Clients with a mortgage loan may prepay a part of their loan before the contractual date. This is called prepayment. In the case of a prepayment, the bank who issued the loan earns less interest than ini- tially agreed. It is therefore essential to build accurate models for predicting prepayment behavior. In this thesis, machine learning models called artificial neural networks are used to predict conditional prepayment rates. In particular, the possibility to quantify the model uncertainty is studied. It is important to acknowledge that the accuracy of a model is not constant over its domain. Most ma- chine learning methods do not provide information about the uncertainty regarding a prediction and, unfortunately, methods that do so are often computationally expensive. This thesis studies uncertainty estimates generated by a neural network with dropout applied. Dropout is a method that randomly drops out neurons in each layer and has been suggested to prevent overfitting. We will see that this method can also be used to extract a measure of uncertainty regarding the prediction of a neural net- work efficiently. This method is used first to evaluate uncertainty for three simple functions under different distributions of the train data. Then the uncertainty estimates are evaluated for a neural network that predicts conditional prepayment rates. It will be illustrated that the uncertainty estimates provide an accurate indication of regions of the domain where predictions are inaccurate. Moreover, using a different model in the regions identified as uncertain can result in higher model performance.

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.

Three interest rate models are researched: Displaced Exponential-Vasicek, Hull-White one factor and Hull-White two factors with time-dependent volatility parameters. The motivation for this is two-fold: firstly, we would like to understand how the capital calculations would be impacted when yield curves are modelled under the three different models. This is done by looking at both magnitude and stability of the risk profiles and scalar risk-measures for three counterparties, which are highly representative for the bank. Secondly, we investigate the benefits and drawbacks of using one model and its corresponding calibration method over the others, with a special attention to the impact on yields correlations.

The first model, calibrated to historical data, is used as a nine-factors model for forward rates and is currently being used within the bank for PFE profiles and CVA regulatory capital. Historical backtest has proven the current model to perform reasonably well on real data and therefore it is used as a benchmark against which the other two models are tested. The two Hull-White models, used as short rate models, are calibrated to the risk-neutral measure (namely, to European swap- tions). Precisely, a two-steps calibration procedure suited for piece-wise constant volatility functions is implemented for both. The stability analysis reveals that the variation of Exposure at Defaults is significant, which might be undesired. On the other side, the two short rate models retain the correlation structure of interest rates better than the current model. This in turn translates into higher capital impact.

This thesis is about pricing European options and forward start options under the Heston LSV model. The impact of conditionally calibrating the Heston parameters on the satisfaction of the Feller condition and thereafter correcting with a local volatility surface is investigated here. The results show that this approach is computationally time efficient and accurate. Efficient numerical approaches for this LSV model, such as the multilevel Monte Carlo method, are also investigated. Furthermore, a comparison of several discretizations schemes for the SV part have been conducted. For the calibration of the local volatility surface, the efficiency of the Particle method and the Bin method are compared. An alternative numerical approach to this problem which builds on these two methods is developed and tested.

This work is on the extension of the SWIFT method to option pricing problems where the sum of lognormals occurs. The SWIFT method (ShannonWavelet Inverse Fourier Technique) is extended to the valuation of geometric Asian options and arithmetic Asian options with a Lévy process as underlying price process and the valuation of European options under SABR dynamics. In both applications a sum of lognormals (or sum of increments) occurs. The main result in this thesis is the SWIFT-SIA method (SWIFT sinc integral approximation), which is applied to the valuation of arithmetic Asian options as well as to the valuation of European options under the SABRmodel. Within the SWIFT-SIA method the recovery of the probability density function is obtained by an approximation, instead of a numerical integration method, which results in a very fast method compared to an alternative method based on cosine expansions as well as high accuracy in the option values.