The purpose of this thesis is to develop an optimization model that chooses the optimal and price efficient combination of underlying assets for a equally weighted basket option. To obtain a price efficient combination of underlying assets a function that calculates the basket option price is needed, for further use in an optimization model. The closed-form basket option pricing is a great challenge, due to the lack of a distribution describing the augmented stochastic price process. Many types of approaches to price an basket option has been made. In this thesis, an analytical approximation of the basket option price has been used, where the analytical approximation aims to develop a method to describe the augmented price process. The approximation is done by moment matching, i.e. matching the first two moments of the real distribution of the basket option with an lognormal distribution. The obtained price function is adjusted and used as the objective function in the optimization model. Furthermore, since the goal is to obtain en equally weighted basket option, the appropriate class of optimization models to use are binary optimization problems. This kind of optimization model is in general hard to solve - especially for increasing dimensions. Three different continuous relaxations of the binary problem has been applied in order to obtain continuous problems, that are easier to solve. The results shows that the purpose of this thesis is fulfilled when formulating and solving the optimization problem - both as an binary and continuous nonlinear optimization model. Moreover, the results from a Monte Carlo simulation for correlated stochastic processes shows that the moment matching technique with a lognormal distribution is a good approximation for pricing a basket option. / Syftet med detta examensarbete är att utveckla ett optimeringsverktyg som väljer den optimala och priseffektiva kombinationen av underliggande tillgångar för en likaviktad aktiekorg. För att kunna hitta en priseffektiv kombination av underliggande tillgångar behöver man finna en passande funktion som bestämmer priset på en likaviktad aktiekorg. Prissättningen av dessa typer av optioner är en stor utmaning. Detta är på grund av bristen av en sannolikhetsfördelning som kan beskriva den utökade och korrelerade stokastiska prisprocess som uppstår för en aktiekorg. Många typer av prissättningar har undersökts och tillämpats. I detta arbete har en analytisk approximation använts för att kunna beskriva den underliggande pris processen approximativt. Uppskattningen görs genom att matcha de tvåförsta momenten av den verkliga fördelningen med motsvarande moment för en lognormal fördelning. Den erhållna prisfunktionen justeras och används som målfunktionen i optimeringsmodellen. Binära ickelinjära optimeringsproblem är i allmänhet svåra att lösa - särskilt för ökande dimensioner av variabler. Tre olika kontinuerliga omformuleringar av det binära optimeringsproblemet har gjorts för att erhålla kontinuerliga problem som är lättare att lösa. Resultaten visar att en optimal och priseffektiv kombination av underliggande aktier är möjlig att hitta genom att formulera ett optimeringsproblem - både som en binär och kontinuerlig ickelinjär optimeringsmodell. Dessutom visar resultaten från en Monte Carlo-simulering, i detta fall för korrelerade stokastiska processer, att moment matching metoden utförd med en lognormal fördelning är en god approximation för prissättningen av aktiekorgar.

The performance of a mobile satellite communications link is conditioned by the characteristics of the propagation path between a satellite and mobile users. The most important propagation effect in land mobile satellite system is roadside attenuation of the signals due to vegetation or urban structures. System designers should have the most reliable information about the statistics of the propagation channel to build reliable systems that can compensate for bad propagation conditions. In 1998, the Virginia Tech Antenna Group developed a simulator, PROSIM, to simulate a propagation channel in the case of roadside tree attenuation in land mobile satellite systems. This thesis describes some improvements to PROSIM, and the adaptation and validation of PROSIM for Digital Audio Radio Satellite systems operating at S-band frequencies. The performance of the simulator for S-band frequencies was evaluated through a measurement campaign conducted with the XM Radio signals at 2.33 GHz in various propagation environments. Finally, additional results on dual satellite systems and fade correlation are described. / Master of Science

