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.

The EAD metric is widely used in the calculations for the capital requirements concerning Counterparty Credit Risk (CCR). In this thesis we compare several methods for calculating this EAD. Basel III gives us two methods, the Standardized Approach for CCR (SA-CCR) and the Internal Model Method (IMM). Furthermore, we introduce an integrated benchmark model, whereby we estimate the EAD by integrating the Wrong Way Risk (WWR) directly, while in SA-CCR and IMM this WWR is captured by an alpha factor of 1.4. In this benchmark model, we derive the formula for backing out the probability of default using CDSs and then, via a Gaussian copula, we include a correlation between the exposures and default probability to model the WWR. The ultimate goal of this thesis is to find out how conservative the SA-CCR is compared to the IMM and the integrated benchmark model, and if the alpha factor of 1.4 is a reasonable value to account for WWR.

The test portfolios consist of interest rate swaps, cross currency swaps and FX forwards, which are the most liquid product types in the market. To value these products we model the interest rate using the Hull-White model and the exchange rate using a GBM, following industry standard.

Testing results suggest that the SA-CCR is at least a factor of 1.5 more conservative than the IMM in the presence of collateral, even with stressed parameters. The level of conservatism is even higher because no diversification is allowed between different asset classes by SA-CCR. Furthermore, we observe that using the parameters backed out from calibration and a default correlation of about 30 to 40\%, our integrated benchmark model based on copula returns more or less comparable EADs of IMM times the alpha factor of 1.4. This indicates that this value of 1.4 is a reasonable value to cover WWR in the IMM framework.

Barrier options, although highly liquid financial derivatives, present notable pricing challenges. In this thesis, we present a novel pricing approach for valuing continuously-monitored knock-out barrier options within the framework of stochastic volatility models.

The underlying process is firstly modelled under geometric Brownian motion and, subsequently, under Heston's stochastic volatility model. A key insight is that the value of a barrier option can be expressed as a single-dimensional integral, whereby the integrand involves the so-called survival density function, which captures the barrier-brea-ching information. Therefore, the option can be valued using the one-dimensional COS method for European options, once the Fourier series coefficients of the survival density are obtained.

The coefficients of the sine series expansion of the survival density function are, in fact, a continuous function, which is closely related to the characteristic function of the density. This motivates us to directly recover that function, which we refer to as the target function herewith, by selecting an appropriate series expansion for it. Thereafter, we insert this series expansion into the partial differential equation (PDE) that the target function should satisfy, which can be derived from the pricing PDE. This results in a linear system, solving which we obtain the coefficients needed to reconstruct the target function. Notably, this approach is particularly advantageous when the reference values of option prices are limited or unavailable, as it relies solely on the PDE to calibrate the series coefficients.

Our choice of series expansion is driven by the need for precise global and local approximations. Our research shows that a proper expansion for the target function can be built up from integrating a two-dimensional Fourier series of its first derivative with respect to time, marking our second pivotal insight. That results in a trigonometric expansion that significantly enhances the accuracy compared to a direct Fourier series expansion on the target function. Finally, our third pivotal insight: applying a change of variables further improves error convergence.

Extensive testing results suggest that, for similar accuracy levels, our approach greatly outperforms Monte Carlo simulations in terms of computation time. It also demonstrates superior computational efficiency and accuracy compared to other advanced numerical methods in the existing literature.
The benefits of this method become even more prominent when pricing a large number of options simultaneously.

Numerical tests reveal algebraic convergence for the series expansion reconstruction. For the option price, theoretical error analysis aligns with our findings, also predicting algebraic convergence.

