings include the superior performance of sparse symbolic regression models over neural network (NN) models in improving the baseline RANS model accuracy. Notably, LASSO and elastic net techniques yielded the most successful models, significantly reducing baseline errors. However, these models did not surpass the Egorov damping approach in terms of accuracy, indicating the need for further refinement.

The developed models were numerically stable and robust, which is important for practical use. However, a main limitation is that the models' accuracy during training did not always correlate with the results when coupled with the RANS equations. Moreover, data from more varied flow conditions is needed to properly assess the generalizability of the models.

Overall, this research highlights the potential of data-driven turbulence modelling to enhance two-phase flow simulations, marking a significant step forward while also identifying areas for future improvement and exploration.

This report focuses on the process of aberration coefficient retrieval for measurements of a
Scanning Electron Microscope (SEM). The main goal is the verification of a modified aberration coefficient retrieval method described in an article by Uno et al. (2005) [1]. This method
first extracts spot profiles, represented by Point Spread Functions (PSFs), from a through-focus
series of SEM images using Fourier analysis. The size and shape of a PSF characterize the
performance of a SEM. Afterward, it determines characteristics from these PSFs to obtain
aberration coefficients.
The process consisted of three parts: probe shape extraction, PSF analysis, and aberration coefficient calculation, which included a consistency test. The original method was changed to
include clarification for the use of complex conjugates of functions in the Fourier domain, the
proper statistical definition of standard deviation as characteristic of the width of a PSF, the
curvature definition used in the method, and contour tracing around a PSF to exclude points
outside the contour. These adaptations were made to clarify the methods described in the reference article.
This modified method was tested against generated PSFs that used the wave-optical description
of five different aberrations. By increasing the value of these individual simulated aberration
coefficients C
′
n with factors s ∈ [1, S], S ∈ N, the consistency of this retrieval method could be
determined by calculating the ratio Q between simulated and retrieved aberration coefficients
as a function of the factor s.
The spot shape extraction method was implemented and assessed successfully, as the shapes of
the generated PSFs were successfully retrieved, and the use of complex conjugates of functions
in the Fourier domain resulted in rotated spot shapes.
Contour tracing around the retrieved PSFs had mixed results. For all aberrations, it presented
a good measure of the spot shape. The implementation of an algorithm that excluded points
pj outside the contour was not accurate for one test aberration: two-fold astigmatism A2. The
contour tracing algorithm excludes important parts of the shapes of the PSFs. From research
into the characteristics asymmetry µ, width σ, and curvature ρ it could be inferred that the first
two characteristics µ and σ represent the mean and standard deviation of discrete probability
mass functions. Curvature ρ has no such direct statistical equivalent, it was concluded however
that this characteristic presented a good measure for the effect of third-order spherical aberration in a PSF.
Results could not indicate that this wave-optical description of aberrations is consistent with
all of the coefficients retrieved using the modified method of aberration coefficient retrieval
as the ratio Q remained consistent for defocus Cdf , three-fold astigmatism A2 and third-order
spherical aberration Cs as a function of the factors s.
Continued research into this method of aberration retrieval is recommended, specifically in the
definitions of the characteristic curvature ρ, more advanced contour tracing, and the digitizations of aberration coefficients. More assessment and modification of this method of aberration
coefficient retrieval can allow for a more comprehensive understanding of the method, which
will eventually contribute to an automatic aberration corrector that could speed up SEM measurements significantly.

Achtergrond: Migraine is een veel voorkomende hoofdpijnaandoening waar neuropeptiden een belangrijke rol lijken te spelen. Dit onderzoek richt zich op het begrijpen van de invloed van zenuwen en gladde spiercellen op de bloeddoorstroming bij vrouwen met en zonder migraine. Hiervoor worden LSCI en LDPI metingen getransformeerd met een Fourier transformatie. Doel: Het doel van dit onderzoek is om verschillen te identificeren in de invloed van zenuwen en gladde spiercellen op de bloeddoorstroming tussen vrouwen met en zonder migraine. Methode: LSCI metingen van de VASCULAR studie worden gebruikt. Fourier analyse wordt toegepast om de invloed van zenuwen en gladde spiercellen te vinden in verschillende fasen: baseline, piek en plateau. Vergelijkingen worden gemaakt tussen vrouwen met migraine (met en zonder aura) en controlepersonen zonder migraine. Resultaten: In de baseline fase zijn minimale verschillen in de invloed van de zenuwen en de gladde spiercellen gevonden tussen vrouwen met en zonder migraine. Tijdens de piek fase lijkt er een verhoogde invloed van de zenuwen en de gladde spiercellen op de bloeddoorstroming bij vrouwen zonder migraine. Conclusie: In ons onderzoek zijn geen evidente verschillen gevonden. Wel lijken er enkele trends in de data te zitten. Verdere studies zijn nodig om deze bevindingen te bevestigen en uit te breiden.

