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.

Analysis and modelling of immune cell behaviour in lymph nodes based on multi-photon imaging

This thesis presents quantitative studies of T cell and dendritic cell (DC) behaviour in mouse lymph nodes (LNs) in the naive state and following immunisation. These processes are of importance and interest in basic immunology, and better understanding could improve both diagnostic capacity and therapeutic manipulations, potentially helping in producing more effective vaccines or developing treatments for autoimmune diseases. The problem is also interesting conceptually as it is relevant to other fields where 3D movement of objects is tracked with a discrete scanning interval. A general immunology introduction is presented in chapter 1. In chapter 2, I apply quantitative methods to multi-photon imaging data to measure how T cells and DCs are spatially arranged in LNs. This has been previously studied to describe differences between the naive and immunised state and as an indicator of the magnitude of the immune response in LNs, but previous analyses have been generally descriptive. The quantitative analysis shows that some of the previous conclusions may have been premature. In chapter 3, I use Bayesian state-space models to test some hypotheses about the mode of T cell search for DCs. A two-state mode of movement where T cells can be classified as either interacting to a DC or freely migrating is supported over a model where T cells would home in on DCs at distance through for example the action of chemokines. In chapter 4, I study whether T cell migration is linked to the geometric structure of the fibroblast reticular network (FRC). I find support for the hypothesis that the movement is constrained to the fibroblast reticular cell (FRC) network over an alternative 'random walk with persistence time' model where cells would move randomly, with a short-term persistence driven by a hypothetical T cell intrinsic 'clock'. I also present unexpected results on the FRC network geometry. Finally, a quantitative method is presented for addressing some measurement biases inherent to multi-photon imaging. In all three chapters, novel findings are made, and the methods developed have the potential for further use to address important problems in the field. In chapter 5, I present a summary and synthesis of results from chapters 3-4 and a more speculative discussion of these results and potential future directions.

The regulation of AMP-activated protein kinase by vascular endothelial growth factor

The function of the vascular endothelium is to maintain vascular homeostasis, by providing an anti-thrombotic, anti-inflammatory and vasodilatory interface between circulating blood and the vessel wall, meanwhile facilitating the selective passage of blood components such as signaling molecules and immune cells. Dysfunction of the vascular endothelium is implicated in a number of pathological states including atherosclerosis and hypertension, and is thought to precede atherogenesis by a number of years. Vascular endothelial growth factor A (VEGF) is a crucial mitogenic signaling molecule, not only essential for embryonic development, but also in the adult for regulating both physiological and pathological angiogenesis. Previous studies by our laboratory have demonstrated that VEGF-A activates AMP-activated protein kinase (AMPK), the downstream component of a signaling cascade important in the regulation of whole body and cellular energy status. Furthermore, studies in our laboratory have indicated that AMPK is essential for VEGF-A-stimulated vascular endothelial cell proliferation. AMPK activation typically stimulates anabolic processes and inhibits catabolic processes including cell proliferation, with the ultimate aim of redressing energy imbalance, and as such is an attractive therapeutic target for the treatment of obesity, metabolic syndromes, and type 2 diabetes. Metabolic diseases are associated with adverse cardiovascular outcomes and AMPK activation is reported to have beneficial effects on the vascular endothelium. The mechanism by which VEGF-A stimulates AMPK, and the functional consequences of VEGF-A-stimulated AMPK activation remain uncertain. The present study therefore aimed to identify the specific mechanism(s) by which VEGF-A regulates the activity of AMPK in endothelial cells, and how this might differ from the activation of AMPK by other agents. Furthermore, the role of AMPK in the pro-proliferative actions of VEGF-A was further examined. Human aortic and umbilical vein endothelial cells were therefore used as a model system to characterise the specific effect(s) of VEGF-A stimulation on AMPK activation. The present study reports that AMPK α1 containing AMPK complexes account for the vast majority of both basal and VEGF-A-stimulated AMPK activity. Furthermore, AMPK α1 is localized to the endoplasmic reticulum when sub-confluent, but translocated to the Golgi apparatus when cells are cultured to confluence. AMPK α2 appears to be associated with a structural cellular component, but neither α1 nor α2 complexes appear to translocate in response to VEGF-A stimulation. The present study confirms previous reports that when measured using the MTS cell proliferation assay, AMPK is required for VEGF-A-stimulated endothelial cell proliferation. However, parallel experiments measuring cell proliferation using the Real-Time Cell Analyzer xCELLigence system, do not agree with these previous reports, suggesting that AMPK may in fact be required for an aspect of mitochondrial metabolism which is enhanced by VEGF-A. Studies into the mitochondrial activity of endothelial cells have proved inconclusive at this time, but further studies into this are warranted. During previous studies in our laboratory, it was suggested that VEGF-A-stimulated AMPK activation may be mediated via the diacylglycerol (DAG)-sensitive transient receptor potential cation channel (TRPCs -3, -6 or -7) family of ion channels. The present study can neither confirm, nor exclude the expression of TRPCs in vascular endothelial cells, nor rule out their involvement in VEGF-A-stimulated AMPK activation; more specific investigative tools are required in order to characterise their involvement. Furthermore, nicotinic acid adenine dinucleotide phosphate (NAADP)-stimulated Ca2+ release from acidic intracellular organelles is not required for AMPK activation by VEGF-A. Despite what is known about the mechanisms by which AMPK is activated, far less is known concerning the downregulation of AMPK activity, as observed in human and animal models of metabolic disease. Phosphorylation of AMPK α1 Ser485 (α2 Ser491) has recently been characterised as a mechanism by which the activity of AMPK is negatively regulated. We report here for the first time that VEGF-A stimulates AMPK α1 Ser485 phosphorylation independently of the previously reported AMPK α1 Ser485 kinases Akt (protein kinase B) and ERK1/2 (extracellular signal-regulated kinase 1/2). Furthermore, inhibition of protein kinase C (PKC), the activity of which is reported to be elevated in metabolic disease, attenuates VEGF-A- and phorbol 12-myristate 13-acetate (PMA)-stimulated AMPK α1 Ser485 phosphorylation, and increases basal AMPK activity. In contrast to this, PKC activation reduces AMPK activity in human vascular endothelial cells. Attempts to identify the PKC isoform responsible for inhibiting AMPK activity suggest that it is one (or more) of the Ca2+-regulated DAG-sensitive isoforms of PKC, however cross regulation of PKC isoform expression has limited the present study. Furthermore, AMPK α1 Ser485 phosphorylation was inversely correlated with human muscle insulin sensitivity. As such, enhanced AMPK α1 Ser485 phosphorylation, potentially mediated by increased PKC activation may help explain some of the reduced AMPK activity observed in metabolic disease.