This thesis presents a comprehensive exploration of the rough Heston model as a means to enhance financial derivative pricing and calibration in the context of the complex behavior of market volatility. Recognizing the limitations of classical models, such as the Black-Scholes and the standard Heston model, which assume constant or mean reverting volatility, this research delves into the application of rough volatility models that account for the empirical ’memory’ effect observed in financial markets. These models, inspired by the fractional Brownian motion with a Hurst parameter less than 0.5, offer a more accurate representation of the volatility surface.
A significant portion of the thesis is dedicated to the development of a novel cosine tensor network that expedites the supervised learning of the characteristic function of the lifted Heston model. This advancement is pivotal for the rapid pricing and calibration of European options under the rough Heston framework. The cosine tensor network, leveraging the characteristic function’s availability, enables the efficient application of Fourier-based methods, such as the COS method, for option pricing. This approach is further extended to the pricing of path-dependent options like barrier and Bermudan options through the 2-dimensional COS method.
The thesis is methodically structured, beginning with a foundational overview of option pricing and volatility, followed by a literature review that situates rough volatility within the broader context of quantitative finance. Subsequent chapters detail the mathematical framework of the rough Heston model, its derivation from market microstructures, and the introduction of the lifted Heston model as a multi-factor approximation.
Empirical analysis is provided to validate the lifted Heston model against traditional methods, demonstrating its superior performance and accuracy. The thesis culminates in the presentation of a new calibration method, supported by real market data, and a novel benchmark for pricing complex derivatives under the rough Heston model.
The research encapsulated in this thesis not only sets a new standard for computational speed and precision in the pricing and calibration of financial derivatives under the rough Heston model but also opens avenues for future research, particularly in the application of rough volatility models to other areas of financial mathematics.

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 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.

The computation of multivariate expectations is a common task in various fields related to probability theory. This thesis aims to develop a generic and efficient solver for multivariate expectation problems, with a focus on its application in the field of quantitative finance, specifically for the quantification of Counterparty Credit Risk (CCR).

The proposed COS-CPD method utilizes the COS method to recover the exposure distribution by its Fourier-cosine series expansion, from which measures such as the PFE and EE can be obtained. The key insight is that the corresponding Fourier coefficients are readily available from the characteristic function, which can be solved using numerical integration methods. However, the efficiency of standard quadrature rules is limited to only a few risk factors, as the dimension of integration is determined by the number of risk factors involved.

To address this limitation, the COS-CPD method reduces the dimension of integration of the characteristic function through two steps. Firstly, the joint density function of the risk factors in the characteristic function is replaced by a dimension-reduced Fourier-cosine series expansion, which is obtained through CPD. With CPD, the computational complexity of computing the Fourier coefficient tensor is reduced to a linear growth with respect to the number of dimensions. Secondly, the portfolio is divided into segments that share the same risk factors. These two steps reduces the evaluation of the characteristic function to the calculation of only one- and two-dimensional integrals, which are solved by the Clenshaw-Curtis quadrature rule. As a result, the COS-CPD method is suitable for portfolios with more than three risk factors.

Numerical comparisons of the COS-CPD method and Monte Carlo (MC) method are made for netting-set PFE and EE profiles of multiple derivative portfolios up to five risk factors. For similar accuracy levels, the COS-CPD method greatly outperforms the Monte Carlo method in computation time. This difference increases for larger portfolios, which makes the COS-CPD method a much more efficient alternative for the MC method, especially for large portfolios.

Furthermore, the COS-CPD method is applied in the context of multi-asset option pricing. A six-dimensional basket option is considered, and the results are compared to a recently developed sparse grid method. The comparison shows that the COS-CPD method outperforms the sparse grid method in both accuracy and computation time. Moreover, the COS-CPD method allows the computation of the option value for multiple strike prices simultaneously, with no significant additional computational cost.