District heating leverages centralised, high efficiency combined heat and power (CHP) systems. It uses waste heat to lower energy consumption and reduce greenhouse emissions. The system also supports renewable energy sources like geothermal and biomass, providing a sustainable heating alternative.

This report examines a nonlinear network of pressure and flow challenges. It focuses on enhancing the Newton-Raphson method and refining direct solving techniques for a single time-step. As net- works grow in complexity, efficient and effective solutions become crucial. Various strategies to speed up the Newton-Raphson algorithm in Gradyent’s heat network solver are discussed, where derivative calculations are straightforward. Both direct and iterative methods to improve the algorithm’s steps are explored. The effectiveness of these enhancements is tested and evaluated across networks of different sizes.sc

Implied volatility is critical in financial markets, especially for option pricing. Traditional methods for its calculation sometimes are not well suited to some scenarios. Recent developments in neural networks have provided more efficient alternatives.

Leveraging advances in quantum computing, our research introduces quantum neural networks for computing implied volatility, assessing the feasibility and characteristics of this novel approach. We focus on two quantum neural network architectures: Dissipative Quantum Neural Networks (DQNN) and Parameterized Variational Quantum Circuits (PVQNN). DQNN, similar to classical neural networks in structure and training ease, faces challenges with quantum state outputs and data decoding, impacting performance negatively. Besides, limited by the reliance on network output states at each layer, DQNN faces challenges in implementation with the current state of quantum hardware.

In contrast, PVQNN offers a more promising solution. Compared to DQNN, PVQNN requires fewer qubits, can apply traditional optimizers to train the model, and can run on NISQ devices. This research thoroughly examines various aspects influencing PVQNN's performance, including training data characteristics, data re-uploading technology, network size, data encoding methods, and quantum circuit design. The selected PVQNN model can achieve high accuracy in implied volatility computation with $R^2$ of approximately 0.999. In addition, we find that the PVQNN can obtain satisfactory results even with limited training data, setting it apart from traditional neural networks.

This thesis not only adopts a new model to compute implied volatility but also deepens the understanding of quantum neural networks in financial modeling. However, due to resource constraints, our experiments are conducted in simulations on traditional computers, and thus our study focuses mainly on the expressive power of QNNs rather than their operational efficiency.

This thesis presents a novel approach to the pricing of green bonds, a growing segment in financial markets with an emphasis on environmental sustainability. Unlike traditional financial instruments, green bonds uniquely incorporate environmental considerations, particularly carbon price (c_t), along with traditional factors like the short rate (r_t), into their valuation. This integration is increasingly relevant in today’s economy, reflecting a shift towards sustainable finance. The core of this research involves applying advanced numerical methods, including the Finite Difference Method, Crank-Nicolson discretization, GMRES and Bi-CGSTAB, in order to develop and analyze pricing models for both green and conventional bonds. The study aims to assess how environmental factors impact the efficiency of these numerical techniques and to compare the outcomes with conventional bond models. The research reveals that green bonds, compared to conventional bonds, present unique numerical challenges, notably requiring more iterations for convergence in iterative methods GMRES and Bi-CGSTAB because of the high carbon price volatility (σc) and the ’Greenium’ phenomenon. Moreover, the comparative analysis showed that while Bi-CGSTAB outperforms GMRES in the green bond model, the opposite is true for conventional bonds. This study not only contributes to the theoretical understanding of green bond pricing but also offers practical insights for financial analysts and investors navigating this evolving market.

Keywords: Green Bonds, Bond Pricing, Zero-Coupon Bond, Short Rate Modeling, Numerical Methods, Crank-Nicolson, GMRES, BiCGSTAB, Environmental Finance, Sustainable Investing, Comparative Analysis, Financial Modeling.

