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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 real markets, to some degree, every trade will incur a non-zero cost and will influence the price of the asset traded. In situations where a dynamic trading strategy is implemented these liquidity effects can play a significant role. In this thesis we examine two situations in which such trading strategies are inherent to the problem; that of pricing a derivative contingent on the asset and that of executing a large portfolio transaction in the asset. The asset's finite liquidity has been incorporated explicitly into its price dynamics using the Bakstein-Howison model [4]. Using this model we have derived the no-arbitrage price of a derivative on the asset and have found a true continuous-time equation when the bid-ask spread in the asset is neglected. Focussing on this pure liquidity case we then employ an asymptotic analysis to examine the price of a European call option near strike and expiry where the liquidity effects are shown to be most significant and closed-form expressions for the price are derived in this region. The asset price model is then extended to incorporate the empirical fact that an asset's liquidity mean reverts stochastically. In this situation the pricing equation is analyzed using the multiscale asymptotic technique developed by Fouque, Papanicolaou, and Sircar [22] and a simplified pricing and calibration framework is developed for an asset possessing liquidity risk. Finally, the derivative pricing framework (both with and without liquidity risk) is applied to a new contract termed the American forward which we present as a possible hedge against an asset's liquidity risk. In the second part of the thesis we investigate how to optimally execute a large transaction of a finitely liquid asset. Using stochastic dynamic programming and attempting only to minimize the transaction's cost, we first find that the optimal strategy is static and contains the naive strategy found in previous studies, but with an extra term to account for interest rates neglected by those studies. Including time risk into the optimization procedure we find expressions for the optimal strategy in the extreme cases when the trader's aversion to this risk is very small and very large. In the former case the optimal strategy is simply the cost-minimization strategy perturbed by a small correction proportional to the trader's level of risk aversion. In the latter case the problem is shown to be much more difficult; we analyze and derive implicit closed-form solutions to the much-simplified perfect liquidity case and show numerical results to demonstrate the agreement of the solution with our intuition.

The deregulation of regional electricity markets has led to more competitive prices but also higher uncertainty in the future electricity price development. Most markets exhibit high volatilities and occasional distinctive price spikes, which results in demand for derivative products which protect the holder against high prices. A good understanding of the stochastic price dynamics is required for the purposes of risk management and pricing derivatives. In this thesis we examine a simple spot price model which is the exponential of the sum of an Ornstein-Uhlenbeck and an independent pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of this spot price process at maturity T. With some restrictions on the set of possible martingale measures we show that the risk neutral dynamics remains within the class of considered models and hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulas for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilised which in turn uses approximations to the conditional density of the spot process. Further contributions of this thesis include a short discussion of interpolation methods to generate a continuous forward curve based on the forward contracts with delivery periods observed in the market, and an investigation into optimal martingale measures in incomplete markets. In particular we present known results of q-optimal martingale measures in the setting of a stochastic volatility model and give a first indication of how to determine the q-optimal measure for q=0 in an exponential Ornstein-Uhlenbeck model consistent with a given forward curve.

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.

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.

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.

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.

In this thesis, we introduce a non-probabilistic model for the short-term interest rate. The key concepts involved in this new approach are the non-diffusive nature of the short rate process and the uncertainty in the model parameters. The model assumes the worst possible outcome for the short rate path when pricing a fixed-income product (from the point of view of the holder) and differs in many important ways from the traditional approaches of fully deterministic or stochastic rates. In this new model, delta hedging and unique pricing play no role, nor does any market price of risk term appear. We present the model and explore the analytical and numerical solutions of the associated partial differential equation. We show how to optimally hedge the interest rate risk of a fixed-income portfolio and price and hedge common and exotic fixed-income products. Finally, we consider extensions to the model and present conclusions and areas for further research.