Barrier options are fundamental financial tools that give rise to pricing challenges, particularly when embedded within stochastic models. This study directs its focus towards Lévy processes as a strategic approach to navigate and resolve these intricate complexities. The model assumption adopted in this thesis is that the underlying log-asset price follows a Levy process. This assumption, in combination with the COS method, enables us to reduce the dimensionality of the pricing problem for barrier options. To be more precise, the strike price and the log-asset price are both factored out from the calculation formula. The traditional COS method, known for rapid European option valuation, depends on the knowledge of the ch.f.. The COS method has been extended to pricing barrier options in [1] and [2]. However, both methods depend on a recursive calculation formula whereby the number of recursion equal to the number of monitoring dates, and thus, the calculation speed lags behind pricing European options , especially with quite a number of monitoring dates. Our key insight lies in the potential of using the traditional COS method for pricing barrier options. As the first step, we integrate the barrier hitting probability into the ch.f., thereby transforming it into a survival ch.f. Although the exact function of the survival ch.f. is unknown, its values on the grid used by the COS method can be accurately solved via Singular Value Decomposition (SVD). Testing results robustly reinforces our findings. Building on this, we work out two techniques to approximate the characteristic function through supervised learning. The model takes Lévy process model parameters and yields approximation function of the ch.f.. The first method, COS-GPR, combines Gaussian Process Regression (GPR) with the traditional COS method for ch.f. estimation. Assuming mutual independence among multivariate model outputs, COS-GPR addresses GPR’s limitations at boundaries with an additional Fourier expansion. Notably, COS-GPR demonstrates significant significantly faster calculation than the existing COS Barrier by seven times while maintaining heightened accuracy, with occasional outliers. The second method, the COS Fourier CPD (CFC) method, replaces the ch.f. in the COS method by a Fourier-series expansion of the ch.f., after which the dimension is strategically reduced using Canonical Polyadic Decomposition (CPD). CFC achieves superior overall accuracy compared to COS Barrier, with a speed increase of 180 times and minimal instances of extreme errors. In comparison to COS-GPR, CFC’s accuracy slightly decreases but exhibits a smaller maximum error. The CFC method has a 180- fold increase in speed compared to the COS Barrier method and maintains linear time complexity as training points per dimension increase. In conclusion, this study introduces efficient methods based on supervised machine learning for barrier option pricing under Levy processes. The fusion of conventional iii iv methods with advanced approaches leads to significant improvements in both speed and precision. These innovations hold the potential to transform financial derivative valuation, enhancing accuracy and efficiency in the process.

This research project, conducted in collaboration between TU Delft and MN, a pension fund asset manager, focuses on the optimal venue selection in FX trading. The objective is to investigate how the venue selection affects trading performance and to improve MN trading execution algorithm, named ALGO. The research aims to propose a new approach for the venue selection problem by allocating weights to different venues instead of solely selecting the best one. It utilizes advanced statistical and machine learning techniques and develops a matching engine capable of reconstructing historical orderbooks for backtesting strategies. The outcomes of this thesis show clear ideas for improving the current venue selection model. The proposed models are anticipated to consistently outperform ALGO, leading to improved trade execution. The insights and methodologies developed in this research will contribute to further investigations in improving venue selection processes and optimizing execution strategies in the FX market. The thesis provides a comprehensive analysis of the problem, explores the mathematical framework, presents real-world data-driven approaches, and discusses the findings and conclusions, offering valuable insights and recommendations for future MN research.

A wide range of practical problems involve computing multi-dimensional integrations. However, in most cases, it is hard to find analytical solutions to these multi-dimensional integrations. Their numerical solutions always suffer from the `curse of dimension', which means the computational complexity grows exponentially with respect to the dimension.

There is an existing approach that approximates multivariate functions by a tensor of truncated multi-dimensional Fourier series coefficients, and uses the Stochastic Gradient Descent method to solve the lower-rank CPD model, which is used to reduce the computational complexity of the coefficient tensor. In contrast to this work, this thesis project extends its application to solve multi-dimensional integrations, utilizing the Fourier-cosine series expansion to represent the integrand. This project also replaces the SGD method with the Conjugate Gradient method, which improves the function matching accuracy significantly and also has great integration accuracy. The computational cost is reduced regardingly as well.