Implied volatility surfaces (IVS) are essential for option pricing and risk management. Recently, generative deep learning models, such as the Denoising Diffusion Probabilistic Model (DDPM), have gained popularity for generating IVS. However, these machine-generated surfaces do not necessarily meet arbitrage-free constraints, e.g., calendar spread and butterfly arbitrage.
This thesis addresses this limitation by integrating arbitrage-free constraints into the DDPM generation process. We introduce a classifier specifically designed to detect violations of arbitrage-free conditions. During the DDPM’s iterative generation process, the classifier guides the formation of the IVS at each step, effectively minimizing potential arbitrage opportunities. To evaluate the effectiveness of our approach, we compare the extent of arbitrage violations in IVS generated by the standard DDPM with those produced by our enhanced DDPM-Classifier model. The results demonstrate that the DDPM-Classifier significantly reduces arbitrage opportunities and improves the quality of IVS.

Counterparty Credit Risk (CCR) refers to the risk that a counterpary involved in a financial contract will default before the final settlement of the contract, resulting in unrealized financial gains. One risk measure for managing counterparty credit risk is the Potential Future Exposure (PFE). The PFE is defined as the 97.5%-quantile of the exposure distribution. The traditional numerical method for computing the PFE is the Monte Carlo simulation method. Recent research has produced a semi-analytical method based on Fourier-cosine series expansion which produce PFE estimates with at least five times the accuracy of the Monte Carlo simulation in one-tenth of the CPU time. In this thesis this COS-PFE framework is combined with Monte Carlo simulation with the goal of reducing the variance of the PFE estimates.
In this thesis three methods are developed. Firstly, the Control Variate method that uses the COS-PFE framework to retrieve the CDF of the portfolio's exposure from which control variates are sampled. The second method developed uses Adaptive Importance Sampling. This method uses the COS-PFE framework to estimate the PFE and expected exposure (EE) of the portfolio. Using the risk factor that has the highest correlation with the exposure of the portfolio a shift is found such that the new joint probability distribution, from which the risk factors are sampled, has an EE that coincides with the portfolio's PFE. The third method developed in this thesis uses Adaptive Importance Sampling based on the Cross-Entropy method. This method aims to find an auxiliary probability distribution which minimizes the Kullbeck-Leibler divergence between itself and the theoretical zero-variance estimator.
The methods were tested using portfolios containing 100, 1000 and 10000 derivatives, both with and without collaterals. Testing showed that the control variate method was not able to produce a variance reduction compared to the straight forward Monte Carlo simulation. However, it did demonstrate the ability to produce a variance reduction for the EE.
The method that finds an optimal shift was successful in producing a variance reduction. The variance of the PFE estimates produced by this method was, on average, 3.5 times lower than the variance found using the straight forward Monte Carlo simulation. However, the algorithm required between 96 and 1773 seconds to produce a PFE estimate.
The adaptive importance sampling method using the cross-entropy approach was tested on all portfolio, both with and without collateral. This method was shown to produce PFE estimates with a significantly lower variance than the straight forward Monte Carlo simulation. For the uncollateralized portfolios containing 100, 1000 and 10000 derivatives the variance of the PFE estimates were on average 35.4, 38.6 and 37.2 times smaller compared to straight forward Monte Carlo simulation. For the portfolios with collateral this performance remains. We conclude that this method produces more accurate PFE estimations with significant lower variance than straight forward Monte Carlo simulation within the same CPU time.