Ce travail présente quelques résultats concernant des méthodes numériques déterministes et probabilistes pour des équations aux dérivées partielles à coefficients aléatoires, avec des applications à l'hydrogéologie. On s'intéresse tout d'abord à l'équation d'écoulement dans un milieu poreux en régime stationnaire avec un coefficient de perméabilité lognormal homogène, incluant le cas d'une fonction de covariance peu régulière. On établit des estimations aux sens fort et faible de l'erreur commise sur la solution en tronquant le développement de Karhunen-Loève du coefficient. Puis on établit des estimations d'erreurs éléments finis dont on déduit une extension de l'estimation d'erreur existante pour la méthode de collocation stochastique, ainsi qu'une estimation d'erreur pour une méthode de Monte-Carlo multi-niveaux. On s'intéresse enfin au couplage de l'équation d'écoulement considérée précédemment avec une équation d'advection-diffusion, dans le cas d'incertitudes importantes et d'une faible longueur de corrélation. On propose l'analyse numérique d'une méthode numérique pour calculer la vitesse moyenne à laquelle la zone contaminée par un polluant s'étend. Il s'agit d'une méthode de Monte-Carlo combinant une méthode d'élements finis pour l'équation d'écoulement et un schéma d'Euler pour l'équation différentielle stochastique associée à l'équation d'advection-diffusion, vue comme une équation de Fokker-Planck. / This work presents some results about probabilistic and deterministic numerical methods for partial differential equations with stochastic coefficients, with applications to hydrogeology. We first consider the steady flow equation in porous media with a homogeneous lognormal permeability coefficient, including the case of a low regularity covariance function. We establish error estimates, both in strong and weak senses, of the error in the solution resulting from the truncature of the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates, from which we deduce an extension of the existing error estimate for the stochastic collocation method along with an error estimate for a multilevel Monte-Carlo method. We finally consider the coupling of the previous flow equation with an advection-diffusion equation, in the case when the uncertainty is important and the correlation length is small. We propose the numerical analysis of a numerical method, which aims at computing the mean velocity of the expansion of a pollutant. The method consists in a Monte-Carlo method, combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation.

Le but de ce mémoire de maîtrise est de décrire les propriétés de la loi double Pareto-lognormale, de montrer comment on peut introduire des variables explicatives dans le modèle et de présenter son large potentiel d'applications dans le domaine de la science actuarielle et de la finance. Tout d'abord, nous donnons la définition de la loi double Pareto-lognormale et présentons certaines de ses propriétés basées sur les travaux de Reed et Jorgensen (2004). Les paramètres peuvent être estimés en utilisant la méthode des moments ou le maximum de vraisemblance. Ensuite, nous ajoutons une variable explicative à notre modèle. La procédure d'estimation des paramètres de ce mo-\\dèle est également discutée. Troisièmement, des applications numériques de notre modèle sont illustrées et quelques tests statistiques utiles sont effectués. / The purpose of this Master's thesis is to describe the double Pareto-lognormal distribution, show how the model can be extended by introducing explanatory variables in the model and present its large potential of applications in actuarial science and finance. First, we give the definition of the double Pareto-lognormal distribution and present some of its properties based on the work of Reed and Jorgensen (2004). The parameters could be estimated by using the method of moments or maximum likelihood. Next, we add an explanatory variable to our model. The procedure of estimation for this model is also discussed. Finally, some numerical applications of our model are illustrated and some useful statistical tests are conducted.

The atmospheric lightwave propagation is considerably influenced by the random variations in the refractive index of air pockets due to turbulence. This undesired effect significantly degrades the performance of free-space optical (FSO) communication systems. Interestingly, the severity of such random degradations is highly related to the range of atmospheric propagation. In this thesis, we introduce relay-assisted FSO communications as a very promising technique to combat the degradation effects of atmospheric turbulence. Considering different configurations of the relays, we quantify the outage behavior of the relay-assisted system and identify the optimum relaying scheme. We further optimize the performance of the relay-assisted FSO system subject to some power constraints and provide optimal power control strategies for different scenarios under consideration. Moreover, an application of FSO relaying technique in quantum communications is investigated. The results demonstrate impressive performance improvements for the proposed relay-assisted FSO systems with respect to the conventional direct transmission whether applied in a classical or a quantum communication channel.

Ce travail présente quelques résultats concernant des méthodes numériques déterministes et probabilistes pour des équations aux dérivées partielles à coefficients aléatoires, avec des applications à l'hydrogéologie. On s'intéresse tout d'abord à l'équation d'écoulement dans un milieu poreux en régime stationnaire avec un coefficient de perméabilité lognormal homogène, incluant le cas d'une fonction de covariance peu régulière. On établit des estimations aux sens fort et faible de l'erreur commise sur la solution en tronquant le développement de Karhunen-Loève du coefficient. Puis on établit des estimations d'erreurs éléments finis dont on déduit une extension de l'estimation d'erreur existante pour la méthode de collocation stochastique, ainsi qu'une estimation d'erreur pour une méthode de Monte-Carlo multi-niveaux. On s'intéresse enfin au couplage de l'équation d'écoulement considérée précédemment avec une équation d'advection-diffusion, dans le cas d'incertitudes importantes et d'une faible longueur de corrélation. On propose l'analyse numérique d'une méthode numérique pour calculer la vitesse moyenne à laquelle la zone contaminée par un polluant s'étend. Il s'agit d'une méthode de Monte-Carlo combinant une méthode d'élements finis pour l'équation d'écoulement et un schéma d'Euler pour l'équation différentielle stochastique associée à l'équation d'advection-diffusion, vue comme une équation de Fokker-Planck.

