This thesis is about pricing European options using a Fourier-based numerical method called the COS method under the rough Heston model. Besides examining the efficiency and accuracy of the COS method for pricing options under the rough Heston model, it is also investigated if the rough Heston model produces the advantages of the so-called rough volatility models. To do so, the characteristic function of the rough Heston model is derived, and the COS method for the rough Heston model and also a Monte Carlo simulation scheme is introduced. Throughout the thesis, the theoretical background of the rough Heston model, the numerical techniques and some numerical experiments on European option prices and implied volatility behaviors are presented. Also, a calibration of the rough Heston model is performed using Artificial Neural Networks. As a result of this thesis, pricing of European options using COS method is succeeded. Moreover, it is shown that the rough Heston model produces the rough volatility behaviors as expected.

Generative adversarial networks (GANs) have shown promising results when applied on partial differential equations and financial time series generation. This thesis investigates if GANs can be used to provide a strong approximation to the solution of stochastic differential equations (SDEs) of the Ito type. Standard GANs are only able to approximate processes in conditional distribution, yielding a weak approximation to the SDE.
A novel GAN architecture is proposed that enables strong approximation, called the constrained GAN. The discriminator of this GAN is informed with the random sample that corresponds to the Brownian motion increment between two time steps. This way, the constrained GAN does not only learn the conditional distribution, but the unique map from a random increment to the next asset value along the path, conditional on the previous value. The architecture was tested on geometric Brownian motion (GBM) and the Cox Ingersoll Ross (CIR) process in one dimension, where it was conditioned on a range of time steps and previous values of the asset process. The constrained GAN was shown to outperform discrete-time schemes in strong error on a discretisation with large time steps. It also outperformed the standard conditional GAN when approximating the conditional distribution. A method is proposed to extend the constrained GAN to general one-dimensional Ito SDEs, beyond the SDEs tested in this work. In future work, the constrained GAN should be conditioned on the SDE parameters as well, allowing it to learn an entire family of solutions at once. Furthermore, the architecture could be extended to higher dimensions, including systems of SDEs, such as the Heston model.

The aim of this thesis is to forecast the evolution of the prepayment rate in a mortgage portfolio. In the Netherlands, people with a loan have the possibility to repay (part of) their outstanding loan before the due date. These prepayments make the length of the portfolio of loans stochastic, which creates problems in the refinancing policy of the bank, and affects the Asset & Liability Management. Moreover, interest rate risk arises from prepayments, meaning that being able to forecast the prepayment rate can increase the performance of the hedging strategy of a bank. Given the magnitude of the mortgage portfolio in the balance sheet of a bank, estimating the prepayment rate is therefore crucial. There are two kinds of models in the literature, the optimal prepayment model, which sees prepayment as a consequence of rational behavior (e.g. prepayments are always exercised at an optimal time), and the exogenous model which also takes into account other macroeconomic variables, client specifics and loan characteristics. Our focus will be on the second kind of techniques, precisely we will approach the problem as a classification task that will be carried out with two different machine learning techniques: Random Forests and Artificial Neural Networks.Since prepayments are rare events, this leads to an imbalanced data set framework. The imbalance between classes creates complications in the development of the algorithm, hence ad hoc corrections are applied to solve them.

Forecasting the prepayments is essential for any financial institution providing mortgages, and it is a crucial step in the hedging of the risk resulting from these unexpected cash flows. The way in which the prepayment rate is predicted impacts on the hedging strategy. For example, if the prepayment model is deterministic only the average prepayments are forecast, and a linear hedge composed of swaps is sufficient. However, in condition of volatile markets the lack of a non-linear hedge can result in losses for the bank. Considering that the there is a correlation between the prepayments and the level of interest rates in the market, we propose a prepayment model which is only based on the refinancing incentive. This way, the prepayments' forecast might be less accurate, but the clear link with the market allows to extend the prepayment model to a stochastic environment. We showed that allowing the notional of a mortgage to be stochastic unveils the non-linear risk embedded in the prepayment option, arising the necessity to include non-linear instruments in the hedging portfolio. The calibration of the refinancing incentive on a data set of more than thirty millions of observations led us to choose the functional form of the prepayments that is able to capture the borrowers' behaviour the most, and it distinguishes the model from full-rational models in which the option to prepay is assumed to be always exercised rationally. Then, the linear and non-linear risks are addressed to a set of tradeable instruments, aiming to build a static hedge. Different combinations of swaps and swaptions are tested, in order to determine which derivatives have the highest replication power. This research can impact considerably the evaluation of the exposure to interest rate risk of mortgage providers, and it can improve the performance of the hedging of the prepayment risk. Moreover, since the linear and non-linear components of the risk embedded in mortgages are distinguished, it can also help in the pricing of the prepayment option, allowing banks to define the mortgage rates with more accuracy.