Geomechanical simulations can give essential insights into subsurface processes, but typically require solving large, ill-conditioned linear systems. An important method for solving these linear systems is the Conjugate Gradient method, but applying this method to ill-conditioned matrices can result in slow convergence. To improve the convergence of the Conjugate Gradient method, the iterative solver is preconditioned using the Algebraic Multigrid method. In Algebraic Multigrid methods, a hierarchy of matrices of different sizes is derived. When applying these methods as a preconditioner to the Conjugate Gradient method, on each level of the multigrid hierarchy fast convergence is observed in particular components of the residual. This leads to much fewer iterations being required in the Conjugate Gradient method, at the cost of the iterations being computationally more expensive. These Algebraic Multigrid methods do require a more problem-specific setup configuration than more simple preconditioners like the Jacobi preconditioner. In this research, the Conjugate Gradient method preconditioned with various Algebraic Multigrid methods is studied and compared with the Jacobi preconditioned Conjugate Gradient method. For this, the Conjugate Gradient method, preconditioned with both the Jacobi and Algebraic Multigrid-based methods, is applied to linear problems derived from geomechanical simulations. Using Algebraic Multigrid preconditioners can reduce the number of iterations required for convergence of the Conjugate Gradient method by a factor of 80. While a single iteration with an Algebraic Multigrid preconditioner is more time-expensive than an iteration with a Jacobi preconditioner, significant reductions, of up to five times, are observed in the runtimes of the linear solver. This comes at the cost of a higher peak memory requirement in the application of the linear solver. The vast reduction of the runtime of the linear solvers makes the studied geomechanical simulations significantly faster. This makes the Algebraic Multigrid preconditioners a valuable addition to these simulations.

We price continuously monitored barrier options under GBM and SABR through a newly developed neural network, the COS-CPD network, based on the COS-CPD method. With the pricing PDE and the COS method, we transform the problem of pricing the barrier options into a problem of finding the survival characteristic function through an initial boundary value problem. By applying trigonometric expansion, derived by integrating out the Fourier expansion on the time derivative of the unknown function, to the survival characteristic function, we find an approximation that can be inserted into the IBVP. Without CPD, this results in a linear system which can be solved to find the expansion coefficients, but this leads to the curse of dimensionality. If we include CPD to remove this curse of dimensionality, resulting in the COS-CPD network, the Alternating Least Squares method must be used to find the factor matrices that replace the original expansion coefficient tensor.

Computational Fluid Dynamics (CFD) offers numerous benefits, notably the ability to study flows that are challenging or costly to investigate using experiments. A central challenge in CFD lies in simulating fluid flow around complex geometries. Additionally, the governing equations follow conservation laws. This thesis aims to establish the foundation for constructing Immersed divergence-conforming finite element spaces that address both challenges. To tackle the first issue, an immersed method is considered. Instead of generating a mesh that conforms to the object where the flow occurs, the approach involves placing the object within a predefined mesh. However, this introduces new challenges, the most significant of which is the ill-conditioning of the associated linear system. The condition number depends on the location of the immersed object and can lead to an extremely large condition number. The second challenge is resolved by discretizing the problem in a way that preserves essential topological and homological structures at a discrete level, utilizing the principles of finite element exterior calculus. Within this thesis, a structure-preserving subcomplex is developed for the de Rham complex in 1D, and its accuracy is validated through numerical experimentation. Optimal convergence is achieved, the discrete inf-sup test is satisfied, and the resulting linear system's conditioning remains unaffected by the placement of the immersed object. However, the constructed structure-preserving subcomplex for the de Rham complex in 2D, known as the immersed divergence-conforming finite element spaces, does not exhibit convergence. For future research, I recommend focusing on simpler vertical and skewed cuts before exploring more intricate immersed shapes. These simpler cuts involve straightforward choices for edge basis functions/1-form basis functions that are relevant. Additionally, if the outcomes remain unsatisfactory, considering a global approach instead of a local approach could be worthwhile.

Erasmus MC is conducting a study on the increased risk of cardiovascular disease in women who have migraine. For this purpose, 50 healthy women and 50 women with migraine in age category of 40-60 years were compared using measurements of their dermal blood flow.
These measurements are done using two different imaging devices: Laser Speckle Imaging and Laser Doppler Imaging. The result is a recording of the fluctuations in the dermal blood flow over a time period influenced by circumstances and various mechanisms and consists of signal like curves. Erasmus MC have processed these measurements and obtained results. However, they are interested in whether applying Fourier Transform could offer additional insights or new (perhaps faster) methods to process the measurements.
The signal like curves can possibly be decomposed into sine and cosine waves using Fourier Transform. The result of the Fourier Transform consists of a visualisation of the intensity of the various components of such a waveform/curve. Based on that, it is possible to estimate the contribution of the mechanisms to blood perfusion. Existing literature and previous studies provide frequency intervals related to several mechanisms that influence the dermal blood flow.
This thesis will research to what extent the mechanisms contribute to the dermal blood flow across the different phases and how their behaviour differs when the circumstances for the measurement are adjusted. Perhaps there are behaviours that are instantly recognizable across all patients. An interesting question to answer would be whether it is possible to directly visualise the risk groups.

