This thesis examines the empirical performance of four Affine Jump Diffusion models in pricing and hedging S&P 500 Index options: the Black Scholes (BS) model, Heston???s Stochastic Volatility (SV) model, a Stochastic Volatility Price Jump (SVJ) model and a Stochastic Volatility Price-Volatility Jump (SVJJ) model. The SVJJ model structure allows for simultaneous jumps in price and volatility processes, with correlated jump size distributions. To the best of our knowledge this is the first empirical study to test the hedging performance of the SVJJ model. As part of our research we derive the SVJJ model minimum variance hedge ratio. We find the SVJ model displays the best price prediction. The SV model lacks the structural complexity to eliminate Black Scholes pricing biases, whereas our results indicate the SVJJ model suffers from overfitting. Despite significant evidence from in and out-of-sample pricing that the SV and SVJ models were better specified than the BS model, this did not result in an improvement in dynamic hedging performance. Overall the BS delta hedge and SV minimum variance hedge produced the lowest errors, although their performance across moneyness-maturity categories differed greatly. The SVJ model???s results were surprisingly poor given its superior performance in out-of-sample pricing. We attribute the inadequate performance of the jump models to the lower hedging ratios these models provided, which may be a result of the negative expected jump sizes.

This thesis presents the modeling of credit risk by using structural approach. Three fundamental questions of credit risk literature are analyzed throughout the research: modeling single firm credit risk, modeling portfolio credit risk and credit risk pricing. First we analyze these questions under the assumptions that firm value follows a geometric Brownian motion and the interest rates are constant. We discuss the weaknesses of the geometric brownian motion assumption in explaining empirical properties of real data. Then we propose a new extended model in which asset value, volatility and interest rates follow affine jump diffusion processes. In our extended model volatility is stochastic, asset value and volatility has correlated jumps and interest rates are stochastic and have jumps. Finally, we analyze the modeling of single firm credit risk and credit risk pricing by using our extended model and show how our model can be used as a solution for the problems we encounter with simple models.

In this study, the geometric Additive market models are considered. In general, these market models are incomplete, that means: the perfect replication of derivatives, in the usual sense, is not possible. In this study, it is shown that the market can be completed by new artificial assets which are called &ldquo / power-jump assets&rdquo / based on the power-jump processes of the underlying Additive process. Then, the hedging portfolio for claims whose payoff function depends on the prices of the stock and the power-jump assets at maturity is derived. In addition to the previous completion strategy, it is also shown that, using a static hedging formula, the market can also be completed by considering portfolios with a continuum of call options with different strikes and the same maturity. What is more, the portfolio optimization problem is considered in the enlarged market. The optimization problem consists of choosing an optimal portfolio in such a way that the largest expected utility of the terminal wealth is obtained. For particular choices of the equivalent martingale measure, it is shown that the optimal portfolio consists only of bonds and stocks.

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This thesis examines the empirical performance of four Affine Jump Diffusion models in pricing and hedging S&P 500 Index options: the Black Scholes (BS) model, Heston???s Stochastic Volatility (SV) model, a Stochastic Volatility Price Jump (SVJ) model and a Stochastic Volatility Price-Volatility Jump (SVJJ) model. The SVJJ model structure allows for simultaneous jumps in price and volatility processes, with correlated jump size distributions. To the best of our knowledge this is the first empirical study to test the hedging performance of the SVJJ model. As part of our research we derive the SVJJ model minimum variance hedge ratio. We find the SVJ model displays the best price prediction. The SV model lacks the structural complexity to eliminate Black Scholes pricing biases, whereas our results indicate the SVJJ model suffers from overfitting. Despite significant evidence from in and out-of-sample pricing that the SV and SVJ models were better specified than the BS model, this did not result in an improvement in dynamic hedging performance. Overall the BS delta hedge and SV minimum variance hedge produced the lowest errors, although their performance across moneyness-maturity categories differed greatly. The SVJ model???s results were surprisingly poor given its superior performance in out-of-sample pricing. We attribute the inadequate performance of the jump models to the lower hedging ratios these models provided, which may be a result of the negative expected jump sizes.

