This thesis explores the modeling of volatility and dependence in forward rates in the fixed income market for the purpose of risk estimation in derivative portfolios. A brief background on popular quantile-based risk measures is given. A short introduction is given to GARCH-type volatility models, as well as copula and vine models for dependence between random variables. Some details on parameter estimation and sampling related to these models are also provided. A backtesting procedure is performed using various combinations of volatility and dependence models. The results of this procedure indicate that the Student's t copula is preferable among the dependence structures considered. Also, none of the choices of conditional distribution for the volatility models provide good results at all the percentiles considered, but the normal distribution appears to be a good choice far into the tails.

This work will study different methods to estimate counterparty credit risk, where the methods represent both analytical approximation and simulation based method. The somewhat more analytical approximation that will be used is the current exposure method from the Bank for International Settlements and is based on simple add-on factor to the current market value. In the simulation part, Monte Carlo methods will be used. The paper will show that Monte Carlo methods enable estimation of the full exposure distribution as a function of time. From that distribution two measures of exposure will be used. The first use the peak at the 95% percentile and the second uses the concept of effective expected exposure. Those three alternative measures will be tested on six different portfolios. The portfolios are based on real data and represent both private persons, small companies, life insurance, investment bank and some of more academic interest. The estimate of exposure in those portfolios will be estimated with and without the establishment of netting agreements in order to see how that affects the exposure. The numerical results indicate that netting results in reduced exposure. In the comparisons between the different exposure measures the results show that the simulation based method in general estimates a lower exposure, but it depends intently on the construction of the portfolio. Based on those observations the main conclusion is that a simulation based approach is preferable since it enables better risk control within the firm as a consequence of enabling anatomizes of the evolution of exposure through time. Keywords: Counterparty Credit Risk, Libor Market Model and Monte Carlo simulation

This thesis studies the estimation of credit exposure arising from a portfolio of interest rate derivatives. The estimation is performed using a Monte Carlo simulation. The results are compared to the exposure obtained under the current exposure method provided by the Bank for International Settlements (BIS). We show that the simulation method provides a much richer set of information for credit risk managers. Also, depending on the current exposure and the nature of the transactions, the BIS method can fail to account for potential exposure. All test portfolios benefit significantly from a netting agreement, but the BIS approach tends to overestimate the risk reduction due to netting. In addition we examine the impact of antithetic variates and different time-discretizations. We find that a discretization based on derivatives' start and maturity dates may reduce simulation time significantly without loosing generality in exposure profiles. Antithetic variates have a small effect.

This thesis seeks to establish a methodology to reveal whether the risk appetite held by investors is dependent on the macroeconomic environment and, if present, to quantify this dependency. To do so a generic model is built and a case study is carried out with data from DnBNOR. The available data consists of the daily profit and losses together with the number and volume of transactions made in a currency portfolio owned by DnBNOR and some selected timeseries on exchange rates, all against NOK. Also, timeseries on the gross national product and consumer price index are collected from Statistics Norway. In the process of building the model, the thesis sets out the theoretical foundation for different risk measurement concepts and gives a presentation of the theory on business cycles as this is used to classify and measure the macroeconomic environment. The model is built with a Bayesian approach and implemented in WinBUGS. The use of Bayesian statistics is motivated by different time resolution of the data; some of the data is observed every day while other parts are observed each quarter. The thesis' main idea is to decompose the relevant part of the economy in one microeconomic and one macroeconomic state. The microeconomic state is unique for each day while the macroeconomic state accounts for one quarter; together they give the expected risk appetite for each day. In this way the impact from the macroeconomic state is quantified and the results show that the macroeconomic state is statistically significant for the risk appetite. As this is a case study one needs more data and research before any universal valid conclusions can be made.

We study the numerical integration of nonholonomic problems. The problems are formulated using Lagrangian and Hamiltonian mechanics. We review briefly the theoretical concepts used in geometric mechanics. We reconstruct two nonholonomic variational integrators from the monograph of Monforte. We also construct two one-step integrators based on a combination of the continuous Legendre transform and the discrete Legendre transform from an article by Marsden and West. Inintially these integrators display promising behavior, but they turn out to be unstable. The variational integrators are compared with a classical Runge-Kutta method. We compare the methods on three nonholonomic systems: The nonholonomic particle from the monograph of Monforte, the nonholonomic system of particles from an article by McLachlan and Perlmutter, and a variation of the Chaplygin sleigh from Bloch.

We present the cryptographic primitives needed in the construction of electronic voting systems based on homomorphic encryptions and on verifiable secret sharing. Then "The theory and implementation of an electronic voting system" by Ivan Damgård, Jens Groth and Gorm Salomonsen is presented as an example of electronic voting systems based on homomorphic encryptions, while "Multi-authority secret-ballot election with linear work" by Ronald Cramer, Matthew Franklin, Berry Schoenmakers and Moti Yung is presented as an example of electronic voting systems based on verifiable secret sharing. Moreover, the mathematical background for these systems are studied with particular emphasis on the security issues of the relevant sub-protocols. Comparing these two examples we find that the presented voting system based on verifiable secret sharing is more secure then the one based on homomorphic encryptions, both in regard to privacy and robustness. On the other hand, we find that the presented voting system based on homomorphic encryptions is more efficient then the one based on verifiable secret sharing.

The location of a reflector or medium in the subsurface is correlated with the high wavenumbers or high frequencies in the velocity field. Indeed, the determination of the high frequencies of the velocity field both normally and laterally is the key step for improving seimic data and then get a better insight of the position of a reflector in the subsurface. This project focus on the velocity data processing part involved in seismic tomography. We describe, compare and implement several highpass operators based on finite-difference and the Hamming window in order to filter a seismic velocity dataset. In fact, we study their behaviour in the frequency domain by examining their spectrums. The main contribution of this project is to construct two dimensional anisotropic operators by rotating a one dimensional operator based on linear interpolation. We test all the operators on a synthetic seismic velocity dataset and compare the results obtained between the isotropic filtering method and the anisotropic filtering method. We show that anisotropic filters can be useful in certain geological circumstances. Finally we attempt to scale the different operators in order to fully incorporate them in the seismic tomography inversion problem by using a Bayesian method. We show that it is possible to decide the strength of the constraint in which we want to filter the seismic dataset by using a regularization parameter.

In this thesis we discuss how a system of ordinary differential equations (ODE) describing electro-chemical processes in a heart cell can be solved by numerical methods. The system is stiff, and explicit numerical solvers are therefore slow. In order to overcome the stiffness, the system is split into a stiff and a non-stiff part. The split system is solved by a Strang splitting method and an exponential integrator, based on a commutator free Lie group method. We outline a theory for estimating the computational cost of a numerical method. The solvers for the split system are compared to implicit solvers for the entire system. The conclusion is that it is possible to take out two components which are responsible for the stiffness of the original system, but that more research needs to be done in order to make efficient methods which take advantage of the fact.

This thesis is intended to present some developments in the theory of algebraic stability. The main topics are stability for triangulated categories and the distinguished slopes of Hille and de la Pena for quiver representations.

We go through the basic theory of simple mehcanical systems on Riemannain manifolds with symmetry, in an attempt to understand some of the main features of configuration space reduction. As a part of this, we will look at some special cases for whom this works out well, and also indicate a direction of further development.