The main objective of this thesis is to simulate, evaluate and discuss several methods for pricing European-style options. The Black-Scholes model has long been considered the standard method for pricing options. One of the downfalls of the Black-Scholes model is that it is strictly continuous and does not incorporate discrete jumps. This thesis will consider two alternate Levy models that include discretized jumps; The Merton Jump Diffusion and Kou's Double Exponential Jump Diffusion. We will use each of the three models to price real world stock data through software simulations and explore the results.Keywords: Levy Processes, Brownian motion, Option pricing, Simulation, Black-Scholes, Merton Jump Diffusion, Kou, Kou's Double Exponential Jump Diffusion. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2013.

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Two optimisation problems were considered, in which market exposure is indirectly controlled. The first one models the capital of a company and an independent portfolio of new businesses, each one represented by a Cram\'r-Lundberg process. The company can choose the proportion of new business it wants to take on and can alter this proportion over time. Here the objective is to find a strategy that maximises the survival probability. We use a point processes framework to deal with the impact of an adapted strategy in the intensity of the new business. We prove that when Cram\'{e}r-Lundberg processes with exponentially distributed claims, it is optimal to choose a threshold type strategy, where the company switches between owning all new businesses or none depending on the capital level. For this type of processes that change both drift and jump measure when crossing the constant threshold, we solve the one and two-sided exit problems. This optimisation problem is also solved when the capital of the company and the new business are modelled by spectrally positive L\'vy processes of bounded variation. Here the one-sided exit problem is solved and we prove optimality of the same type of threshold strategy for any jump distribution. The second problem is a stochastic variation of the work done by Taylor about underwriting in a competitive market. Taylor maximised discounted future cash flows over a finite time horizon in a discrete time setting when the change of exposure from one period to the next has a multiplicative form involving the company's premium and the market average premium. The control is the company's premium strategy over a the mentioned finite time horizon. Taylor's work opened a rich line of research, and we discuss some of it. In contrast with Taylor's model, we consider the market average premium to be a Markov chain instead of a deterministic vector. This allows to model uncertainty in future conditions of the market. We also consider an infinite time horizon instead of finite. This solves the time dependency in Taylor's optimal strategies that were giving unrealistic results. Our main result is a formula to calculate explicitly the value function of a specific class of pricing strategies. Further we explore concrete examples numerically. We find a mix of optimal strategies where in some examples the company should follow the market while in other cases should go against it.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Imported from http://etd.sun.ac.za / ENGLISH ABSTRACT: In this thesis, we propose to use Levy processes to model the dynamics of asset prices. In the first part, we deal with single asset options and model the log stock prices with a Levy process. We employ pure jump Levy processes of infinite activity, in particular variance gamma and CGMY processes. We fit the log-returns of six stocks to variance gamma and CGMY distributions and check the goodness of fit using statistical tests. It is observed that the variance gamma and the CGMY distributions fit the financial market data much better than the normal distribution. Calibration shows that at given maturity time the two models fit into the option prices very well. In the second part, we investigate the effect of dependence structure to multivariate option pricing. We use the new concept of Levy copula introduced in the literature by Tankov [40]. Levy copulas allow us to separate the dependence structure from the behavior of the marginal components. We consider bivariate variance gamma and bivariate CGMY models. To model the dependence structure between underlying assets we use the Clayton Levy copula. The empirical results on six stocks indicate a strong dependence between two different stock prices. Subsequently, we compute bivariate option prices taking into account the dependence structure. It is observed that option prices are highly sensitive to the dependence structure between underlying assets, and neglecting tail dependence will lead to errors in option pricing. / AFRIKAANSE OPSOMMING: In hierdie proefskrif word Levy prosesse voorgestel om die bewegings van batepryse te modelleer. Levy prosesse besit die vermoe om die risiko van spronge in ag te neem, asook om die implisiete volatiliteite, wat in finansiele opsie pryse voorkom, te reproduseer. Ons gebruik suiwer–sprong Levy prosesse met oneindige aktiwiteit, in besonder die gamma– variansie (Eng. variance gamma) en CGMY–prosesse. Ons pas die log–opbrengste van ses aandele op die gamma–variansie en CGMY distribusies, en kontroleer die resultate met behulp van statistiese pasgehaltetoetse. Die resultate bevestig dat die gamma–variansie en CGMY modelle die finansiele data beter pas as die normaalverdeling. Kalibrasie toon ook aan dat vir ’n gegewe verstryktyd die twee modelle ook die opsiepryse goed pas. Ons ondersoek daarna die gebruik van Levy prosesse vir opsies op meervoudige bates. Ons gebruik die nuwe konsep van Levy copulas, wat deur Tankov[40] ingelei is. Levy copulas laat toe om die onderlinge afhanklikheid tussen bateprysspronge te skei van die randkomponente. Ons bespreek daarna die simulasie van meerveranderlike Levy prosesse met behulp van Levy copulas. Daarna bepaal ons die pryse van opsies op meervoudige bates in multi–dimensionele exponensiele Levy modelle met behulp van Monte Carlo–metodes. Ons beskou die tweeveranderlike gamma-variansie en – CGMY modelle en modelleer die afhanklikheidsstruktuur tussen onderleggende bates met ’n Levy Clayton copula. Daarna bereken ons tweeveranderlike opsiepryse. Kalibrasie toon aan dat hierdie opsiepryse baie sensitief is vir die afhanlikheidsstruktuur, en dat prysbepaling foutief is as die afhanklikheid tussen die sterte van die onderleggende verdelings verontagsaam word.