The atmospheric lightwave propagation is considerably influenced by the random variations in the refractive index of air pockets due to turbulence. This undesired effect significantly degrades the performance of free-space optical (FSO) communication systems. Interestingly, the severity of such random degradations is highly related to the range of atmospheric propagation. In this thesis, we introduce relay-assisted FSO communications as a very promising technique to combat the degradation effects of atmospheric turbulence. Considering different configurations of the relays, we quantify the outage behavior of the relay-assisted system and identify the optimum relaying scheme. We further optimize the performance of the relay-assisted FSO system subject to some power constraints and provide optimal power control strategies for different scenarios under consideration. Moreover, an application of FSO relaying technique in quantum communications is investigated. The results demonstrate impressive performance improvements for the proposed relay-assisted FSO systems with respect to the conventional direct transmission whether applied in a classical or a quantum communication channel.

Norovirus (NoV) is a significant cause of gastroenteritis globally, and the consumption of oysters is frequently linked to outbreaks. Depuration is the principle means employed to reduce levels of potentially harmful agents or toxins in shellfish. The aim of this thesis is to construct mathematical models which can describe the depuration dynamics of water-borne pathogens, and specifically examine the dynamics of NoV during depuration for a population shellfish. Legislation is currently under consideration within the EU by the Directorate-General for Health and Consumers (DG SANCO) to limit the maximum level of NoV that consumers are exposed to via this route. Therefore it is important to the utility of the thesis that any models constructed should incorporate control measures which could be used to implement minimum NoV levels. Doing so allowed calculation of minimum depuration times that would be required to adhere to the control measures incorporated into the models. In addition to modelling the impact on pathogens during the depuration, we wished to gain some insight into how the variability, and not just the mean levels, of water-borne pathogens can be as important with respect to the length of depuration required to minimise any food safety risks to the consumer. This proved difficult in the absence of any data sets that can be used to calculate variability measures, as little data is currently available to inform these values for NoV. However, our modelling techniques were able to calculate an upper limit on the variability of water-borne pathogens that can be well approximated by lognormal distributions. Finally we construct a model which provided linkage between the depuration process and the accretion of pathogens by shellfish while still within farming waters. This model proposed that the pulses of untreated waste waters released by sewage treatment works due to high levels of rainfall would be transmitted into shellfish whilst filter-feeding.

