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This thesis examines two distinct classes of problem in which nonlinearities arise in option pricing theory. In the first class, we consider the effects of the inclusion of finite liquidity into the Black-Scholes-Merton option pricing model, which for the most part result in highly nonlinear partial differential equations (PDEs). In particular, we investigate a model studied by Schonbucher and Wilmott (2000) and furthermore, show how many of the proposed existing models in the literature can be placed into a unified analytical framework. Detailed analysis reveals that the form of the nonlinearities introduced can lead to serious solution difficulties for standard (put and call) payoff conditions. One is associated with the infinite gamma and in such regimes it is necessary to admit solutions with discontinuous deltas, and perhaps even more disturbingly, negative option values. A second failure (applicable to smoothed payoff functions) is caused by a singularity in the coefficient of the diffusion term in the option-pricing equation. It is concluded in this case is that the model irretrievably breaks down and there is insufficient 'financial modelling' in the pricing 'equation. The repercussions for American options are also considered. In the second class of problem, we investigate the properties of the recently introduced British option (Peskir and Samee, 2008a,b), a new non-standard class of early exercise option, which can help to mediate the effects of a finitely liquid market, since the contract does not require the holder to enter the market and hence incur liquidation costs. Here we choose to focus on the interesting nonlinear behaviour of the early-exercise boundary, specifically for times close to, and far from, expiry. In both classes, detailed asymptotic analysis, coupled with advanced numerical techniques (informed by the asymptotics) are exploited to extract the relevent dynamics.

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This thesis makes explicit, develops and critically discusses a concept of causality that is assumed in structural models in econometrics. The thesis begins with a development of Herbert Simon's (1953) treatment of causal order for linear deterministic, simultaneous systems of equations to provide a fully explicit mechanistic interpretation for these systems. Doing this allows important properties of the assumed causal reading to be discussed including: the invariance of mechanisms to intervention and the role of independence in interventions. This work is then extended to basic structural models actually used in econometrics, linear models with errors-in-the-equations. This part of the thesis provides a discussion of how error terms are to be interpreted and sets out a way to introduce probabilistic concepts into the mechanistic interpretation set out earlier. The resulting analysis is then critically compared with similar work by economists, Stephen LeRoy (1995) and Kevin Hoover (2001a) who both develop Simon's work on causal order in different ways. In the latter part of the thesis, the mechanistic interpretation set out at the beginning is used to interpret identification conditions. Typically, these are presented in econometrics as mathematical conditions for determining whether unknown parameters in equations can be measured from observation. In the thesis it is shown that the identification conditions imposed on sets of equations when interpreted mechanistically require a sparseness of causal structure that ensures that experiments are hypothetically possible of the causal structure. It also analyses the role of identifiability conditions in causal inference. The final part of the thesis shows that the mechanistic interpretation developed in the thesis succeeds, unlike Simon's own methods for analysing spurious correlation, in avoiding important criticisms by Nancy Cartwright (1989) whose own approach to inferring causal structure from observations is also critically analysed.

Health systems in most high-income countries provide protection against the financial risks associated with ill health on a broadly universal basis, although many impose measures that lower financial protection, for example, by offering a limited package of benefits or requiring beneficiaries to pay part of the cost of health care at the point of use. This last measure, known as cost sharing or user charges, is a policy tool applied mainly to raise revenue for the health system and to enhance efficiency. The neo-classical economic argument for user fees posits that moral hazard exists in health insurance markets, and user fees help combat this "overconsumption" of care. The simultaneous policy argument is that cost sharing reduces unnecessary consumption of prescription drugs, which leads to both expenditure reductions and health improvements. While literature investigating these cost sharing arguments exists, there are still unanswered questions as many of the papers are outdated, others only focus on specific populations, and there are methodological issues surrounding some studies. As estimates vary widely between studies, a methodological approach that obtains an "adjusted" or "composite" price elasticity from the literature by pooling the existing estimates would provide a broad measure of elasticity. Updated estimates of the price elasticity of demand for the general population, the elderly, and low-income individuals in the United States would also be useful for American policymakers given recent changes to public and private insurance coverage. The calculation of estimates for elderly Americans also provides a useful backdrop for comparing estimates from the elderly in British Columbia, Canada. There is a need for economic and policy discussions related to economic efficiency, policy arguments for efficiency in user fees, and equity in the literature. We contribute to filling these gaps by estimating the price elasticity of demand in three main settings: a collection of elasticity estimates from the existing literature, the American population, and the older population in British Columbia, Canada. Based on our results, we determine that the price elasticity of demand is relatively low in all of these settings, even among the low-income group and the general population. Our adjusted price elasticity estimate is an insignificant -0.16, while the elasticity estimates from the American analyses range from an insignificant -0.11 for the elderly to -0.25 for the general population. We obtain an estimate of -0.30 for the elderly in British Columbia. Overall, the sensitivity to user fees depends on the institutional setting, the level of cost sharing, the specific subpopulation examined, and other factors. The implications of these relatively low estimates can be viewed from both an economics and policy perspective. While cost sharing leads to greater efficiency when we define efficiency in a neo-classical economic sense, from a policy perspective user fees may negatively affect beneficiaries' health and equity. The implication is that policymakers should set transparent policy goals and openly discuss whether any detrimental effects of cost sharing are considered acceptable from a policy standpoint.

