Dutch pension funds are under the supervision by De Nederlandsche Bank (DNB) and they must adhere to the Financial Assessment Framework (FTK), which outlines the methods for calculating liabilities, buffer reserves, and risk factors. As part of the FTK, a feasibility test must be performed, which is a scenario-based analysis of pension funds' investment strategies and their pension policies based on economic scenario sets projected 100 years into future. Every quarter, DNB publishes these economic scenarios (P-scenarios) and equivalent risk-neutral scenarios (Q-scenarios) are used to calculate the effects of the pension fund reform in the Netherlands. These scenarios are generated by a stochastic model, which is revised every 5 years by a commission appointed by the Dutch government, called the Commission Parameters. The model currently in use is based on Commission Parameters 2022 and is referred to as CP2022.

The CP2022 model is an affine and arbitrage-free model framework for correlated interest rate, inflation rate, stock index and consumer price index, each being driven by Heston's stochastic volatility process. To generate realistic economic scenarios, the model is calibrated quarterly using bond and option market data. Currently, as part of the calibration process, option prices are computed under the dynamics stated by CP2022 using a Monte Carlo-based approach. As an alternative, we resort to a Fourier method called the COS method, which relies on the availability of the characteristic function of the state factors. We propose an efficient option valuation scheme for pricing European call and put options, and cliquet options under CP2022. Compared to the Monte Carlo method, our scheme demonstrates superior accuracy and computational efficiency. ... Read more

Energy markets are characterized by unique features such as seasonality, mean reversion, and sudden price spikes, which make the valuation of related derivatives considerably more challenging than in traditional financial markets. This thesis investigates the use of the COS method, a Fourier-based numerical technique, for the efficient valuation of energy derivatives within a Markov-modulated framework.

The study begins with energy quanto options, whose payoff depends on two correlated underlyings incorporating jumps and regime-switching dynamics. By deriving the characteristic functions for two benchmark models and implementing the COS method, the results demonstrate high accuracy and significant computational speed-ups compared to FFT- and Monte Carlo-based benchmarks.

The research work then extends to electricity storage contracts, which feature early-exercise opportunities and operational constraints. In line with the methodology presented by [1], this section replicates their integration of the COS method into a dynamic programming framework to evaluate the option values across multiple contract types and volatility regimes. The obtained prices fall within the confidence intervals of the LSMC benchmark, confirming the robustness of the COS approach in handling path-dependent and constrained problems.

Finally, the pricing framework is generalized to electricity storage under a two-state Markov-modulated model, to better capture the behavior of electricity prices. The extended COS-based algorithm accurately reproduces complex probability densities and achieves stable convergence of the prices while maintaining high computational efficiency even under this more complex setting.

In conclusion, this thesis demonstrates that the COS method is a powerful and computationally efficient alternative to stochastic simulation methods for pricing energy derivatives. It is capable of handling jump dynamics, regime switching, early-exercise features and operational constraints, highlighting its potential for broader applications in energy finance.

[1] Boris C Boonstra and Cornelis W Oosterlee. “Valuation of electricity storage contracts using the COS method”. In: Applied Mathematics and Computation 410 (2021), p. 126416 ... Read more

This thesis investigates the use of path signatures as feature representations for derivative pricing within a regression-based framework. Motivated by the universal approximation theorem for signatures, which ensures that any continuous path functional can be approximated by a linear functional of its signature, the study evaluates the practical performance of this approach for a range of path-dependent derivatives. The analysis focuses on two research questions: how does signature-based regression compare with standard Monte Carlo simulation in terms of accuracy and stability, and whether more compact signature representations can alleviate the exponential growth in the signature dimension with respect to the number of assets. To address these questions, the work introduces an alternative representation of the signature features, the filtered signature, which uses the signatures of individual time-augmented asset paths rather than the full joint multi-asset signature, with the aim of reducing dimensionality while retaining essential path information. The effectiveness of this representation, along with the overall performance of the regression framework, is then evaluated through a series of numerical experiments. For the financial products analyzed, numerical results indicate that, when combined with analytically derived expected signatures, the price estimator obtained through signature-based regression exhibits markedly lower variance than standard Monte Carlo. As a result, the required number of simulated paths to achieve the same level of accuracy is reduced. The results further show that the filtered signature representation attains pricing accuracy and stability comparable to the full signature, while relying on significantly fewer terms. These findings suggest that essential and enough information about the path is encoded within the individual signatures of the time-augmented underlying assets, supporting the adoption of compact signature representations for multi-asset derivative pricing. ... Read more

In recent years, high-frequency trading has shaped the dynamics of equity markets. Within this context, this thesis develops and evaluates two proprietary strategies applied to the constituents of the FTSE MIB index, the main benchmark of the Italian equity market. The first contribution is a market making strategy, where quoted prices incorporate a premium to account for expected market impact, while a hedging offset is introduced to control exposures to systematic factors. These factors are extracted through principal component analysis, after cleaning the empirical covariance matrix of returns with Random Matrix Theory to separate signal from noise. The discussion then turns to the closing auction, where another strategy is designed to exploit the unique opportunities specific to this phase. In particular, the strategy determines the optimal quantities to submit by solving a convex quadratic optimization problem that balances expected profitability with the variance of the residual portfolio, which is computed through interior point methods implemented via Newton iterations to achieve computational efficiency. Both strategies are tested in a proprietary event-driven backtester that reproduces order-by-order market microstructure, and the results show consistent profitability, robustness across stocks and conditions, and effective control of inventory imbalances. Future research may extend these methodologies to other indices and to strategies operating simultaneously across multiple benchmarks, thereby exploiting overlaps to offset risks more efficiently and to investigate cross-market dynamics. ... Read more

Statistical inference of low-frequency time series is a challenge present in various fields, such as financial risk management and weather forecasting. Practical difficulties arise due to the scarcity of non-overlapping observations. The “direct method”, which directly uses the available low-frequency data to construct estimators, often results in inaccurate estimations.

In this thesis, we propose a novel “simulation-based method” for statistical inference of low-frequency time series that result from the aggregation of a higher-frequency time series over a period of time. We start by estimating the distribution of this higher-frequency process. We then simulate a large number of paths from this estimated distribution. By independently aggregating each simulated path, we generate corresponding low-frequency data. This provides us with a large simulated dataset of the low-frequency process, which enables us to apply estimation procedures and bypass the limitations posed by the shortage of original low-frequency data.

We also provide a theoretical framework and propose three families of estimators constructed from the estimated higher-frequency distribution, analyzing their properties under additional assumptions. Through a comprehensive simulation study, we compare the simulation-based method with the traditional direct method across different scenarios and objectives. While our study focuses on the marginal distributions of low-frequency processes, the simulation-based method’s applicability extends to joint distributions across multiple time points. This research offers a robust method for parameter estimation when faced with limited low-frequency data. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more

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.
... Read more

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.
... Read more