Rough volatility models have become a prominent tool in quantitative finance due to their ability to cap- ture the rough nature of financial time series. However, these models typically have a non-Markovian structure, and this poses significant computational challenges. Existing methods for approximating these models often involve either complex quadrature techniques or require optimization algorithms op- erating in high-dimensional spaces, making them difficult and computationally intensive to implement in practice. This work introduces a novel and highly efficient quadrature rule for a Markovian approx- imation of rough volatility models, along with its theoretical error bounds. We verify the error bounds through a detailed error analysis and conduct several experiments on option pricing, demonstrating that the proposed method outperforms other state-of-the-art methodologies in terms of efficiency and accuracy across a range of roughness parameters. A further advantage of the approach is its ease to implement it in practice.

Electronic trading algorithms are at the centre of every buy-side equity trading desk. These algorithms rely often on market impact models, which are stochastic models for the stock prices that account for the feedback effects of trading. Propagator models are central tools for describing the evolution of market impact during and after a trade. This thesis extends the linear propagator model by proposing a new variant that incorporates time-varying liquidity and general decay kernels. To bridge the gap between theory and practice, we use Robeco's proprietary order data base to calibrate the model and validate its performance. The main findings reveal a two-stage decay pattern of market impact, the absence of a single best admissible decay kernel, and a model performance which is in line with our expectations based on the low-signal to noise ratio of the data. The main application of the model in this research is its use in the optimal execution problem, in both a intraday and multiday setting. In an intraday setting we formulate the problem for a risk aware trader and incorporate short-term alpha signals. The discrete analogs of these problems are solved analytically and we highlight significant cost reduction compared to industry benchmarks. In the multiday framework, we quantify the expected cost of trading two adjacent orders and use this to find optimal multiday execution strategies. In a final simulation study we quantify the expected cost of rebalancing two similar investment accounts on consecutive days with a varying number of overlapping stocks. The simulation study accentuate a significant additional cost for the account trading on the second day, which stresses the importance of multiday cost management in rebalancing investment accounts.

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.

Market making, the act of providing liquidity to the market by simultaneously buying and selling, is a difficult problem to solve. The use of reinforcement learning to solve for market making is increasing, as academics and practitioners alike look for novel ways to approximate for optimal policies in increasingly complex markets. This thesis examines the use of reinforcement learning to solve the market making problem in limit order books. To this end, we provide the theoretical background on market making, modelling limit order books, and reinforcement learning. Furthermore, we implement and compare ’classical’ algorithms, such as Q-learning and value and policy iteration, and compare their policies with newer techniques involving deep learning, namely deep Q-networks and double deep Q-networks. To train and compare these models, we use two models to simulate the dynamics of an order book. Experiment results and the ultimately learned policies are presented and discussed. We propose several ideas that could be worked on in the future.

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.

Banks issue mortgages with an embedded option for borrowers to prepay a part of the loan. However, this behaviour poses a risk to banks as it disrupts the level and timing of mortgage cash flows. From an earning perspective, when interest rates decrease, customers are financially incentivised to prepay their mortgages, resulting in a decrease in the bank’s income when the cash proceeds are reinvested at a lower rate. Conversely, from a value perspective, with an increase in interest rates, reducing the financial incentive to prepay, cash flows are moved further ahead in time, thereby increasing the duration of the mortgage. These two scenarios highlight the instability in the bank’s margin and value caused by prepayments. To address this risk, banks employ hedging strategies to mitigate the prepayment risk and achieve margin and value stability. This research aims to identify an effective hedging strategy that can accomplish both.

The research utilised the one-factor Hull-White model to simulate various interest rate scenarios, while an interest rate-dependent logistic prepayment model provided monthly prepayment rates based on the mortgagors’ refinancing incentives. Ten different hedging techniques were explored, including the internal funding, a static and dynamic notional hedge, and a static and dynamic value hedge. Additionally, a calibrated receiver swaption was included in each of these five hedging approaches. Subsequently, each of these hedging approaches was assessed for its margin stability, measured by the variance of the net interest margin, and its value stability, evaluated through the variance of the net present value, the average basis point value, and the NPV-at-Risk in ±200 basis point shocked interest rate scenarios.

