Many systems studied in the biological, physical, and social sciences are composed of multiple interacting components. Often the number of components and interactions is so large that attaining an understanding of the system necessitates some form of simplication. A common representation that captures the key connection patterns is a network in which the nodes correspond to system components and the edges represent interactions. In this thesis we use network techniques and more traditional clustering methods to coarse-grain systems composed of many interacting components and to identify the most important interactions. This thesis focuses on two main themes: the analysis of financial systems and the study of network communities, an important mesoscopic feature of many networks. In the first part of the thesis, we discuss some of the issues associated with the analysis of financial data and investigate the potential for risk-free profit in the foreign exchange market. We then use principal component analysis (PCA) to identify common features in the correlation structure of different financial markets. In the second part of the thesis, we focus on network communities. We investigate the evolving structure of foreign exchange (FX) market correlations by representing the correlations as time-dependent networks and investigating the evolution of network communities. We employ a node-centric approach that allows us to track the effects of the community evolution on the functional roles of individual nodes and uncovers major trading changes that occurred in the market. Finally, we consider the community structure of networks from a wide variety of different disciplines. We introduce a framework for comparing network communities and use this technique to identify networks with similar mesoscopic structures. Based on this similarity, we create taxonomies of a large set of networks from different fields and individual families of networks from the same field.

In this thesis, we study the problems of risk measurement, valuation and hedging of financial positions in incomplete markets when an insufficient number of assets are available for investment (real options). We work closely with three measures of risk: Worst-Case Scenario (WCS) (the supremum of expected values over a set of given probability measures), Value-at-Risk (VaR) and Average Value-at-Risk (AVaR), and analyse the problem of hedging derivative securities depending on a non-traded asset, defined in terms of the risk measures via their acceptance sets. The hedging problem associated to VaR is the problem of minimising the expected shortfall. For WCS, the hedging problem turns out to be a robust version of minimising the expected shortfall; and as AVaR can be seen as a particular case of WCS, its hedging problem is also related to the minimisation of expected shortfall. Under some sufficient conditions, we solve explicitly the minimal expected shortfall problem in a discrete-time setting of two assets driven by correlated binomial models. In the continuous-time case, we analyse the problem of measuring risk by WCS, VaR and AVaR on positions modelled as Markov diffusion processes and develop some results on transformations of Markov processes to apply to the risk measurement of derivative securities. In all cases, we characterise the risk of a position as the solution of a partial differential equation of second order with boundary conditions. In relation to the valuation and hedging of derivative securities, and in the search for explicit solutions, we analyse a variant of the robust version of the expected shortfall hedging problem. Instead of taking the loss function $l(x) = [x]^+$ we work with the strictly increasing, strictly convex function $L_{\epsilon}(x) = \epsilon \log \left( \frac{1+exp\{−x/\epsilon\} }{ exp\{−x/\epsilon\} } \right)$. Clearly $lim_{\epsilon \rightarrow 0} L_{\epsilon}(x) = l(x)$. The reformulation to the problem for L_{\epsilon}(x) also allow us to use directly the dual theory under robust preferences recently developed in [82]. Due to the fact that the function $L_{\epsilon}(x)$ is not separable in its variables, we are not able to solve explicitly, but instead, we use a power series approximation in the dual variables. It turns out that the approximated solution corresponds to the robust version of a utility maximisation problem with exponential preferences $(U(x) = −\frac{1}{\gamma}e^{-\gamma x})$ for a preferenes parameter $\gamma = 1/\epsilon$. For the approximated problem, we analyse the cases with and without random endowment, and obtain an expression for the utility indifference bid price of a derivative security which depends only on the non-traded asset.

Cette thèse traite de plusieurs sujets en mathématiques financières: risque de crédit, optimisation de portefeuille et modélisation des taux d’intérêts. Le chapitre 1 consiste en trois études dans le domaine du risque de crédit. La plus innovante est la première dans laquel nous construisons un modèle tel que la propriété d’immersion n’est vérifiée sous aucune mesure martingale équivalente. Le chapitre 2 étudie le problème de maximisation de la somme d’une utilité de la richesse terminale et d’une utilité de la consommation. Le chapitre 3 étudie l’évaluation des produits dérivés de taux d’intérêt dans un cadre multicourbe, qui prend en compte la différence entre une courbe de taux sans risque et des courbes de taux Libor de différents tenors. / This thesis deals with several topics in mathematical finance: credit risk, portfolio optimization and interest rate modeling. Chapter 1 consists of three studies in the field of credit risk. The most innovative is the first one, where we construct a model such that the immersion property does not hold under any equivalent martingale measure. Chapter 2 studies the problem of maximization of the sum of the utility of the terminal wealth and the utility of the consumption, in a case where a sudden jump in the risk-free interest rate induces market incompleteness. Chapter 3 studies the valuation of Libor interest rate derivatives in a multiple-curve setup, which accounts for the spreads between a risk-free discount curve and Libor curves of different tenors.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