The most suitable model that idealizes random sequences of shock and impacts on vibratory systems is that of a random train of pulses (or impulses), whose arrivals are characterized in terms of stochastic point processes. Most of the existing methods of stochastic dynamics are relevant to random impulsive excitations driven by Poisson processes and there exist some methods for Erlang renewal-driven impulse processes. Herein, two classes of random impulse processes are considered. The first one is the train of impulses whose interarrival timesare driven by an Erlang renewal process. The second class is obtained by selecting some impulses from the train driven by an Erlang renewal process. The selection is performed with the aid of the jump, zero-one, stochastic process governed by the stochastic differential equation driven by the independent Erlang renewal processes. The underlying counting process, driving the arrival times of the impulses, is fully characterized. The expressions for the probability density functions of the first and second waiting times are derived and by means of these functions it is proved that the underlying counting process is a renewal (non-Erlang) process. The probability density functions of the interarrival times are evaluated for four different cases of the driving process and the results obtained for some example sets of parameters are shown graphically. The advantage of modeling the interarrival times using the class of non-Erlang renewal processes analyzed in the present dissertation, rather than the Poisson or Erlang distributions is that it is possible to deal with a broader class of the interarrival probability density functions. The non-Erlang renewal processes considered herein, obtained from two independent Erlang renewal processes, are characterized by four parameters that can be chosen to fit more closely the actual data on the distribution of the interarrival times. As the renewal counting process is not the one with independent increments, the state vector of the dynamic system under a renewal impulse process excitation is not a Markov process. The non-Markov problem may be then converted into a Markov one at the expense of augmenting the state vector by auxiliary discrete stochastic variables driven by a Poisson process. Other than the existing in literature (Iwankiewicz and Nielsen), a novel technique of conversion is devised here, where the auxiliary variables are all zero-one processes. In a considered class of non-Erlang renewal impulse processes each of the driving Erlang processes is recast in terms of the Poisson process, the augmented state vector driven by two independent Poisson processes becomes a non-diffusive Markov process. For a linear oscillator, under a considered class of non-Erlang renewal impulse process, the equations for response moments are obtained from the generalized Ito’s differential rule and the mean value and variance of the response are evaluated and shown graphically for some selected sets of parameters. For a non-linear oscillator under both Erlang renewal-driven impulses and the considered class of non-Erlang renewal impulse processes, the technique of equations for moments together with a modified closure technique is devised. The specific physical properties of an impulsive load process allow to modify the classical cumulant-neglect closure scheme and to develop a more efficient technique for the class of excitations considered. The joint probability density of the augmented state vector is expressed as sum of contributions conditioned on the ‘on’ and ‘off’ states of the auxiliary variables. A discrete part of the joint probability density function accounts for the fact that there is a finite probability of the system being in a deterministic state (for example at rest) from the initial time to the occurrence of the first impulse. The continuous part, which is the conditional probability given that the first impulse has occurred, can be expressed in terms of functions of the displacement and velocity of the system. These functions can be viewed as unknown probability densities of a bi-variate stochastic process, each of which originates a set of ‘conditional moments’. The set of relationships between unconditional and conditional moments is derived. The ordinary cumulant neglect closure is then performed on the conditional moments pertinent to the continuous part only. The closure scheme is then formulated by expressing the ‘unconditional’ moments of order greater then the order of closure, in terms of unconditional moments of lower order. The stochastic analysis of a Duffing oscillator under the the random train of impulses driven by an Erlang renewal processes and a non-Erlang renewal process R(t), is performed by applying the second order ordinary cumulant neglect closure and the modified second order closure approximation and the approximate analytical results are verified against direct Monte Carlo simulation. The modified closure scheme proves to give better results for highly non-Gaussian train of impulses, characterized by low mean arrival rate.