The analysis indicated that relying solely on internal funding performs poorly in terms of both margin and value stability. Dynamic hedges were found to generally outperform their static counterparts, due to their ability to respond to market changes. Furthermore, the notional hedge demonstrates superior margin stability, while the value hedge exhibits the best value stability. Additionally, the analysis revealed that the incorporation of a receiver swaption significantly improves the NPV-at-Risk but has limited impact on the other risk metrics. Based on the conducted research, it is concluded that for a bank aiming for both value and margin stability, the most effective hedge strategy is the dynamic value hedge without the utilisation of a swaption. However, it should be noted that the ultimate choice for a hedging strategy depends heavily on the risk appetite of each bank. If a bank prioritises attaining margin stability, the recommended choice would be the dynamic notional hedge without the incorporation of the receiver swaption. On the other hand, for a bank that prefers value stability over margin stability, the dynamic value hedge without the inclusion of a swaption should be considered. Moreover, the final decision may also be influenced by mandatory requirements imposed by financial regulators, such as the European Central Bank.

This thesis investigates the application of neural stochastic differential equations (NSDEs) in financial modeling. It begins by presenting existing theoretical interpretation of NSDEs and investigates the properties of their solutions. By establishing a solid foundation, the thesis sets the stage for further explorations. A significant focus of the research is on the calibration and joint calibration of options using Markovian-type neural SDEs. This modeling approach is carefully chosen after conducting an extensive analysis that considers both the supporting evidence for the rough volatility hypothesis and the counterarguments in favor of Markovian volatility models. By undertaking a thorough study of these perspectives, the thesis aims to provide an in-depth understanding of the advantages and disadvantages associated with each modeling approach.
Additionally, the thesis addresses the challenge of data scarcity, a common issue faced in financial research. Recognizing the limited availability of real-world data and its potential impact on model development and testing, the research explores various existing methods for generating synthetic financial data.

The increasing number of Renewable Sources (RES) in the European electric grid has resulted in the necessity for producers to adjust their position with respect to the change in weather forecasting. Therefore, the European Power Exchange (EPEX SPOT) has seen an expansion of the Intraday Market (ID) where market participants can trade up until a few minutes before delivery. However, literature in Electricity Price Forecasting (EPF) primarily concerns the Day-ahead market (DA). The goal of this thesis is to develop a model to predict the prices of hourly contracts in the ID market using stochastic processes. This would facilitate the obtainment of a probability density function (pdf) for the prices of the contracts at any time during the day. This thesis focuses on the German Energy market which is the most liquid among the European countries. I then propose 4 different Levy processes for the forecasting of the process: a simple Brownian Motion, a Jump Diffusion Process, a Normal Inverse Gaussian (NIG), and a Generalised Hyperbolic model (GH). The Normal Inverse Gaussian model is selected according to the Akaike Information Criterion and the Bayesian Information Criterion.
In order to integrate other variables into the model, I then conducted a fundamental analysis. I find a low correlation between the volume traded during the ID phase and the change in Wind, solar, and con- sumption forecasting. Moreover, the volatility of the contract shows a low correlation with the changes in forecasting and the traded volume.
Additionally, I conduct an augmented Dickey-Fueller test and a Mann-Kendall test on the price of the contracts to verify the presence of either mean reversion or drift. Results show that the latter cannot be refused for most days.
Finally, alongside the NIG model (noise model) I propose one model including a drift component (noise-drift model) and a second model adding both a drift and a volatility component (noise-drift-volatility model). The three models are compared in the accuracy of forecasting the tail probabilities of the ID1 and ID3 indexes. It is found that the noise model and the noise-drift model cast a better prediction than the noise-drift-volatility model. This suggests that the volatility during the first part of a day of trading is a poor predictor of volatility in the hours close to delivery.