Although energy commodity price forecasting has been around for quite some time, up until recently, especially in Europe, it mostly concerned other energy commodities than gas. That is why in this work, a forecasting model is presented for the single day forecast of the daily VWAP price of the TTF month-ahead gas contract from 2020/01/02 until 2023/08/03. A model combining machine learning and data preprocessing is proposed. First a decomposition of the gas price is produced by a combination of Variational Mode Decomposition (VMD) and Independent Component Analysis (ICA). With these decompositions an initial volatility regime split is chosen because of the unusual characteristics of the dataset. During the forecasting period the Russo-Ukrainian war started, and the economy was recovering from the COVID-19 crisis, causing the gas price to surge to levels previously unknown. Using the decompositions as input, a final price prediction has been made using a Gated Recurrent Unit Neural Network (GRUNN) and Support Vector Regression (SVR), amongst others. The proposed model is compared to a selection of benchmarks, one of which is the naive forecast. To conclude, a selection of exogenous variables is added to the model to improve the performance. Gas storage and the UK NBP gas price are chosen for their specific characteristics. The best performance the model exhibits is between 0.38% MAPE for the low volatility regime, and 1.4% MAPE for the high volatility regime.

This thesis investigates the behaviour of one-dimensional blood flow in the human arterial tree. It focuses on modeling the change in velocity, cross-sectional area, and concentration. The equations governing this flow are derived using the physical laws of conservation of mass, conservation of momentum, and the
advection-diffusion equation. To solve this set of equations numerically, the Finite Volume Method in combination with a flux difference splitting approach is employed. The study begins with an examination of a single artery, revealing wave-like behavior in pressure and concentration. Pressure propagates along the vessel, while concentration gradually dilutes over time. This
behaviour is also visible when expanding to the 55-artery model of the entire human body. The results of this model were found to be very comparable to the literature. Additionally, the propagation of a chemical
species injected into an artery near the heart is investigated. It takes around 25 seconds to propagate to
the legs but the amount of concentration is greatly reduced. The strength of this wave upon reaching the
legs is investigated for different values of the diffusion coefficient. It is found that when
D = 0.02 cm2/s, the strength is around the 80% of its original value while a value of D = 2 cm2/s reduces
it to 40%. Further research should focus on expanding this model and improving its accuracy. Especially
the exact concentration waves in single arteries are very prone to spurious oscillations. By optimizing the
numerical code, the accuracy can be greatly increased. Moreover, more realistic additions can be made to
this model. Such as expanding to a two-way model enabling to inject at various points around the arterial
tree. Nevertheless, this model provides insight into the propagation of blood and concentration in the
human circulatory system, highlighting the need for refinement.

Prisons serve as amplifiers of Tuberculosis (TB) transmission due to overpopulation, lack of hygiene, and bad ventilation [Mabud et al., 2019] [Baussano et al., 2010]. The risk of TB is elevated during and after incarceration, only returning to the general population’s risk seven years after liberation. This is an indication that formerly-incarcerated individuals are a highrisk group. By preventing TB among ex-prisoners, the burden and transmission to the general population can remain limited. This study aims to analyze the cost-effectiveness of various preventive measures among formerly-incarcerated individuals in Brazil. With the imprisonment rate growing in the country and the high TB incidence rates, Brazil can benefit greatly from proposed measures. The research mathematically models the effect of several interventions and quantifies associated health benefits and costs with the use of Disability Adjusted Life Years and Incremental Cost-Effectiveness Ratios. The parameters of the model are estimated with Bayesian statistics, using likelihood functions and prior distributions to calculate a posterior distribution. The results of the research indicate how the different interventions behave and which is most advisable for implementation in the Brazilian health system. Our conclusion is that shorter preventive therapies in combination with the skin test Tuberculin Skin Test perform better than longer therapies and/or the blood test Interferon Gamma Release Assays. This research was conducted in collaboration with the Harvard T.H. Chan School of Public Health and the TU Delft, with help from the Brazilian ministry of health.