The object of this thesis is the study of some new financial models. The common feature is that they all involve optimal decisions. Some of the decisions take the form of a control and we enter the theory of stochastic optimal control and of Hamilton-Jacobi-Bellman (HJB) equations. Other decisions are "binary" and we deal with the theory of optimal stopping and free boundary problems. Throughout the thesis we will prefer a heuristic and intuitive approach to a too technical one which could hide the underlying ideas. In the first part we introduce the reader to option pricing, HJB equations and free boundary problems, and we review briefly the use of these mathematical tools in finance. The second part of the thesis deals with passport options. The pricing of these exotic options involves stochastic optimal control and free boundary problems. Finally, in the last part we study the end-of-the-year bonus for traders: how to optimally reward a trader?

The major objective of this thesis is to study optimization problems in finance. Most of the effort is directed towards studying the impact of transaction costs in those problems. In addition, we study dynamic meanvariance asset allocation problems. Stochastic HJB equations, Pontryagin Maximum Principle and perturbation analysis are the major mathematical techniques used. In Chapter 1, we introduce the background literature. Following that, we use the Pontryagin Maximum Principle to tackle the problem of dynamic mean-variance asset allocation and rediscover the doubling strategy. In Chapter 2, we present one of the major results of this thesis. In this chapter, we study a financial optimization problem based on a market model without transaction costs first. Then we study the equivalent problem based on a market model with transaction costs. We find that there is a relationship between these two solutions. Using this relationship, we can obtain the solution of one when we have the solution of another. In Chapter 3, we generalize the results of chapter 2. In Chapter 4, we use Pontryagin Maximum Principle to study the problem limit of the no-transaction region when transaction costs tend to 0. We find that the limit is the no-transaction cost solution.

Multi-asset credit derivatives trade in huge volumes, yet no models exist that are capable of properly accounting for the spread behaviour of dependent companies. In this thesis we consider new ways of incorporating a richer and more realistic dependence structure into multi-firm models. We focus on the structural framework in which firm value is modelled as a geometric Brownian motion, with default as the first hitting time of an exponential default threshold. Specification of a dependence structure consisting of a common driving influence and firm-specific inter-company ties allows for both default causality and default asymmetry and we incorporate default contagion in the first passage framework for the first time. Building on the work by Zhou (2001a), we propose an analytical model for corporate bond yields in the presence of default contagion and two-firm credit default swap baskets. We derive closed-form solutions for credit spreads, and results clearly highlight the importance of dependence assumptions. Extending this framework numerically, we calculate CDS spreads for baskets of three firms with a wide variety of credit dependence specifications. We examine the impact of firm value correlation and credit contagion for symmetric and asymmetric baskets, and incorporate contagion that has a declining impact over time.

Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multi–asset (high dimensional) American options is still more difficult. We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the open–source derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library. We propose a new type of multi–asset option, the “Radial Barrier Option” for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multi–asset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes. We investigate the asymptotic solution of the single asset Black–Scholes–Merton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options.

This thesis is divided into three parts; an intensity based network model of firm default, an agent based network model of firm default, and an agent based model of feedback effects from dynamic hedging. The common theme among all three parts is the application of ideas from both physics and mathematics to the solution of problems motivated by the financial markets. Less broadly, in the first two parts, the common themes are credit markets, networks, and dependent defaults. Part one tackles the problem of default dependence from a probabilistic perspective, modeling the default of companies as generalised Poisson processes, with the default dependence structure given by a network. We present a mathematical framework to solve a generalised version of the Jarrow Yu model of looping defaults [27] and study the relationship between network structure and the resilience of a network of firms to default events. Using this model we then show how to price simple multi-name credit products such as kth to default baskets. Part two again considers dependent defaults, but here the network is dynamic and firms are modelled as simple agents, defined by strategies, whose interactions determine a network of trading links. Using our agent based network model of firm default we study network structure and their degree distributions, firm lifetimes, and look for evidence of agent learning and default clustering. We then study the effect of default on a network of firms and the response of remaining firms to that default event. Part three considers a relatively more established agent based framework, called the Minority Game. We first describe in detail the Minority Game and discuss its suitability as a market model. We then show how it may be applied to modelling the actions of traders delta hedging a short option position. We show that for a variety of option positions, in a sufficiently illiquid market feedback effects arise from the actions of the traders as their trades impact upon the underlying market.