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.

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.

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.

The right to use a certain amount of capacity in an electrical cable between two countries for the purpose of trading energy is an asset that can be bought. Each hour of capacity can be seen as a real spread option with the energy prices of each country being the underlying processes. In this thesis we build a model to find the fair value of a month of cross-border capacity and use it to predict this value for the Germany-France andNetherlands-Germany border for the months of July-October 2021. Our model is implemented as an IOHMM with 4 states following a multinomial distribution and a non-stationary Normal Inverse Gaussian distribution for each output distribution. We use SPOT prices of each bordering country to create a ’spread process’, which are then used as target values. As inputs various market drivers for each respective country were used. The model produces a mixture probability distribution for the spread for each hour of the month. Taking the discounted expected value then gives the hourly capacity value; certain Greek calculation is also possible. The results show that the capacities follow a comparable trend to the market. The predicted values for September and October fall short of the realized prices; we expect this to be due to unprecedented volatility in the energy market. Finally, we tested whether certain key drivers were mapped to the spreads as one would expect; this was the case for the Dutch and French drivers, but not for the German drivers.

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.

We analyze three different methods that can approximate the expected shortfall of a financial portfolio in a nested simulation. In this simulation process, the outer simulation generates risk scenarios, and the inner simulation approximates the value of the financial portfolio under each risk scenario. The first method is the most standard one, and therefore we call it ’the standard Monte Carlo method’. This method uses the same amount of computational cost for each inner simulation. The second method adapts the computational cost of the inner simulation to the output of the outer simulation. Therefore, we call it ’the adaptive sampling method’. This technique has already been proven to work more efficiently than the standard Monte Carlo method. The third method is called ’the multilevel Monte Carlo method’ (MLMC) and is based on the technique that has been introduced by Giles. This method approximates the expected shortfall multiple times and with different levels of accuracy. These estimators are used to give an accurate overall approximation of the expected shortfall. To the best of our knowledge, it has never been examined how efficiently the expected shortfall can be approximated in this manner by the MLMC method. This thesis will thoroughly explain how each method can be applied in a nested simulation to approximate the expected shortfall. In addition, an analysis is given on how to ensure that each method is used as efficiently as possible in the nested simulation. We also perform a numerical experiment in which we simulate a simplified version of a financial portfolio using stochastic processes. With the help of the numerical experiment, we examine which method is most efficient to approximate the expected shortfall.

The problem of calibrating time-inhomogeneous exponential Lévy models with finite jump activity based on market prices of plain vanilla options is studied. Belomestny and Reiß introduced an estimation procedure for calibration in the homogeneous case with one maturity. The open-ended question that will be addressed is if we can extend this model to use all listed plain vanilla options with multiple maturities. This opens a way to use all available data to create a time-inhomogeneous model with time-dependent Lévy triplets between subsequent intervals based on the maturities.

We establish via an adapted Lévy-Khintchine representation an estimation procedure on the explicit inversion of the option price formulas in the spectral domain with a cut-off scheme for the regularisation of high frequencies. The estimation procedure will be shown to be well-defined and the parameters to be normally distributed asymptotically.

Practical implications imply that using the asymptotic variances of the normal distributions leads to insufficient confidence intervals for non-asymptotical cases. We, therefore, construct confidence intervals using an approximated finite sample variance.

Monte Carlo simulations are implemented in the computational software R to evaluate the stability and accuracy of the estimation procedure. Furthermore, the finite sample confidence intervals are assessed with coverage probabilities.

In the end, the estimation procedure is evaluated by calibrating to market data of plain vanilla S\&P500 options, which contain numerous maturities. The estimated parameters with confidence intervals between maturities support time-dependency and the constructed time-inhomogeneous exponential L{\'e}vy models appear favorable in contrast to the time-homogeneous model.