Dans cette thèse, on considère une équation différentielle stochastique gouvernée par un processus de Lévy de saut pur dont l’indice d’activité des sauts α ∈ (0, 2) et on observe des données haute fréquence de ce processus sur un intervalle de temps fixé. Cette thèse est consacrée tout d’abord à l’étude du comportement de la densité du processus en temps petit. Ces résultats permettent ensuite de montrer la propriété LAMN (Local Asymptotic Mixed Normality) pour les paramètres de dérive et d’échelle. Enfin, on étudie des estimateurs de l’indice α du processus.La première partie traite du comportement asymptotique de la densité en temps petit du processus. Le processus est supposé dépendre d’un paramètre β = (θ,σ) et on étudie, dans cette partie, la sensibilité de la densité par rapport à ce paramètre. Cela étend les résultats de [17] qui étaient restreints à l’indice α ∈ (1,2) et ne considéraient que la sensibilité par rapport au paramètre de dérive. En utilisant le calcul de Malliavin, on obtient la représentation de la densité, de sa dérivée et de sa dérivée logarithmique comme une espérance et une espérance conditionnelle. Ces formules de représentation font apparaître des poids de Malliavin dont les expressions sont données explicitement, ce qui permet d’analyser le comportement asymptotique de la densité en temps petit, en utilisant la propriété d’autosimilarité du processus stable.La deuxième partie de cette thèse concerne la propriété LAMN (Local Asymptotic Mixed Normality) pour les paramètres. Le coefficient de dérive et le coefficient d’échelle dépendent tous les deux de paramètres inconnus et on étend les résultats de [17]. On identifie l’information de Fisher asymptotique ainsi que les vitesses optimales de convergence. Ces quantités dépendent de l’indice αLa troisième partie propose des estimateurs pour l’indice d’activité des sauts α ∈ (0,2) basés sur des méthodes de moments qui généralisent les résultats de Masuda [53]. On montre la consistence et la normalité asymptotique des estimateurs et on illustre les résultats par des simulations numériques / In this thesis, we consider a stochastic differential equation driven by a truncated pure jump Lévy process with index α ∈(0,2) and observe high frequency data of the process on a fixed observation time. We first study the behavior of the density of the process in small time. Next, we prove the Local Asymptotic Mixed Normality (LAMN) property for the drift and scaling parameters from high frequency observations. Finally, we propose some estimators of the index parameter of the process.The first part deals with the asymptotic behavior of the density in small time of the process. The process is assumed to depend on a parameter β = (θ,σ) and we study, in this part, the sensitivity of the density with respect to this parameter. This extends the results of [17] which were restricted to the index α ∈ (1,2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density, its derivative and its logarithm derivative as an expectation and a conditional expectation. These representation formulas involve some Malliavin weights whose expressions are given explicitly and this permits to analyze the asymptotic behavior in small time of the density, using the self-similarity property of the stable process.The second part of this thesis concerns the Local Asymptotic Mixed Normality property for the parameters. Both the drift coefficient and scale coefficient depend on the unknown parameters. Extending the results of [17], we compute the asymptotic Fisher information and find that the rate in the Local Asymptotic Mixed Normality property depends on the index α.The third part proposes some estimators of the jump activity index α ∈ (0,2) based on the method of moments as in Masuda [53]. We prove the consistency and asymptotic normality of the estimators and give some simulations to illustrate the finite-sample behaviors of the estimators

