The presented dissertation is focused on the applications of statistical methods and ap-proaches applied in the energy business. The need for the modeling of energy risks arose only recently when the energy business was opened to competition. Therefore, the prima-ry aim of the dissertation is to clarify the main principles of the energy business which are necessary for understanding both risk principles and motivation of the proposed models. I am largely focused on retail risks, i.e., the risks associated with delivery to end-consumers. In particular, I deal with energy contracts providing volume flexibility, recalled as swing options in the literature. Therefore, the second issue on which I am focusing is a group of demand-driven swing options whose more systematic analysis in the portfolio context has not been published so far. Examining the risk, I apply the deductive (probabil-istic) analysis which reveals interesting relations between correlations. The practical ap-plications also require inductive considerations resulting in the construction of statistical estimators relying on historical data. I propose an estimator of the volumetric correlation based on a classical theory whose bias is investigated via MC simulation. To analyze a par-ticular volume-price correlation, I introduced the notion of robust dependency. Applying bootstrap procedures, robust dependency can be used both for testing purposes and for sensitivity analysis of the sample correlation. There are many works available devoted to energy price models which are different from the price models applied on financial markets. Therefore, the third target of the dis-sertation is an empirical statistical analysis of both power and natural gas Czech spot pric-es which can serve as a basis for the development of price models adapted to the Czech market environment. Finally, the fourth aim is the evaluation of power contracts which is very specific. The outputs of the model are both a synthetic market price and a hedging strategy. The model is designed to provide flexibility in practical applications.

Les Partenariats Publics Privés (PPPs) peuvent être un outil efficace pour optimiser et moderniser la commande publique dans un contexte où les besoins en investissement public ne cessent d’accroître. Les fréquences importantes de renégociation et les difficultés à estimer correctement les revenus futurs demeurent un défi majeur lors de la structuration financière des PPPs. Ce travail propose d’incorporer des clauses financières flexibles afin de remédier à ce problème. L’approche développée se base sur les théories d’options réelles et d’optimisation multi-objectif. Dans un premier temps, une méthodologie adéquate pour la gestion des risques est développée. La volatilité du projet est déterminée par le biais de la simulation de Monte Carlo et un déflateur stochastique est introduit afin de conduire les différentes valorisations d’options sous la probabilité historique. Ce travail développe dans un second temps, quatre formes de flexibilité qui permettent de réajuster l'équilibre financier du projet, si le revenu est insuffisant. Enfin une approche d’optimisation multi-objectif est développée afin de permettre de visualiser les différents compromis auxquels l’introduction de la flexibilité donne lieu. / Public private partnerships can be a solution to the dilemma of how to do more with less available funds that public entities are constantly financing in the last decades. If implemented properly, Public Private Partnerships can contribute to the modernization of public service provision and can constitute efficient vehicles for the delivery of optimal value for money. The high incidence of renegotiation as well as the difficulty of accurately predicting the future demand on the projects is a matter of concern when it comes to the financial structuring of Public Private Partnerships. This work proposes a real-option- based optimization framework to boost the financial viability of the projects. This is done by introducing flexible financial clauses. First, an adequate framework for risk management, where volatility is derived by Monte Carlo simulation and the valuation is made without switching to the risk neutral measure, is presented. Four families of flexible clauses are, afterwards, investigated. Such clauses are triggerred, if the revenue level of the projet is not sufficient to guarnatee its financiel viability. Finally, this work develops a multi-objective optimization approach in order to assess the different trade-offs that the introduction of flexibility leads to. The proposed optimization problem is solved via multi-objective evolutionary algorithms.

Zentraler Gegenstand dieser Dissertation ist die Entwicklung von mathematischen Methoden zur Charakterisierung und Implementierung von optimalen Investmentstrategien eines Kleininvestors auf einem Finanzmarkt. Zur Behandlung dieser Probleme ziehen wir als Hauptwerkzeug Stochastische Rückwärts-Differenzialgleichungen (BSDEs) mit nicht-linearen Drifts heran. Diese Nicht-Lineariäten ordnen sie außerhalb der Standardklasse der Lipschitz-stetigen BSDEs ein und treten häufig in finanzmathematischen Kontrollproblemen auf. Wir charakterisieren das optimale Vermögen und die optimale Investmentstrategie eines Kleininvestors mit Hilfe einer sog. Stochastischen Vorwärts-Rückwärts-Differenzialgleichung (FBSDE), einem System bestehend aus einer stochastischen Vorwärtsgleichung, die vollständig gekoppelt ist an eine Rückwärtsgleichung. Die Festlegung bestimmter Nutzenfunktionen führt uns schließlich zu einer weiteren Klasse von nicht-standard BSDEs, die in unmittelbarem Zusammenhang zu dem sog. Ansatz der stochastischen partiellen Rückwärts-Differenzialgleichungen (BSPDEs) steht. Anschließend entwickeln wir eine Methode zur numerischen Behandlung von quadratischen BSDEs, die auf einem stochastischen Analogon der Cole-Hopf-Transformation basiert. Wir studieren weiterhin eine Klasse von BSDEs, deren Drifts explizite Pfadabhängigkiten aufweisen und leiten mehrere analytische Eigenschaften her. Schließlich studieren wir Dualdarstellungen für Optimalen Mehrfachstoppprobleme. Wir leiten Martingal-Dualdarstellungen her, die die Grundlage für die Entwicklung von Regressions-basierten Monte Carlo Simulationsalgorithmen bilden, die schnell und effektiv untere und obere Schranken berechnen. / This thesis elaborates on the wealth maximization problem of a small investor who invests in a financial market. Key tools for our studies come across in the form of several classes of BSDEs with particular non-linearities, casting them outside the standard class of Lipschitz continuous BSDEs. We first give a characterization of a small investor''s optimal wealth and its associated optimal strategy by means of a systems of coupled equations, a forward-backward stochastic differential equation (FBSDE) with non-Lipschitz coefficients, where the backward component is of quadratic growth. We then examine how specifying concrete utility functions give rise to another class of non-standard BSDEs. In this context, we also investigate the relationship to a modeling approach based on random fields techniques, known by now as the backward stochastic partial differential equations (BSPDEs) approach. We continue with the presentation of a numerical method for a special type of quadratic BSDEs. This method is based on a stochastic analogue to the Cole-Hopf transformation from PDE theory. We discuss its applicability to numerically solve indifference pricing problems for contingent claims in an incomplete market. We then proceed to BSDEs whose drifts explicitly incorporate path dependence. Several analytical properties for this type of non-standard BSDEs are derived. Finally, we devote our attention to the problem of a small investor who is equipped with several exercise rights that allow her to collect pre-specified cashflows. We solve this problem by casting it into the language of multiple optimal stopping and develop a martingale dual approach for characterizing the optimal possible outcome. Moreover, we develop regression based Monte Carlo algorithms which simulate efficiently lower and upper price bounds.