In this project, we study an asymptotic expansion method for solving stochastic volatility European option pricing problems. We explain the backgrounds and details associated with the method. Specifically, we present in full detail the arguments behind the derivation of the pricing PDEs and detailed calculation in deriving asymptotic option pricing formulas using our own model specifications. Finally, we discuss potential difficulties and problems in the implementation of the methods.

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Ce travail s'inscrit dans le cadre d'un partenariat scientifique entre le Laboratoire Navier et la société STRAINS. Convaincue du besoin de renouveler les méthodes actuelles de calcul de structures, STRAINS développe un nouveau logiciel dédié à l'analyse des ouvrages d'art. Dans ce contexte, cette thèse propose de nouveaux outils pour l'analyse des structures élancées. Le modèle élastique de poutre d'ordre supérieur développé par cite{Ferradi2016} est d'abord adapté au cas des déformations imposées, permettant ainsi au modèle de représenter un grand nombre de phénomènes physiques tels que le fluage, la précontrainte ou les chargements thermiques. Différents exemples viennent souligner la précision numérique du modèle ainsi que ses performances en temps de calcul. Le modèle est également étendu au cadre de la théorie de la plasticité. Considérant les déformations plastiques comme des déformations imposées, les résultats précédemment obtenus sont utilisés pour développer une nouvelle cinématique d'ordre supérieur. Enfin, un nouvel élément de poutre élastoplastique pour le béton armé est proposé. Le béton est décrit grâce au modèle élastoplastique et les ferraillages sont modélisés par des éléments barres à une dimension. Cette méthode permet une description précise du comportement du béton et une représentation fidèle des renforcements. La validité des calculs est évaluée par des considérations de dissipation énergétique / This thesis work is presented in the framework of a scientific partnership between Laboratoire Navier and the french start-up STRAINS. Believing in the need for new methodologies in structural analysis, STRAINS is developing a new software dedicated to the structural analysis of bridges. In this context, this work suggests new tools for the analysis of slender structures.The higher-order elastic beam element developed by cite{Ferradi2016} is first extended to the case of eigenstrains, enabling the model to deal with various physical phenomena such as creep, prestress or thermal loads. An enriched kinematics is used to capture the local response of the structure. Different examples highlight the local accuracy of the model and its fast computational performances. The model is also extended to plasticity in small perturbations. Considering the plastic strains developing in the structure as eigenstrains, the previous works are used to derive a higher-order elastoplastic kinematics.Finally, a new elastoplastic beam element for reinforced concrete is suggested. The concrete material is described by using the elastoplastic beam model developed previously while steel rebars are modeled by one dimensional bar elements. This method enables a fine local description of the concrete behavior and an accurate representation of the reinforcement. The validity of computations is assessed thanks to energy considerations

The double-mean-reverting model by Gatheral is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci, we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

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In this Master thesis, we use a singular and regular perturbation theory to derive an analytic approximation formula for the expected discounted penalty function. Our model is an extension of Cramer–Lundberg extended classical model because we consider a more general insurance risk model in which the compound Poisson risk process is perturbed by a Brownian motion multiplied by a stochastic volatility driven by two factors- which have mean reversion models. Moreover, unlike the classical model, our model allows a ruin to be caused either by claims or by surplus’ fluctuation. We compute explicitly the first terms of the asymptotic expansion and we show that they satisfy either an integro-differential equation or a Poisson equation. In addition, we derive the existence and uniqueness conditions of the risk model with two stochastic volatilities factors.

Modern financial engineering is a part of applied mathematics that studies market models. Each model is characterized by several parameters. Some of them are familiar to a wide audience, for example, the price of a risky security, or the risk free interest rate. Other parameters are less known, for example, the volatility of the security. This parameter determines the rate of change of security prices and is determined by several factors. For example, during the periods of stable economic growth the prices are changing slowly, and the volatility is small. During the crisis periods, the volatility significantly increases. Classical market models, in particular, the celebrated Nobel Prize awarded Black–Scholes–Merton model (1973), suppose that the volatility remains constant during the lifetime of a financial instrument. Nowadays, in most cases, this assumption cannot adequately describe reality. We consider a model where both the security price and the volatility are described by random functions of time, or stochastic processes. Moreover, the volatility process is modelled as a sum of two independent stochastic processes. Both of them are mean reverting in the sense that they randomly oscillate around their average values and never escape neither to very small nor to very big values. One is changing slowly and describes low frequency, for example, seasonal effects, another is changing fast and describes various high frequency effects. We formulate the model in the form of a system of a special kind of equations called stochastic differential equations. Our system includes three stochastic processes, four independent factors, and depends on two small parameters. We calculate the price of a particular financial instrument called European call option. This financial contract gives its holder the right (but not the obligation) to buy a predefined number of units of the risky security on a predefined date and pay a predefined price. To solve this problem, we use the classical result of Feynman (1948) and Kac (1949). The price of the instrument is the solution to another kind of problem called boundary value problem for a partial differential equation. The resulting equation cannot be solved analytically. Instead we represent the solution in the form of an expansion in the integer and half-integer powers of the two small parameters mentioned above. We calculate the coefficients of the expansion up to the second order, find their financial sense, perform numerical studies, and validate our results by comparing them to known verified models from the literature. The results of our investigation can be used by both financial institutions and individual investors for optimization of their incomes.

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Simulations using the Finite Element Method (FEM) play an increasingly important role in the design process of joints and fasteners in the aerospace industry. In order to utilize the potential of such adhesive bonding, there is an increasing need for effective and accurate computational methods. The geometry and the nature of an adhesive joint are, however, not so simple to describe effectively using standard FEM-codes. To overcome this difficulty, special FEM-elements can be developed that provide a material surface treatment of the adhesive and the joined parts. In order to create a model that reflects the above features, one may introduce proper scalings on the geometry and on the material properties in terms of a perturbation parameter. Within the framework of three-dimensional elasticity, together with an asymptotic expansion method, a material surface model is obtained through a systematic procedure. In such a derivation, no a priori assumptions for the displacements or stress fields are needed. The final result is a variational equation posed over a single reference surface which forms the basis of a structural element for the compound joint. Through the usage of continuum damage mechanics and the framework of a generalized standard material, the linear elastic model is extended to include an elastic-plastic material model with damage for the adhesive. The model is FE-discretized and an important implication is that the (quasi-static) propagation of the local failure zone in the adhesive layer can be simulated. The failure load is obtained as a computational result and consequently no postulated failure criterion is needed. The derived FE-method opens up the possibility to efficiently model and analyze the mechanical behavior of large bonded structures. / At the time the thesis was defended paper I. was in fact two manuscripts, which later were combined to give the published article.