Quantification of air quality in space and time and its effects on health

The long-term adverse effects on health associated with air pollution exposure can be estimated using either cohort or spatio-temporal ecological designs. In a cohort study, the health status of a cohort of people are assessed periodically over a number of years, and then related to estimated ambient pollution concentrations in the cities in which they live. However, such cohort studies are expensive and time consuming to implement, due to the long-term follow up required for the cohort. Therefore, spatio-temporal ecological studies are also being used to estimate the long-term health effects of air pollution as they are easy to implement due to the routine availability of the required data. Spatio-temporal ecological studies estimate the health impact of air pollution by utilising geographical and temporal contrasts in air pollution and disease risk across $n$ contiguous small-areas, such as census tracts or electoral wards, for multiple time periods. The disease data are counts of the numbers of disease cases occurring in each areal unit and time period, and thus Poisson log-linear models are typically used for the analysis. The linear predictor includes pollutant concentrations and known confounders such as socio-economic deprivation. However, as the disease data typically contain residual spatial or spatio-temporal autocorrelation after the covariate effects have been accounted for, these known covariates are augmented by a set of random effects. One key problem in these studies is estimating spatially representative pollution concentrations in each areal which are typically estimated by applying Kriging to data from a sparse monitoring network, or by computing averages over modelled concentrations (grid level) from an atmospheric dispersion model. The aim of this thesis is to investigate the health effects of long-term exposure to Nitrogen Dioxide (NO2) and Particular matter (PM10) in mainland Scotland, UK. In order to have an initial impression about the air pollution health effects in mainland Scotland, chapter 3 presents a standard epidemiological study using a benchmark method. The remaining main chapters (4, 5, 6) cover the main methodological focus in this thesis which has been threefold: (i) how to better estimate pollution by developing a multivariate spatio-temporal fusion model that relates monitored and modelled pollution data over space, time and pollutant; (ii) how to simultaneously estimate the joint effects of multiple pollutants; and (iii) how to allow for the uncertainty in the estimated pollution concentrations when estimating their health effects. Specifically, chapters 4 and 5 are developed to achieve (i), while chapter 6 focuses on (ii) and (iii). In chapter 4, I propose an integrated model for estimating the long-term health effects of NO2, that fuses modelled and measured pollution data to provide improved predictions of areal level pollution concentrations and hence health effects. The air pollution fusion model proposed is a Bayesian space-time linear regression model for relating the measured concentrations to the modelled concentrations for a single pollutant, whilst allowing for additional covariate information such as site type (e.g. roadside, rural, etc) and temperature. However, it is known that some pollutants might be correlated because they may be generated by common processes or be driven by similar factors such as meteorology. The correlation between pollutants can help to predict one pollutant by borrowing strength from the others. Therefore, in chapter 5, I propose a multi-pollutant model which is a multivariate spatio-temporal fusion model that extends the single pollutant model in chapter 4, which relates monitored and modelled pollution data over space, time and pollutant to predict pollution across mainland Scotland. Considering that we are exposed to multiple pollutants simultaneously because the air we breathe contains a complex mixture of particle and gas phase pollutants, the health effects of exposure to multiple pollutants have been investigated in chapter 6. Therefore, this is a natural extension to the single pollutant health effects in chapter 4. Given NO2 and PM10 are highly correlated (multicollinearity issue) in my data, I first propose a temporally-varying linear model to regress one pollutant (e.g. NO2) against another (e.g. PM10) and then use the residuals in the disease model as well as PM10, thus investigating the health effects of exposure to both pollutants simultaneously. Another issue considered in chapter 6 is to allow for the uncertainty in the estimated pollution concentrations when estimating their health effects. There are in total four approaches being developed to adjust the exposure uncertainty. Finally, chapter 7 summarises the work contained within this thesis and discusses the implications for future research.

Pricing derivatives with stochastic volatility

This Ph.D. thesis contains 4 essays in mathematical finance with a focus on pricing Asian option (Chapter 4), pricing futures and futures option (Chapter 5 and Chapter 6) and time dependent volatility in futures option (Chapter 7). In Chapter 4, the applicability of the Albrecher et al.(2005)'s comonotonicity approach was investigated in the context of various benchmark models for equities and com- modities. Instead of classical Levy models as in Albrecher et al.(2005), the focus is the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and the Schwartz (1997) two-factor model. It is shown that the method delivers rather tight upper bounds for the prices of Asian Options in these models and as a by-product delivers super-hedging strategies which can be easily implemented. In Chapter 5, two types of three-factor models were studied to give the value of com- modities futures contracts, which allow volatility to be stochastic. Both these two models have closed-form solutions for futures contracts price. However, it is shown that Model 2 is better than Model 1 theoretically and also performs very well empiri- cally. Moreover, Model 2 can easily be implemented in practice. In comparison to the Schwartz (1997) two-factor model, it is shown that Model 2 has its unique advantages; hence, it is also a good choice to price the value of commodity futures contracts. Fur- thermore, if these two models are used at the same time, a more accurate price for commodity futures contracts can be obtained in most situations. In Chapter 6, the applicability of the asymptotic approach developed in Fouque et al.(2000b) was investigated for pricing commodity futures options in a Schwartz (1997) multi-factor model, featuring both stochastic convenience yield and stochastic volatility. It is shown that the zero-order term in the expansion coincides with the Schwartz (1997) two-factor term, with averaged volatility, and an explicit expression for the first-order correction term is provided. With empirical data from the natural gas futures market, it is also demonstrated that a significantly better calibration can be achieved by using the correction term as compared to the standard Schwartz (1997) two-factor expression, at virtually no extra effort. In Chapter 7, a new pricing formula is derived for futures options in the Schwartz (1997) two-factor model with time dependent spot volatility. The pricing formula can also be used to find the result of the time dependent spot volatility with futures options prices in the market. Furthermore, the limitations of the method that is used to find the time dependent spot volatility will be explained, and it is also shown how to make sure of its accuracy.