This thesis also tests an expectation operator related to the COS method, which can be used to compute the expectation of functions of several random variables with much less computational complexity. This method filters out insignificant Fourier-cosine basis functions of the marginal distribution functions, and uses the selected `principal' basis functions to compute Fourier-cosine coefficients of the joint density function by the high-dimensional COS method. The test results show that more than half of the Fourier-cosine series terms can be dropped per dimension while the expectation accuracy is kept, and the correlation does not influence the expectation accuracy significantly for target functions of normally distributed random variables.

Since the introduction of rough volatility there have been numerous attempts at combining it with existing models in order to better approximate the volatility surface with a low number of parameters. The drawback of rough volatility is usually the time needed to compute a volatility surface. We compare three major rough volatility models and compare their ability to fit to the market volatility surface. We implement the rough Bergomi, rough Heston and lifted Heston models
and introduce a fourth model by taking a Nelson-Siegel parameterization for the instantaneous forward variance curve in the rough Bergomi model. We then minimize the volatility surfaces of the models to that of the market and compare these results through minimization time, fit and hedging possibilities. We find that for both Bergomi models, a single volatility surface is seven times faster to compute than for the lifted Heston model, which in turn is five times faster to compute than for the rough Heston model. Minimization times for the rough Heston models are comparable, however significantly higher than for the rough Bergomi models. We also find that the rough Bergomi model is under-parameterised, this is however fixed by the Nelson-Siegel parameterization, which has a minimization error in line with that of both Heston models. Finally, hedging specifically against a parameter rarely improves the outcome, although a hedge against the instantaneous forward variance curve is consistently good and can improve the error between only a delta hedge by 25-50\% throughout all models.

This thesis investigates the application of machine learning models on foreign exchange data around the WM/R 4pm Closing Spot Rate (colloquially known as the WMR Fix). Due to the nature of the market dynamics around the WMR Fix, inefficiencies can occur and therefore some predictability might be expected. We aim to find these inefficiencies. This is done by applying machine learning models, specifically recurrent neural networks, on limit order book data of foreign exchange (FX). The focus will be on the Euro - US dollar exchange rate.

With the emergence of more complex option pricing models, the demand for fast and accurate numerical pricing techniques is increasing. Due to a growing amount of accessible computational power, neural networks have become a feasible numerical method for approximating solutions to these pricing models. This work concentrates on analysing various neural network architectures on option pricing optimisation problems in a supervised and semi-supervised learning setting. We compare the mean-squared error (MSE) and computational training time of a multilayer perceptron (MLP), highway architecture and a recently developed DGM network (Sirignano et al., 2018) along with slight variations on the Black-Scholes and Heston European call option pricing problem as well as the implied volatility problem. We find that on nearly all the supervised learning problems, the generalised highway architecture outperforms its counterparts in terms of MSE relative to computation time. On the Black-Scholes problem, we noticed a reduction of 9.8% in MSE for the generalised highway network while containing 96.2% fewer parameters compared to the MLP considered in (Liu et al., 2019).

On the semi-supervised learning problem, where we directly optimise the neural network to fit the partial differential equation (PDE) and boundary/initial conditions, we concluded that the network architecture of the DGM allows for optimisation of both the interior condition as well as the non-smooth terminal condition. As this was not the case for the MLP and highway networks, the DGM network turned out to be the best performing network architecture on the semi-supervised learning problems. Additionally, we found indications that on the semi-supervised learning problem the performance of the DGM network remained consistent when increasing the dimensionality of the problem.

In this research, we consider neural network-algorithms for option pricing. We use the Black-Scholes model and the lifted Heston model. We derive the option pricing partial differential equation (PDE), which we solve with a neural network, and the conditional characteristic function of the stock price which leads to the option price with the COS method. We consider two neural network-algorithms: the Deep Galerkin Method (DGM) and the Time Deep Nitsche Method (TDNM). We extend the TDNM to be able to solve the option pricing PDE by splitting the PDE operator in a symmetric part and an asymmetric part. The splitting method is more stable than transforming the Black-Scholes option pricing PDE to the symmetric heat equation. The DGM can predict options prices perfectly in the Black-Scholes model even when $r$ and $\sigma$ are added as variables to the neural network. In the lifted Heston model, the DGM can predict option prices perfectly for small dimensions, but has larger errors for larger dimensions. The TDNM is as good as DGM for small times to maturity and volatilities, but has larger errors for large times to maturity and volatilities.