Biodiversity is determined by a myriad of complex processes acting at different scales. Given the current rates of biodiversity loss and change, it is of paramount importance that we improve our understanding of the underlying structure of ecological communities. In this thesis, I focused on Species Abundance Distributions (SAD), as a synthetic measure of biodiversity and community structure, and on Beta (β) diversity patterns, as a description of the spatial variation of species composition. I systematically assessed the effect of scale on both these patterns, analysing a broad range of community data, including different taxa and habitats, from the terrestrial, marine and freshwater realms. Knowledge of the scaling properties of abundance and compositional patterns must be fully integrated in biodiversity research if we are to understand biodiversity and the processes underpinning it, from local to global scales. SADs depict the relative abundance of the species present in a community. Although typically described by unimodal logseries or lognormal distributions, empirical SADs can also exhibit multiple modes. However, the existence of multiple modes in SADs has largely been overlooked, assumed to be due to sampling errors or a rare pattern. Thus, we do not know how prevalent multimodality is, nor do we have an understanding of the factors leading to this pattern. Here, I provided the first global empirical assessment of the prevalence of multimodality across a wide range of taxa, habitats and spatial extents. I employed an improved method combining two model selection tools, and (conservatively) estimated that ~15% of the communities were multimodal with strong support. Furthermore, I showed that the pattern is more common for communities at broader spatial scales and with greater taxonomic diversity (i.e. more phylogenetically diverse communities, since taxonomic diversity was measured as number of families). This suggests a link between multimodality and ecological heterogeneity, broadly defined to incorporate the spatial, environmental, taxonomic and functional variability of ecological systems. Empirical understanding of how spatial scale affects SAD shape is still lacking. Here, I established a gradient in spatial scale spanning several orders of magnitude by decomposing the total extent of several datasets into smaller subsets. I performed an exploratory analysis of how SAD shape is affected by area sampled, species richness, total abundance and taxonomic diversity. Clear shifts in SAD shape can provide information about relevant ecological and spatial mechanisms affecting community structure. There was a clear effect of area, species richness and taxonomic diversity in determining SAD shape, while total abundance did not exhibit any directional effect. The results supported the findings of the previous analysis, with a higher prevalence of multimodal SADs for larger areas and for more taxonomically diverse communities, while also suggesting that species spatial aggregation patterns can be linked to SAD shape. On the other hand, there was a systematic departure from the predictions of two important macroecological theories for SAD across scales, specifically regarding logseries distributions being selected only for smaller scales and when species richness and number of families were proportionally much smaller than the total extent. β diversity quantifies the variation in species composition between sites. Although a fundamental component of biodiversity, its spatial scaling properties are still poorly understood. Here, I tested if two conceptual types of β diversity showed systematic variation with scale, while also explicitly accounting for the two β diversity components, turnover and nestedness (species replacement vs species richness differences). I provided the first empirical analysis of β diversity scaling patterns for different taxa, revealing remarkably consistent scaling curves. Total β diversity and turnover exhibit a power law decay with log area, while nestedness is largely insensitive to scale changes. For the distance decay of similarity analysis, while area sampled affected the overall dissimilarity values, rates of similarity were consistent across large variations in sampled area. Finally, in both these analyses, turnover was the main contributor to compositional change. These results suggest that species are spatially aggregated across spatial scales (from local to regional scales), while also illustrating that substantial change in community structure might occur, despite species richness remaining relatively stable. This systematic and comprehensive analysis of SAD and community similarity patterns highlighted spatial scale, ecological heterogeneity and species spatial aggregation patterns as critical components underlying the results found. This work expanded the range of scales at which both theories deriving SAD and community similarity studies have been developed and tested (from local plots to continents). The results here showed strong departures from two important macroecological theories for SAD at different scales. In addition, the overall findings in this thesis clearly indicate that unified theories of biodiversity (or assuming a set of synthetic minimal assumptions) are unable to accommodate the variability in SADs shape across spatial scales reported here, and cannot fully reproduce community similarity patterns across scales. Incorporating more realistic assumptions, or imposing scale dependent assumptions, may prove to be a fruitful avenue for ecological research regarding the scaling properties of SAD and community similarity patterns. This will allow deriving new predictions and improving the ability of theoretical models to incorporate the variability in abundance and similarity patterns across scales.

This thesis considers the price and characteristics of an exotic option called the Volatility-Cap-Target-Level(VCTL) option. The payoff function is a simple European option style but the underlying value is a dynamic portfolio which is comprised of two components: A risky asset and a non-risky asset. The non-risky asset is a bond and the risky asset can be a fund or an index related to any asset category such as equities, commodities, real estate, etc. The main purpose of using a dynamic portfolio is to keep the realized volatility of the portfolio under control and preferably below a certain maximum level, denoted as the Volatility-Cap-Target-Level (VCTL). This is attained by a variable allocation between the risky asset and the non-risky asset during the maturity of the VCTL-option. The allocation is reviewed and if necessary adjusted every 15th day. Adjustment depends entirely upon the realized historical volatility of the risky asset. Moreover, it is assumed that the risky asset is governed by a certain group of stochastic differential equations called affine jump-diffusion models. All models will be calibrated using out-of-the money European call options based on the Deutsche-Aktien-Index(DAX). The numerical implementation of the portfolio diffusions and the use of Monte Carlo methods will result in different VCTL-option prices. Thus, to price a nonstandard product and to comply with good risk management, it is advocated that the financial institution use several research models such as the SVSJ- and the Seppmodel in addition to the Black-Scholes model. Keywords: Exotic option, basket option, risk management, greeks, affine jumpdiffusions, the Black-Scholes model, the Heston model, Bates model with lognormal jumps, the Bates model with log-asymmetric double exponential jumps, the Stochastic-Volatility-Simultaneous-Jumps(SVSJ)-model, the Sepp-model.