In this study, general geometric Levy market models are considered. Since these models are, in general, incomplete, that is, all contingent claims cannot be replicated by a self-financing portfolio consisting of investments in a risk-free bond and in the stock, it is suggested that the market should be enlarged by artificial assets based on the power-jump processes of the underlying Levy process. Then it is shown that the enlarged market is complete and the explicit hedging portfolios for claims whose payoff function depends on the prices of the stock and the artificial assets at maturity are derived. Furthermore, the portfolio optimization problem is considered in the enlarged market. The problem consists of choosing an optimal portfolio in such a way that the largest expected utility of the terminal wealth is obtained. It is shown that for particular choices of the equivalent martingale measure in the market, the optimal portfolio only consists of bonds and stocks. This corresponds to completing the market with additional assets in such a way that they are superfluous in the sense that the terminal expected utility is not improved by including these assets in the portfolio.

Nous étudions certains processus de Lévy à valeurs dans les groupes d'isométries respectifs des espace-temps de Minkowski, de De Sitter et de Anti-De-Sitter. Le groupe d'isométries est vu comme le fibré des repères de l'espace-temps et les processus de Lévy considérés se projettent sur le fibré unitaire en un processus markovien relativiste ; c'est-à-dire que les trajectoires dans l'espace-temps sont de genre temps et que le générateur est invariant par les isométries. Dans la première partie nous adaptons pour les diffusions hypoelliptiques générales un résultat de Ben Arous et Gradinaru concernant la singularité de la fonction de Green hypoelliptique. Nous déduisons de cela un critère d'effilement de Wiener local pour les diffusions relativistes dans le groupe de Poincaré, groupe des isométries de l'espace-temps de Minkowski. Dans les deux dernières parties nous nous intéressons au comportement asymptotique du flot stochastique associé à ces processus de Lévy dans les différents groupes d'isométries. Sous une condition d'intégrabilité de la mesure de Lévy nous calculons explicitement les coefficients de Lyapounov des processus dans le groupe de Poincaré. Nous effectuons un travail similaire pour les espace-temps de De Sitter et Anti-De-Sitter en nous limitant au cas des diffusions. Nous explicitons de plus la frontière de Poisson pour la diffusion dans le groupe d'isométries de l'espace-temps de De Sitter. / We study some Lévy processes with values in the isometry group of Minkowski, De Sitter and Anti-de-Sitter space-times. The isometry group is seen as the frame bundle of the space-time and the Lévy processes we consider are some lift of relativistic markovian processes with values in the unitary tangent bundle of the space-time. Theses processes are relativistic in the sense that theirs trajectories are time-like and their generators are invariant by the isometries of the space-time. In the first part of this work we adapt to the case of a general hypoelliptic diffusion a result of Ben Arous and Gradinaru concerning the singularity of the hypoelliptic Green function. We deduce of this a local Wiener criterion for the relativistic diffusion in the isometry group of Minkowski space-time. In the two last parts we are interested to the asymptotic behavior of the stochastic flow associated to these Lévy processes in the different considered space-times. Under a integrability condition on the Lévy measure we compute explicitly the Lyapunov coefficient for such flows in the isometry group of Minkowski space-time. Then, we do a similar work in the context of de Sitter and Anti-de-Sitter space-times limiting ourselves to the case of diffusions. In fine, we explicit the Poisson boundary of the diffusion in the isometry group of de Sitter space-time.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-28T13:00:07Z No. of bitstreams: 1 arquivo total.pdf: 809257 bytes, checksum: 7406ef37d18bbaf1d9cdd5649f5cff19 (MD5) / Made available in DSpace on 2016-03-28T13:00:07Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 809257 bytes, checksum: 7406ef37d18bbaf1d9cdd5649f5cff19 (MD5) Previous issue date: 2015-12-11 / Let X = fX 0;X0 = 0g be a mean zero -stable random walk on Z with inhomogeneous jump rates f 􀀀1 i ; i 2 Zg, with 2 (1; 2] and f i : i 2 Zg is a family of independent random walk variables with common marginal distribution in the basis of attraction of an -stable law with 2 (0; 2]. In this paper we derive results about the long time behavior of this process, we obtain the scaling limit. To this end, rst we will approach probability on metric spaces, speci cally treat the D space of the functions that are right-continuous and have left-hand limits. We will also expose some results dealing with stable laws that are directly related to the above problem. / Seja X = fX 0;X0 = 0g um passeio aleat orio de m edia zero -est avel sobre Z com taxas de saltos n~ao homog^eneas f 􀀀1 i ; i 2 Zg, com 2 (1; 2] e f i : i 2 Zg uma fam lia de vari aveis aleat orias independentes com distribui c~ao marginal comum na bacia de atra c~ao de uma lei -est avel com 2 (0; 2]. Neste trabalho, obtemos resultados sobre o comportamento a longo prazo deste processo obtendo seu limite de escala. Para isso, faremos previamente um estudo sobre probabilidade em espa cos m etricos, mais especi camente sobre o espa co D das fun coes cont nuas a direita com limite a esquerda. Tamb em iremos expor alguns resultados que tratam de leis est aveis que est~ao relacionadas diretamente ao problema supracitado.

