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.

Development of dendritic cells in the intestine

The intestinal tract is exposed to a large variety of antigens such as food proteins, commensal bacteria and pathogens and contains one of the largest arms of the immune system. The intestinal immune system has to discriminate between harmless and harmful antigens, inducing tolerance to harmless antigens and active immunity towards pathogens and other harmful materials. Dendritic cells (DC) in the mucosal lamina propria (LP) are central to this process, as they sample bacteria from the local environment and constitutively migrate to the draining mesenteric lymph nodes (MLN), where they present antigen to naïve T cells in order to direct an appropriate immune response. Despite their crucial role, understanding the function and phenotype of LP DC has been hampered by the fact that they share phenotypic markers with macrophages (mφ), which are the dominant population of mononuclear phagocyte (MP) in the LP. Recent work in our own and other laboratories has established gating strategies and phenotyping panels that allow precise discrimination between intestinal DC and mφ using the mφ specific markers CD64 and F4/80. In this way four bona fide DC subsets with distinct functions have been identified in adult LP based on their expression of CD11b and CD103 and a major aim of my project was to understand how these subsets might develop in the neonatal intestine. At the beginning of my PhD, the laboratory had used these new methods to show that signal regulatory protein α (SIRPα), an inhibitory receptor expressed by myeloid cells, was expressed by mφ and most DC in the intestine, except for those expressing CD103 alone. In addition, mice carrying a non-signalling mutation in SIRPα (SIRPα mt) had a selective reduction in CD103+CD11b+ DC, a subset which is unique to the intestinal LP. This was the basis for the initial experiments of my project, described in Chapter 3, where I investigated if the phenotype in SIRPα mt mice was intrinsic to haematopoietic cells or not. To explore this, I generated bone marrow (BM) chimeric mice by reconstituting irradiated WT mice with SIRPα mt BM, or SIRPα mt animals with WT BM. These experiments suggested that the defect in CD103+CD11b+ DC was not replicated in DC derived from BM of SIRPα origin. However as this seemed inconsistent with other data, I considered the possibility that 18 the phenotype may have been lost with age, as the BM chimeric mice were considerably older than those used in the original studies of SIRPα function. However a comparison of DC subsets in the intestine of WT and SIRPα mt mice as they aged provided no conclusive evidence to support this idea. As these experiments did show age-dependent effects on DC subsets, in Chapter 4, I went on to investigate how the DC populations appeared in the intestine and other tissues in the neonatal period. These experiments showed there were few CD103+CD11b+ DC present in the LP and migratory DC compartment of the MLN in the neonate and that as this population gradually increased in proportion with age, there was a reciprocal decrease in the relative proportion of CD103-CD11b+ DC. Interestingly, most of the changes in DC numbers in the intestine were found during the second or third week of life when the weaning process began. To validate my findings that there were few CD103+CD11b+ DC in the neonate and that this was not merely an absence of CD103 upregulation, I examined the expression of CD101 and Trem-1, markers that other work in the laboratory had suggested were specific to the CD103+CD11b+ DC lineage. My work showed that CD101 and Trem-1 were co- expressed by most CD103+CD11b+ DC in small intestine (SI) LP, as well as a small subset of CD103-CD11b+ DC in this tissue. Interestingly, Trem-1 was highly specific to the SI LP and migratory DC in the MLN, but absent from the colon and other tissues. CD101 expression was also only found on CD11b+ DC, but showed a less restricted pattern of distribution, being found in several tissues as well as the SI LP. The relative timing of their development suggested there might be a relationship between CD103+CD11b+ and CD103-CD11b+ DC and this was supported by microarray analysis. I hypothesised that the CD103-CD11b+ DC that co-expressed CD101 and Trem-1 may be the cells that developed into CD103+CD11b+ DC. To investigate this I analysed how CD101 and Trem-1 expression changed with age amongst the DC subsets in SI LP, colonic LP (CLP) and MLN. The proportion of CD101+Trem-1+ cells increased amongst CD103+CD11b+ DC in the SI LP and MLN with age, while amongst CD103+CD11b+ DC in the CLP this decreased. This was not the same in CD103-CD11b+ DC, where CD101 and Trem-1 expression was more varied with age in all tissues. CD101 and Trem-1 were not expressed to any great extent on CD103+CD11b- or CD103-CD11b- DC. The phenotypic development of the 19 intestinal DC subsets was paralleled by the gradual upregulation of CD103 expression, while the production of retinoic acid (RA), as assessed by the AldefluorTM assay, was low early in life and did not attain adult levels until after weaning. Thus DC in the neonatal intestine take some time to acquire the adult pattern of phenotypic subsets and are functionally immature compared with their adult counterparts. In Chapter 5, I used CD101 and Trem-1 to explore the ontogeny of intestinal DC subsets in CCR2-/- and SIRPα mt mice, both of which have selective defects in one particular group of DC. The selective defect seen amongst CD103+CD11b+ DC in adult SIRPα mt mice was more profound in mice at D7 and D14 of age, indicating that it may be intrinsic to this population and not highly dependent on environmental factors that change after birth. The expression of CD101 and Trem-1 by both CD103+CD11b+ and CD103-CD11b+ DC was reduced in SIRPα mt mice, again indicating that this entire lineage was affected by the lack of SIRPα signalling. However there was also a generalised defect in the numbers of all DC subsets in many tissues from early in life, suggesting there was compromised development, recruitment or survival of DC in the absence of SIRPα signalling. In contrast to the findings in SIRPα mt mice, more CD103+CD11b+ DC co-expressed CD101 and Trem-1 in CCR2-/- mice, while there were no differences in the expression of these molecules amongst CD103-CD11b+ DC. This may suggest that CCR2+ CD103-CD11b+ DC are not the cells that express CD101 and Trem-1 that are predicted to be the direct precursors of CD103+CD11b+ DC. I also examined the expression of DC growth factor receptors on DC subsets from mice of different ages, but no clear age or subset- related patterns of the expression of mRNA for Csf2ra, Irf4, Tgfbr1 and Rara could be observed. Next, I investigated whether Trem-1 played any role in DC development. Preliminary experiments in Trem-1-/- mice show no differences between any of the DC subsets, nor were there any selective effects on individual subsets when DC development from Trem-1-/- KO and WT BM was compared in competitive chimeras. However these experiments were difficult to interpret due to viability problems and because I found an unexpected defect in the ability of Trem-1-/- BM to generate all DC, irrespective of whether they expressed Trem-1 or not. 20 The final experiments I carried out were to examine the role of the microbiota in driving the differentiation of intestinal DC subsets, based on the hypothesis that this could be one of the environmental factors that might influence events in the developing intestine. To this end I performed experiments in both antibiotic treated and germ free adult mice, both of which showed no significant phenotypic differences amongst any of the DC subsets. However the study of germ free mice was compromised by recent contamination of the colony and may not be the conclusive answer. Together the data in this thesis have shown that the population of CD103+CD11b+ DC, which is unique to the intestine, is not present at birth. These cells gradually increase in frequency over time and as this occurs there is a reciprocal decrease in the frequency of CD103-CD11b+ DC. Along with other results, this leads to the idea that there may be a linear developmental pathway from CD103-CD11b+ DC to CD103+CD11b+ DC that is driven by non-microbial factors that are located preferentially in the small intestine. My project indicates that markers such as CD101 and Trem-1 may assist the dissection of this process and highlights the importance of the neonatal period for these events.