Quantifying uplift and denudation of a thermally heterogeneous crust: a detailed multi-thermochronometeric study of central west Britain

Topography is often thought as exclusively linked to mountain ranges formed by plates collision. It is now, however, known that apart from compression, uplift and denudation of rocks may be triggered by rifting, like it happens at elevated passive margins, and away from plate boundaries by both intra-plate stress causing reactivation of older structures, and by epeirogenic movements driven by mantle dynamics and initiating long-wavelength uplift. In the Cenozoic, central west Britain and other parts of the North Atlantic margins experienced multiple episodes of rock uplift and denudation that have been variable both at spatial and temporal scales. The origin of topography in central west Britain is enigmatic, and because of its location, it may be related to any of the processes mentioned above. In this study, three low temperature thermochronometers, the apatite fission track (AFT) and apatite and zircon (U-Th-Sm)/He (AHe and ZHe, respectively) methods were used to establish the rock cooling history from 200◦C to 30◦C. The samples were collected from the intrusive rocks in the high elevation, high relief regions of the Lake District (NW England), southern Scotland and northern Wales. AFT ages from the region are youngest (55–70 Ma) in the Lake District and increase northwards into southern Scotland and southwards in north Wales (>200 Ma). AHe and ZHe ages show no systematic pattern; the former range from 50 to 80 Ma and the latter tend to record the post-emplacement cooling of the intrusions (200–400 Ma). The complex, multi-thermochronometric inverse modelling suggests a ubiquitous, rapid Late Cretaceous/early Palaeogene cooling event that is particularly marked in Lake District and Criffell. The timing and rate of cooling in southern Scotland and in northern Wales is poorly resolved as the amount of cooling was less than 60◦C. The Lake District plutons were at >110◦C prior to the early Palaeogene; cooling due to a combined effect of high heat flow, from the heat producing granite batholith, and the blanketing effect of the overlying low conductivity Late Mesozoic limestones and mudstones. Modelling of the heat transfer suggests that this combination produced an elevated geothermal gradient within the sedimentary rocks (50–70◦C/km) that was about two times higher than at the present day. Inverse modelling of the AFT and AHe data taking the crustal structure into consideration suggests that denudation was the highest, 2.0–2.5 km, in the coastal areas of the Lake District and southern Scotland, gradually decreasing to less than 1 km in the northern Southern Uplands and northern Wales. Both the rift-related uplift and the intra-plate compression poorly correlate with the timing, location and spatial distribution of the early Palaeogene denudation. The pattern of early Palaeogene denudation correlates with the thickness of magmatic underplating, if the changes of mean topography, Late Cretaceous water depth and eroded rock density are taken into consideration. However, the uplift due to underplating alone cannot fully justify the total early Palaeogene denudation. The amount that is not ex- plained by underplating is, however, roughly spatially constant across the study area and can be referred to the transient thermal uplift induced by the mantle plume arrival. No other mechanisms are required to explain the observed pattern of denudation. The onset of denudation across the region is not uniform. Denudation started at 70–75 Ma in the central part of the Lake District whereas the coastal areas the rapid erosion appears to have initiated later (65–60 Ma). This is ~10 Ma earlier than the first vol- canic manifestation of the proto-Iceland plume and favours the hypothesis of the short period of plume incubation below the lithosphere before the volcanism. In most of the localities, the rocks had cooled to temperatures lower than 30◦C by the end of the Palaeogene, suggesting that the total Neogene denudation was, at a maximum, several hundreds of metres. Rapid cooling in the last 3 million years is resolved in some places in southern Scotland, where it could be explained by glacial erosion and post-glacial isostatic uplift.