Prof. F. Lombard

Cette thèse étudie des modèles de populations branchantes appelés arbres de ramification, dans lesquels les individus évoluent indépendamment les uns des autres, ont des durées de vie indépendantes, identiquement distribuées (non nécessairement exponentielles), et donnent naissance à taux constant au cours de leur vie. On enrichit ces modèles en supposant que chaque individu porte un type et peut subir à la naissance une mutation, qui lui confère un nouveau type. On démontre dans le premier chapitre des résultats théoriques de convergence en loi pour des processus de Lévy bivariés sans sauts négatifs. Ces résultats sont ensuite exploités dans le deuxième chapitre pour établir un principe d'invariance pour l'arbre généalogique des populations décrites ci-dessus, enrichi de leur historique mutationnel, dans une asymptotique de grande taille de population. Enfin, on étudie dans le troisième chapitre la structure généalogique et le spectre de fréquence par site (nombre de mutations portées par un nombre donné d'individus) d'échantillons uniformes dans des populations branchantes critiques dont la limite d'échelle est un arbre brownien (par exemple, des arbres de naissance et mort critiques). Des perspectives d'applications de ces résultats à la génétique des populations sont présentées dans le quatrième chapitre. / This thesis studies branching population models called splitting trees, where individuals evolve independently from one another, have independent and identically distributed lifetimes (that are not necessarily exponential), and give birth at constant rate during their lives. We further assume that each individual carries a type, and possibly undergoes a mutation at her birth, that changes her type into a new one. In the first chapter, we prove convegence results for bivariate Lévy processes with non negative jumps. These theoretical results are used in the second chapter to establish an invariance principle for the genealogical tree of the populations described above, enriched with their mutational history, in a large population size asymptotic. Finally we study in the third chapter the genealogical structure and the site frequency spectrum (number of mutations carried by a given number of individuals) for uniform samples in critical branching populations whose scaling limit is a Brownian tree (e.g., critical birth-death trees). Possible future applications of these results to population genetics are presented in the fourth chapter.

<p>The goal of this thesis is to establish generalized isoperimetric inequalities on first exit time distributions as well as expectations of L\'evy processes.</p> <p>Firstly, we prove inequalities on first exit time distributions in the case that the L\'evy process is an $\alpha$-stable symmetric process $A_t$ on $\R^d$, $\alpha\in(0,2]$. Given $A_t$ and a bounded domain $D\subset\R^d$, we present a proof, based on the classical Brascamp-Lieb-Luttinger inequalities for multiple integrals, that the distribution of the first exit time of $A_t$ from $D$ increases under Steiner symmetrization. Further, it is shown that when a sequence of domains $\{D_m\}$ each contained in a ball $B\subset\R^d$ and satisfying the $\varepsilon$-cone property converges to a domain $D'$ with respect to the Hausdorff metric, the sequence of distributions of first exit times for Brownian motion from  $D_m$  converges to the distribution of the exit time of Brownian motion from $D'$. The second set of results in this thesis extends the theorems from \cite{BanMen} by proving generalized isoperimetric inequalities on expectations of L\'evy processes in the case of Steiner symmetrization.% using the Brascamp-Lieb-Luttinger inequalities used above. </p> <p>These results will then be used to establish inequalities involving distributions of first exit times of $\alpha$-stable symmetric processes $A_t$ from triangles and quadrilaterals. The primary application of these inequalities is verifying a conjecture from Ba\~nuelos for these planar domains. This extends a classical result of P\'olya and Szeg\"o to the fractional Laplacian with Dirichlet boundary conditions.</p>