In this research a new method for pricing continuous Arithmetic averaged Asian options is proposed. The computation is based on Fourier-cosine expansion, namely the COS method. Therefore, we derive the characteristic function of Integrated Geometric Brownian Motion based on Bougerol's identity.

Extensive numerical error analysis on the CDF recovery of IGBM and the option prices is performed. Via numerical tests, the convergence of errors using our new method has been proved. We are able to price continuous Arithmetic averaged Asian options with a minimal error of order 10^-2, and a maximum precision of order 10^-5 within seconds.

Implied volatility surfaces are integral to option pricing and risk management but often display missing data. Prior research has typically engaged mathematical models or data-driven methods for generating or completing these surfaces. Given the similarity between implied volatility surfaces and images, our paper introduces the Denoising Diffusion Probabilistic Model (DDPM), a novel deep learning image generation model, for this task. A distinctive aspect of DDPM's training involves progressively adding noise to the surfaces until they resemble pure noise, and then learning to denoise back to the original surfaces. We employ the Heston model to simulate implied volatility surfaces, then train the DDPM using this synthetic data. Additionally, a Variational Autoencoder model (VAE) is implemented as a comparative benchmark for assessing DDPM's efficacy. Our experiments demonstrate that DDPM excels in generating and reconstructing missing areas in implied volatility surfaces, highlighting its potential in this field. Looking to the future, combining DDPM with VAE could provide more interpretable results, enhancing the model's utility and applicability in financial analysis.

After the financial crisis, the standards for the valuation of financial derivatives were reviewed and several adjustments were made to these valuations, of which Credit Value Adjustment (CVA) is the most important one. CVA represents the price of counterparty credit risk that should be added to the default-free fair price of a financial derivative. Nowadays, banks and other financial institutions have dedicated CVA desks that are responsible for estimating CVA and its sensitivities to many market parameters for each counterparty. It is therefore important to have accurate and efficient methods to compute CVA and its sensitivities. This can, however, be a challenge.

Multiple methods to compute CVA sensitivities already exist in the literature, but these methods have certain drawbacks, such as a high computational expense or a payoff dependency. An efficient approach for computing CVA sensitivities is the Likelihood Ratio Method. One of the main advantages is its payoff independence, making the method applicable to sensitivities of any payoff, even payoffs with discontinuities. Furthermore, the Likelihood Ratio Method stands out by its ability to derive multiple sensitivities within a single Monte Carlo simulation. In this research, an innovative approach is developed, which involves using the Hull-White short rate model to model interest rates while applying the Likelihood Ratio Method. A drawback of the Likelihood Ratio Method is its susceptibility to high variance. To mitigate this issue, variance reduction techniques are explored, including antithetic sampling, control variates, and Quasi-Monte Carlo methods.

The performance of the Likelihood Ratio Method is compared to the Bump & Reprice method in computing first-order CVA sensitivities of three different over-the-counter derivatives. It is shown that for a certain range of model parameter values, the Likelihood Ratio Method is able to match the Bump & Reprice sensitivities. However, the Likelihood Ratio Method exhibits high variance for some extreme parameter values.

PDEs, like HJB-equations, can be solved using grid-based methods. These methods are inefficient for solving high-dimensional HJB-equation, because they suffer from the Curse of Dimensionality. Neural networks may overcome this problem. In this research, we solve high dimensional Partial Integro Differential Equations (PIDE) using neural networks. PIDE are PDEs that are associated with a jump-diffusion process. In this work, we only use finite activity jump processes. This means that the jump has a compensation component that to make it a martingale. We show two methods to solve PIDEs: a forward method (H-dBSDE, dBSDE-Jump) and a backward method (DBDP-MC). Both methodologies use neural networks to regress the solution and its derivative. The DBDP-MC is extended to jumps by calculating the compensation of the jumps with an offline Monte Carlo simulation. We tested this methodology on Bermudan basket options with 50 dimensions. The method was able to price them correctly. The dBSDE was extended by adding a new set of neural networks. These networks are learned with a different extra loss function. We argue that we can learn the two losses in a hierarchical way, leading to the Hierarchical dBSDE (H-dBSDE) method. Other work was done by minimizing the two loss functions simultaneously by using the sum of them. Easier problems like pricing European option can be solved correctly by the dBSDE-Jump method. However, we show that this can lead to wrong terminal fits, which makes it difficult to solve complex problems efficiently.