This thesis consists of three chapters that investigate the importance of frictions in insurance and labour markets and their effects on macroeconomic outcomes. It asks how the behavior of aggregate employment and unemployment are affected, or the behavior of a planner who sets benefits to maximize welfare, when agents possess a number of risk sharing opportunities and luck in the labour market is the principal component of idiosyncratic risks. Chapter one deals with the technical aspects of this question. I introduce wealth accumulation in a battery of familiar search models and explore the implications for wages, allocations and the amount of risk sharing that firms can provide to their workforce. The second chapter investigates how the government should optimally set unemployment benefits depending on the range of private insurance opportunities in the economy. I consider a class of models that feature heterogeneous agents and wealth accumulation and contrast their properties with another where firms can provide additional insurance to their workforce. I show that the role of public policy is substantially different between the two economies. The third chapter is joint work with Jochen Mankart. We consider another margin of insurance, namely family self insurance, whereby household members can adjust jointly their labour supply to insure against income losses. We investigate how this feature can affect the cyclical behavior of key labour market statistics. In the US data we find that insurance within the family is important in explaining why the labour force is acyclical and not volatile but when we turn to the model we get the converse prediction. We then evaluate what important additions need to be made to our framework to make the model consistent with the data.

This thesis examines the impact of different risk sharing arrangements under incomplete financial markets on macroeconomic outcomes. The first two chapters are joint work with Giacomo Rodano. In the first chapter, we examine the effects of Chapter 7 of the US bankruptcy law on entrepreneurs. The latter are subject to production risk. They can borrow and in case they fail they can default on their debt. We examine the optimal wealth exemption level and the optimal credit market exclusion duration in this environment. In addition to unsecured credit, entrepreneurs can also obtain secured credit in the second chapter. Secured credit lowers the cost of a generous bankruptcy regime because agents who are rationed out of the unsecured credit market can still obtain secured credit. Therefore, the optimal exemption level is relatively high. In the third chapter, I investigate the effects of wealth exemptions on interest rates if entrepreneurs can choose the riskiness of their project. The default possibility leads to a kink in the value function which makes agents locally risk-loving. In the fourth chapter, I focus on consumers only. In particular, I show that wealth exemptions are of particular importance in a model with expense shocks. Wealth exemptions encourage people to save more so that aggregate savings rise. The model is also consistent with the fact that consumer bankruptcy cases are not correlated with wealth exemption levels. The fifth chapter is joint work with Rigas Oikonomou. We compare two environments: on the one hand the standard one in which a household consists of one member and, on the other hand, one in which a household consists of two members who share their risks perfectly. We investigate the differences between the two models in labor market flows and volatilities of labor market statistics in response to productivity shocks.

We discuss the estimation of the order of integration of a fractional process that may be contaminated by a time-varying deterministic component, or subject to a break in the dynamics of the zero-mean stochastic component, and the estimation of the cointegrating parameter in a bivariate system generated by fractionally integrated processes and by additive polynomial trends. In Chapter 1 we review the theoretical literature on fractional integration and cointegration, and we analyse a situation in which a fractional model reconciles two apparently conflicting economic theories. In Chapter 2 we consider local Whittle estimation of the order of integration when the process is contaminated by a deterministic trend or by a break in the mean. We propose a simple condition to assess whether the asymptotic properties of the estimate are unaffected by the time-varying mean, and a test, with asymptotically normal test statistic under the null, to detect if that condition is met. In Chapter 3 we discuss local Whittle estimation when the zero-mean stochastic component is subject to a break: we show that the estimate is robust to instability in the short term dynamics, while in presence of a break in the long term dynamics only the highest order of integration is consistently estimated. We propose a test to detect that break: the limit distribution of the test statistic under the null is not standard, but it is well known in the literature. We also propose a procedure to estimate the location of a break when it is present. In Chapter 4 we consider a cointegrating relation in which a nonstationary, bivariate process is augmented by a deterministic trend. We derive the limit properties of the Ordinary Least Squares and Generalised Least Squares estimates: these depend on the comparison between the deterministic and the stochastic components.