Climate variability of the last 1000 years in the NW Pacific: high resolution, multi-biomarker records from Lake Toyoni

The East Asian Monsoon (EAM) is an active component of the global climate system and has a profound social and economic impact in East Asia and its surrounding countries. Its impact on regional hydrological processes may influence society through industrial water supplies, food productivity and energy use. In order to predict future rates of climate change, reliable and accurate reconstructions of regional temperature and rainfall are required from all over the world to test climate models and better predict future climate variability. Hokkaido is a region which has limited palaeo-climate data and is sensitive to climate change. Instrumental data show that the climate in Hokkaido is influenced by the East Asian Monsoon (EAM), however, instrumental data is limited to the past ~150 years. Therefore down-core climate reconstructions, prior to instrumental records, are required to provide a better understanding of the long-term behaviour of the climate drivers (e.g. the EAM, Westerlies, and teleconnections) in this region. The present study develops multi-proxy reconstructions to determine past climatic and hydrologic variability in Japan over the past 1000 years and aid in understanding the effects of the EAM and the Westerlies independently and interactively. A 250-cm long sediment core from Lake Toyoni, Hokkaido was retrieved to investigate terrestrial and aquatic input, lake temperature and hydrological changes over the past 1000-years within Lake Toyoni and its catchment using X-Ray Fluorescence (XRF) data, alkenone palaeothermometry, the molecular and hydrogen isotopic composition of higher plant waxes (δD(HPW)). Here, we conducted the first survey for alkenone biomarkers in eight lakes in the Hokkaido, Japan. We detected the occurrence of alkenones within the sediments of Lake Toyoni. We present the first lacustrine alkenone record from Japan, including genetic analysis of the alkenone producer. C37 alkenone concentrations in surface sediments are 18µg C37 g−1 of dry sediment and the dominant alkenone is C37:4. 18S rDNA analysis revealed the presence of a single alkenone producer in Lake Toyoni and thus a single calibration is used for reconstructing lake temperature based on alkenone unsaturation patterns. Temperature reconstructions over the past 1000 years suggest that lake water temperatures varies between 8 and 19°C which is in line with water temperature changes observed in the modern Lake Toyoni. The alkenone-based temperature reconstruction provides evidence for the variability of the EAM over the past 1000 years. The δD(HPW) suggest that the large fluctuations (∼40‰) represent changes in temperature and source precipitation in this region, which is ultimately controlled by the EAM system and therefore a proxy for the EAM system. In order to complement the biomarker reconstructions, the XRF data strengthen the lake temperature and hydrological reconstructions by providing information on past productivity, which is controlled by the East Asian Summer monsoon (EASM) and wind input into Lake Toyoni, which is controlled by the East Asian Winter Monsoon (EAWM) and the Westerlies. By combining the data generated from XRF, alkenone palaeothermometry and the δD(HPW) reconstructions, we provide valuable information on the EAM and the Westerlies, including; the timing of intensification and weakening, the teleconnections influencing them and the relationship between them. During the Medieval Warm Period (MWP), we find that the EASM dominated and the EAWM was suppressed, whereas, during the Little Ice Age (LIA), the influence of the EAWM dominated with time periods of increased EASM and Westerlies intensification. The El Niño Southern Oscillation (ENSO) significantly influenced the EAM; a strong EASM occurred during El Niño conditions and a strong EAWM occurred during La Niña. The North Atlantic Oscillation, on the other hand, was a key driver of the Westerlies intensification; strengthening of the Westerlies during a positive NAO phase and weakening of the Westerlies during a negative NAO phase. A key finding from this study is that our data support an anti-phase relationship between the EASM and the EAWM (e.g. the intensification of the EASM and weakening of the EAWM and vice versa) and that the EAWM and the Westerlies vary independently from each other, rather than coincide as previously suggested in other studies.