Cette thèse porte sur l'étude probabiliste des réponses démographique et génétique de populations à certains événements ponctuels. Dans une première partie, nous étudions l'impact de catastrophes tuant une fraction de la population et survenant de manière répétée, sur le comportement en temps long d'une population modélisée par un processus de branchement. Dans un premier temps nous construisons une nouvelle classe de processus, les processus de branchement à états continus avec catastrophes, en les réalisant comme l'unique solution forte d'une équation différentielle stochastique. Nous déterminons ensuite les conditions d'extinction de la population. Enfin, dans les cas d'absorption presque sûre nous calculons la vitesse d'absorption asymptotique du processus. Ce dernier résultat a une application directe à la détermination du nombre de cellules infectées dans un modèle d'infection de cellules par des parasites. En effet, la quantité de parasites dans une lignée cellulaire suit dans ce modèle un processus de branchement, et les "catastrophes" surviennent lorsque la quantité de parasites est partagée entre les deux cellules filles lors des divisions cellulaires. Dans une seconde partie, nous nous intéressons à la signature génétique laissée par un balayage sélectif. Le matériel génétique d'un individu détermine (pour une grande partie) son phénotype et en particulier certains traits quantitatifs comme les taux de naissance et de mort intrinsèque, ou sa capacité d'interaction avec les autres individus. Mais son génotype seul ne détermine pas son ``adaptation'' dans le milieu dans lequel il vit : l'espérance de vie d'un humain par exemple est très dépendante de l'environnement dans lequel il vit (accès à l'eau potable, à des infrastructures médicales,...). L'approche éco-évolutive cherche à prendre en compte l'environnement en modélisant les interactions entre les individus. Lorsqu'une mutation ou une modification de l'environnement survient, des allèles peuvent envahir la population au détriment des autres allèles : c'est le phénomène de balayage sélectif. Ces événements évolutifs laissent des traces dans la diversité neutre au voisinage du locus auquel l'allèle s'est fixé. En effet ce dernier ``emmène'' avec lui des allèles qui se trouvent sur les loci physiquement liés au locus sous sélection. La seule possibilité pour un locus de ne pas être ``emmené'' est l'occurence d'une recombination génétique, qui l'associe à un autre haplotype dans la population. Nous quantifions la signature laissée par un tel balayage sélectif sur la diversité neutre. Nous nous concentrons dans un premier temps sur la variation des proportions neutres dans les loci voisins du locus sous sélection sous différents scénarios de balayages. Nous montrons que ces différents scenari évolutifs laissent des traces bien distinctes sur la diversité neutre, qui peuvent permettre de les discriminer. Dans un deuxième temps, nous nous intéressons aux généalogies jointes de deux loci neutres au voisinage du locus sous sélection. Cela nous permet en particulier de quantifier des statistiques attendues sous certains scenari de sélection, qui sont utilisées à l'heure actuelle pour détecter des événements de sélection dans l'histoire évolutive de populations à partir de données génétiques actuelles. Dans ces travaux, la population évolue suivant un processus de naissance et mort multitype avec compétition. Si un tel modèle est plus réaliste que les processus de branchement, la non-linéarité introduite par les compétitions entre individus en rend l'étude plus complexe / This thesis is devoted to the probabilistic study of demographic and genetical responses of a population to some point wise events. In a first part, we are interested in the effect of random catastrophes, which kill a fraction of the population and occur repeatedly, in populations modeled by branching processes. First we construct a new class of processes, the continuous state branching processes with catastrophes, as the unique strong solution of a stochastic differential equation. Then we describe the conditions for the population extinction. Finally, in the case of almost sure absorption, we state the asymptotical rate of absorption. This last result has a direct application to the determination of the number of infected cells in a model of cell infection by parasites. Indeed, the parasite population size in a lineage follows in this model a branching process, and catastrophes correspond to the sharing of the parasites between the two daughter cells when a division occurs. In a second part, we focus on the genetic signature of selective sweeps. The genetic material of an individual (mostly) determines its phenotype and in particular some quantitative traits, as birth and intrinsic death rates, and interactions with others individuals. But genotype is not sufficient to determine "adaptation" in a given environment: for example the life expectancy of a human being is very dependent on his environment (access to drinking water, to medical infrastructures,...). The eco-evolutive approach aims at taking into account the environment by modeling interactions between individuals. When a mutation or an environmental modification occurs, some alleles can invade the population to the detriment of other alleles: this phenomenon is called a selective sweep and leaves signatures in the neutral diversity in the vicinity of the locus where the allele fixates. Indeed, this latter "hitchhiking” alleles situated on loci linked to the selected locus. The only possibility for an allele to escape this "hitchhiking" is the occurrence of a genetical recombination, which associates it to another haplotype in the population. We quantify the signature left by such a selective sweep on the neutral diversity. We first focus on neutral proportion variation in loci partially linked with the selected locus, under different scenari of selective sweeps. We prove that these different scenari leave distinct signatures on neutral diversity, which can allow to discriminate them. Then we focus on the linked genealogies of two neutral alleles situated in the vicinity of the selected locus. In particular, we quantify some statistics under different scenari of selective sweeps, which are currently used to detect recent selective events in current population genetic data. In these works the population evolves as a multitype birth and death process with competition. If such a model is more realistic than branching processes, the non-linearity caused by competitions makes its study more complex

Nessa tese apresentamos uma nova classe de modelos, os processos de saltos com memória de alcance variável, uma generalização a tempo contínuo do processo introduzido em Galves e Löcherbach (2013). Desenvolvemos um novo estimador para a árvore de contexto imersa no processo de salto com memória de alcance variável, considerando mais parâmetros fornecidos pela amostra. Obtivemos também uma cota superior da taxa de convergência da árvore estimada para árvore real, provando a convergência quase certa do estimador. / In this work we deal with a new class of models: the jump processes with variable length memory. This is a continuous-time generalization of the process introduced in Galves and Löcherbach (2013). We present a new estimator for the tree context embedded in this process, considering all information provided by the sample. We also present an exponential upper bound for the rate of convergence, proving then the almost sure convergence of the estimator.

Nessa tese apresentamos uma nova classe de modelos, os processos de saltos com memória de alcance variável, uma generalização a tempo contínuo do processo introduzido em Galves e Löcherbach (2013). Desenvolvemos um novo estimador para a árvore de contexto imersa no processo de salto com memória de alcance variável, considerando mais parâmetros fornecidos pela amostra. Obtivemos também uma cota superior da taxa de convergência da árvore estimada para árvore real, provando a convergência quase certa do estimador. / In this work we deal with a new class of models: the jump processes with variable length memory. This is a continuous-time generalization of the process introduced in Galves and Löcherbach (2013). We present a new estimator for the tree context embedded in this process, considering all information provided by the sample. We also present an exponential upper bound for the rate of convergence, proving then the almost sure convergence of the estimator.