Applied evolution: An integrated approach to studying life history traits in response to drug selection

The use of chemical control measures to reduce the impact of parasite and pest species has frequently resulted in the development of resistance. Thus, resistance management has become a key concern in human and veterinary medicine, and in agricultural production. Although it is known that factors such as gene flow between susceptible and resistant populations, drug type, application methods, and costs of resistance can affect the rate of resistance evolution, less is known about the impacts of density-dependent eco-evolutionary processes that could be altered by drug-induced mortality. The overall aim of this thesis was to take an experimental evolution approach to assess how life history traits respond to drug selection, using a free-living dioecious worm (Caenorhabditis remanei) as a model. In Chapter 2, I defined the relationship between C. remanei survival and Ivermectin dose over a range of concentrations, in order to control the intensity of selection used in the selection experiment described in Chapter 4. The dose-response data were also used to appraise curve-fitting methods, using Akaike Information Criterion (AIC) model selection to compare a series of nonlinear models. The type of model fitted to the dose response data had a significant effect on the estimates of LD50 and LD99, suggesting that failure to fit an appropriate model could give misleading estimates of resistance status. In addition, simulated data were used to establish that a potential cost of resistance could be predicted by comparing survival at the upper asymptote of dose-response curves for resistant and susceptible populations, even when differences were as low as 4%. This approach to dose-response modeling ensures that the maximum amount of useful information relating to resistance is gathered in one study. In Chapter 3, I asked how simulations could be used to inform important design choices used in selection experiments. Specifically, I focused on the effects of both within- and between-line variation on estimated power, when detecting small, medium and large effect sizes. Using mixed-effect models on simulated data, I demonstrated that commonly used designs with realistic levels of variation could be underpowered for substantial effect sizes. Thus, use of simulation-based power analysis provides an effective way to avoid under or overpowering a study designs incorporating variation due to random effects. In Chapter 4, I 3 investigated how Ivermectin dosage and changes in population density affect the rate of resistance evolution. I exposed replicate lines of C. remanei to two doses of Ivermectin (high and low) to assess relative survival of lines selected in drug-treated environments compared to untreated controls over 10 generations. Additionally, I maintained lines where mortality was imposed randomly to control for differences in density between drug treatments and to distinguish between the evolutionary consequences of drug treatment versus ecological processes affected by changes in density-dependent feedback. Intriguingly, both drug-selected and random-mortality lines showed an increase in survivorship when challenged with Ivermectin; the magnitude of this increase varied with the intensity of selection and life-history stage. The results suggest that interactions between density-dependent processes and life history may mediate evolved changes in susceptibility to control measures, which could result in misleading conclusions about the evolution of heritable resistance following drug treatment. In Chapter 5, I investigated whether the apparent changes in drug susceptibility found in Chapter 4 were related to evolved changes in life-history of C. remanei populations after selection in drug-treated and random-mortality environments. Rapid passage of lines in the drug-free environment had no effect on the measured life-history traits. In the drug-free environment, adult size and fecundity of drug-selected lines increased compared to the controls but drug selection did not affect lifespan. In the treated environment, drug-selected lines showed increased lifespan and fecundity relative to controls. Adult size of randomly culled lines responded in a similar way to drug-selected lines in the drug-free environment, but no change in fecundity or lifespan was observed in either environment. The results suggest that life histories of nematodes can respond to selection as a result of the application of control measures. Failure to take these responses into account when applying control measures could result in adverse outcomes, such as larger and more fecund parasites, as well as over-estimation of the development of genetically controlled resistance. In conclusion, my thesis shows that there may be a complex relationship between drug selection, density-dependent regulatory processes and life history of populations challenged with control measures. This relationship could have implications for how resistance is monitored and managed if life histories of parasitic species show such eco-evolutionary responses to drug application.

