Efficiently managing hedging portfolios on behalf of pension funds is key in achieving the target hedging strategy, which can significantly impact coverage ratios. A new optimization approach to fixed income portfolio management for pension funds is proposed that finds interest rate risk hedging strategies while incorporating additional requirements. These are relevant requirements for pension funds such as country allocations, low transaction costs and reasonable investment costs. In doing so, pension fund regulations and common practices are investigated in a rigorous mathematical framework. The hedging strategies are shown to perform well when back-testing. In addition, simulation of the interest rate and cash flows in a Defined Benefits pension scheme displays the good performance of the strategies. These strategies are further tailored to specific pension funds by considering the trade-off between yield and risk, which could contribute to increasing a pension fund’s coverage ratio. Alternatively, a procedure is also proposed to generate more diversified albeit less optimal hedging portfolios using the optimization approach.

This thesis showcases a rather contemporary method of solving a generalized system of stochastic differential equations (SDE's) comparable to the SABR model. The solution is derived from a stochastic-local volatility (SLV) model in which the local volatility (LV) component is kept general. This generality is maintained throughout all derivations, eventually yielding a model containing an undefined LV function. This function can then be specified however the user of the model deems suitable, as long as minor constraints are satisfied. Obviously, this is a very valuable quality of the model as it is highly customizable. The solution consists of a set of pricing functions that seemingly possess all the aforementioned desirable properties, i.e. fast in evaluation, computational tractability, flexibility etc., with little disadvantages. The generalized SLV model that is used is typically denoted in the form of two SDE's, though in the majority of this thesis an atypical three SDE form is used. This extended system is used to isolate the LV component, in turn enabling for appropriate application of an SDE transformation called the Lamperti transform, which will provide the key to solving the entire system. The Lamperti transform is a highly versatile method for transforming SDE's into new equations typically more suitable for simulation and parameter estimation procedures, and its inner-workings and various applications will be the main focus of this thesis.

This thesis captures the calibration of a FX hybrid model: The FX Black-Scholes Hull-White model. The main focus is on the calibration of the parameters in the Hull-White process: The mean reversion and the volatility parameter. The latter is commonly calibrated as a time-dependent parameter, whilst the mean reversion parameter is not. This thesis covers the calibration of the mean reversion as a time-dependent parameter. A known method from the Literature is researched, where we calibrate the mean reversion independently from the volatility parameter to the ratio of two swaptions with the same expiry but different tenor. In our research this method is extended to the negative interest rate environment by assuming that the swap rate follows a shifted lognormal distribution instead of a lognormal distribution. We show that a specific set of swaptions can be chosen, so that the calibration problem is simplified. This choice leads to sequential calibration of convex optimization problems. Numerical results of calibration to artificial and market data are presented, where we compare our method to picking the mean reversion parameter arbitrarily. The findings suggest that the choice of mean reversion parameter affects the calibration procedure. Therefore, we argue that a calibration method for the mean reversion would be appropriate. Besides the focus on the Hull-White process, the calibration of the volatility parameter in the FX Black-Scholes process of the hybrid model is investigated. For the calibration of the FX volatility parameter ATM options on FX rates are used. Numerical results of calibration to artificial and market data are shown. A typical problem of calibration in the industry is discussed: Precision for late maturities. The results in this thesis suggest that this problem could be solved. For research on homogeneity constraints on the time-dependent parameters, the performance of delta-hedging is investigated. The delta-hedging is performed in a simple Black-Scholes world with time-dependent volatility. We show that given a fixed amount hedges beforehand and interest rate zero, there exists an optimal distribution of time points that will lead to equal variance on the profit and loss for every volatility function that has equal implied volatility from t0 to maturity T. Using this strategy, the homogeneity of the volatility parameter has no impact on the performance of delta-hedging. Therefore, in this thesis no homogeneity constraints are used for the calibration of the time-dependent parameters.

We study the impact of wrong-way risk (WWR) on the credit valuation adjustment (CVA) of a portfolio of interest rate swaps (IRSs), using an intensity-based reduced form model. To model WWR in IRSs we create a dependence between he underlying market risk factor of the IRS and the survival probability of the counterparty. The focus lies on modeling the credit part of the CVA, choosing the most suitable default model. Using a Monte Carlo (MC) framework, we correlate the Brownian increments of the stochastic processes that describe the interest rate and the hazard rate. Assuming the correlation parameter is chosen based on historical data, we solve the problem of leaking correlation. Two stochastic processes for the credit part are chosen, the CIR++ and the JCIR++, such that the influence of the model choice can be separated from the impact of WWR. With a case study, we vary the correlation parameter to quantify the impact of WWR on CVA of the portfolio. We show that the introduced dependence has a significant impact on the CVA, that depends on the calibration, the portfolio and the chosen correlation. Solving the leaking correlation problem can increase the impact of WWR on the CVA, especially using JCIR++ dynamics.

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.

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.

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.

The yield curve represents market supply and demand implied expectations of future interest rates and is calibrated from the most liquidly traded interest rate derivatives like cash deposits, forward rate agreeents, swaps and futures. Due to the daily margining mechanism of futures contracts, interest rate futures require the substraction of a convexity adjustment in order for them to be used in curve calibration. It is common practice to use externally computed convexity adjustments, which treats the convexity adjustment as a black-box parameter. We will argue the inherent relationship between the convexity adjustment and cap/floor volatility smiles and derive a nested calibration algorithm for the simultaneous calibration of the yield curve to futures and the convexity adjustment to cap/floor volatility surfaces. This introduces dependencies of the yield curve to option volatilities and we will argue that for simple interest rate derivatives the implied vegas are negligible. .

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.

This thesis studies advanced and accurate discretization schemes for relevant partial differential equations (PDEs) in finance. We start with techniques which may be particularly useful for the pricing of so-called vanilla financial options, European or American, and then move on to more complex models for the pricing of exotic options. @en