To fulfil the need in the industry for fast and accurate PFE calculations in practice, a new, semi-analytical method of calculating the PFE metric for CCR has been developed, tested and analyzed in this thesis. Herewith we focus on the calculation of PFEs for liquid IR and FX portfolios involving up to three correlated risk-factors: a domestic and foreign short rate and the exchange rate of this currency pair. Both netting-set level and counterparty level PFEs are covered in our research. The short rates are modelled under the one-factor Hull-White (HW1F) model and for the exchange rate we assume they follow geometric Brownian motion. The key insight is that the cumulative distribution function (CDF) can be recovered semi-analytically using Fourier-cosine expansion, whereby the series coefficients are readily available from the characteristic function of the total exposure. The characteristic function in turn can be solved numerically via quadrature rules. Risk metrics, such as the potential future exposure (PFE), can be attained once the CDF is reconstructed using the Fourier series.

Our theoretical error analysis predicts stable convergence of the COS method and observed exponential convergence of the COS method for both netting-set and counterparty level PFE calculations. For three artificial portfolios of different sizes, it was observed that the COS method is at least five times more accurate than the Monte Carlo (MC) simulation method but takes only one-tenth of the CPU time of the MC method. The advantage of the COS method becomes even more prominent when the number of derivatives in a portfolio increases. We conclude that the COS method is a much more efficient alternative for MC method for PFE calculations, at least for portfolios involving three risk factors.
Our theoretical error analysis predicts stable convergence of the COS method and observed exponential convergence of the COS method for both netting-set and counterparty level PFE calculations. For three artificial portfolios of different sizes, it was observed that the COS method is at least five times more accurate than the Monte Carlo (MC) simulation method but takes only one-tenth of the CPU time of the MC method. The advantage of the COS method becomes even more prominent when the number of derivatives in a portfolio increases. We conclude that the COS method is a much more efficient alternative for MC method for PFE calculations, at least for portfolios involving three risk factors.
We conducted theoretical analysis on the error convergence and observed exponential convergence of the COS method for both netting-set and counterparty level PFE calculations. For three artificial portfolios of different sizes, it was observed that the COS method is at least five times more accurate than the Monte Carlo (MC) simulation method but takes only one-tenth of the CPU time of the MC method. The advantage of the COS method becomes even more prominent when the number of derivatives in a portfolio increases. We conclude that the COS method is a much more efficient alternative for MC method for PFE calculations, at least for portfolios involving three risk factors.

Computing portfolio credit losses and associated risk sensitivities is crucial for the financial industry to help guard against unexpected events. Quantitative models play an instrumental role to this end. As a direct consequence of their probabilistic nature, portfolio losses are usually simulated using Monte Carlo copula models, which in turn play a decisive role in their measurement of risk metrics such as the Value-at-Risk (VaR). Semi-analytical numerical methods are alternatives to the Monte Carlo simulations to compute the distribution of the portfolio credit losses, the VaR and the VaR sensitivities. We find that numerical approaches such as the COS method, based on a Fourier cosine series expansion are superior to the Monte Carlo based computations in terms of both, the computational speed and the accuracy. Several studies have demonstrated these results, using the examples of various copula models in a single threaded environment. In this study, we extend that scope and critically examine modelling approaches for improving the computing efficiency by investigating and validating, a multi-threaded GPU based algorithm for the COS method. In this process, we demonstrate the suitability of COS algorithm for parallelization on the GPU and highlight the performance improvements over existing methods.

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