Exploring coordination-driven self-assembly with autonomous chemical robots

The work presented herein focused on the automation of coordination-driven self assembly, exploring methods that allow syntheses to be followed more closely while forming new ligands, as part of the fundamental study of the digitization of chemical synthesis and discovery. Whilst the control and understanding of the principle of pre-organization and self-sorting under non-equilibrium conditions remains a key goal, a clear gap has been identified in the absence of approaches that can permit fast screening and real-time observation of the reaction process under different conditions. A firm emphasis was thus placed on the realization of an autonomous chemical robot, which can not only monitor and manipulate coordination chemistry in real-time, but can also allow the exploration of a large chemical parameter space defined by the ligand building blocks and the metal to coordinate. The self-assembly of imine ligands with copper and nickel cations has been studied in a multi-step approach using a self-built flow system capable of automatically controlling the liquid-handling and collecting data in real-time using a benchtop MS and NMR spectrometer. This study led to the identification of a transient Cu(I) species in situ which allows for the formation of dimeric and trimeric carbonato bridged Cu(II) assemblies. Furthermore, new Ni(II) complexes and more remarkably also a new binuclear Cu(I) complex, which usually requires long and laborious inert conditions, could be isolated. The study was then expanded to the autonomous optimization of the ligand synthesis by enabling feedback control on the chemical system via benchtop NMR. The synthesis of new polydentate ligands has emerged as a result of the study aiming to enhance the complexity of the chemical system to accelerate the discovery of new complexes. This type of ligand consists of 1-pyridinyl-4-imino-1,2,3-triazole units, which can coordinate with different metal salts. The studies to test for the CuAAC synthesis via microwave lead to the discovery of four new Cu complexes, one of them being a coordination polymer obtained from a solvent dependent crystallization technique. With the goal of easier integration into an automated system, copper tubing has been exploited as the chemical reactor for the synthesis of this ligand, as it efficiently enhances the rate of the triazole formation and consequently promotes the formation of the full ligand in high yields within two hours. Lastly, the digitization of coordination-driven self-assembly has been realized for the first time using an in-house autonomous chemical robot, herein named the ‘Finder’. The chemical parameter space to explore was defined by the selection of six variables, which consist of the ligand precursors necessary to form complex ligands (aldehydes, alkineamines and azides), of the metal salt solutions and of other reaction parameters – duration, temperature and reagent volumes. The platform was assembled using rounded bottom flasks, flow syringe pumps, copper tubing, as an active reactor, and in-line analytics – a pH meter probe, a UV-vis flow cell and a benchtop MS. The control over the system was then obtained with an algorithm capable of autonomously focusing the experiments on the most reactive region (by avoiding areas of low interest) of the chemical parameter space to explore. This study led to interesting observations, such as metal exchange phenomena, and also to the autonomous discovery of self assembled structures in solution and solid state – such as 1-pyridinyl-4-imino-1,2,3-triazole based Fe complexes and two helicates based on the same ligand coordination motif.