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.

Investigating Angiotensin II in cardiac remodelling and the counter-regulatory actions of Angiotensin-(1-9)

Cardiovascular diseases (CVDs) including, hypertension, coronary heart disease and heart failure are the leading cause of death worldwide. Hypertension, a chronic increase in blood pressure above 140/90 mmHg, is the single main contributor to deaths due to heart disease and stroke. In the heart, hypertension results in adaptive cardiac remodelling, including LV hypertrophy to normalize wall stress and maintain cardiac contractile function. However, chronic increases in BP results in the development of hypertensive heart disease (HHD). HHD describes the maladaptive changes during cardiac remodelling which result in reduced systolic and diastolic function and eventually heart failure. This includes ventricular dilation due to eccentric hypertrophy, cardiac fibrosis which stiffens the ventricular wall and microvascular rarefaction resulting in a decrease in coronary blood flow albeit an increase in energy demand. Chronic activation of the renin-angiotensin-system (RAS) with its effector peptide angiotensin (Ang)II plays a key role in the development of hypertension and the maladaptive changes in HHD. Ang II acts via the angiotensin type 1 receptor (AT1R) to mediate most of its pathological actions during HHD, including stimulation of cardiomyocyte hypertrophy, activation of cardiac fibroblasts and increased collagen deposition. The counter-regulatory axis of the RAS which is centred on the ACE2/Ang-(1-7)/Mas axis has been demonstrated to counteract the pathological actions of Ang II in the heart and vasculature. Ang-(1-7) via the Mas receptor prevents Ang II-induced cardiac hypertrophy and fibrosis and improves cardiac contractile function in animal models of HHD. In contrast, less is known about Ang-(1-9) although evidence has demonstrated that Ang-(1-9) also antagonises Ang II and is anti-hypertrophic and anti-fibrotic in animal models of acute cardiac remodelling. However, so far it is not well documented whether Ang-(1-9) can reverse established cardiac dysfunction and remodelling and whether it is beneficial when administered chronically. Therefore, the main aim of this thesis was to assess the effects of chronic Ang-(1-9) administration on cardiac structure and function in a model of Ang II-induced cardiac remodelling. Furthermore, this thesis aimed to investigate novel pathways contributing to the pathological remodelling in response to Ang II. First, a mouse model of chronic Ang II infusion was established and characterised by comparing the structural and functional effects of the infusion of a low and high dose of Ang II after 6 weeks. Echocardiographic measurements demonstrated that low dose Ang II infusion resulted in a gradual decline in cardiac function while a high dose of Ang II induced acute cardiac contractile dysfunction. Both doses equally induced the development of cardiac hypertrophy and cardiac fibrosis characterised by an increase in the deposition of collagen I and collagen III. Moreover, increases in gene expression of fibrotic and hypertrophic markers could be detected following high dose Ang II infusion over 6 weeks. Following this characterisation, the high dose infusion model was used to assess the effects of Ang-(1-9) on cardiac structural and functional remodelling in established disease. Initially, it was evaluated whether Ang-(1-9) can reverse Ang II-induced cardiac disease by administering Ang-(1-9) for 2-4 weeks following an initial 2 week infusion of a high dose of Ang II to induce cardiac contractile dysfunction. The infusion of Ang-(1-9) for 2 weeks was associated with a significant improvement of LV fractional shortening compared to Ang II infusion. However, after 4 weeks fractional shortening declined to Ang II levels. Despite the transient improvement in cardiac contractile function, Ang-(1-9) did not modulate blood pressure, LV hypertrophy or cardiac fibrosis. To further investigate the direct cardiac effects of Ang-(1-9), cardiac contractile performance in response to Ang-(1-9) was evaluated in the isolated Langendorff-perfused rat heart. Perfusion of Ang-(1-9) in the paced and spontaneously beating rat heart mediated a positive inotropic effect characterised by an increase in LV developed pressure, cardiac contractility and relaxation. This was in contrast to Ang II and Ang-(1-7). Furthermore, the positive inotropic effect to Ang-(1-9) was blocked by the AT1R antagonist losartan and the protein kinase A inhibitor H89. Next, endothelial-to-mesenchymal transition (EndMT) as a novel pathway that may contribute to Ang II-induced cardiac remodelling was assessed in Ang II-infused mice in vivo and in human coronary artery endothelial cells (HCAEC) in vitro. Infusion of Ang II to mice for 2-6 weeks resulted in a significant decrease in myocardial capillary density and this was associated with the occurrence of dual labelling of endothelial cells for endothelial and mesenchymal markers. In vitro stimulation of HCAEC with TGFβ and Ang II revealed that Ang II exacerbated TGF-induced gene expression of mesenchymal markers. This was not correlated with any changes in SMAD2 or ERK1/2 phosphorylation with co-stimulation of TGFβ and Ang II. However, superoxide production was significantly increased in HCAEC stimulated with Ang II but not TGFβ. Finally, the role of Ang II in microvesicle (MV)-mediated cardiomyocyte hypertrophy was investigated. MVs purified from neonatal rat cardiac fibroblasts were found to contain detectable Ang II and this was increased by stimulation of fibroblasts with Ang II. Treatment of cardiomyocytes with MVs derived from Ang II-stimulated fibroblasts induced cardiomyocyte hypertrophy which could be blocked by the AT1R antagonist losartan and an inhibitor of MV synthesis and release brefeldin A. Furthermore, Ang II was found to be present in MVs isolated from serum and plasma of Ang II-infused mice and SHRSP and WKY rats. Overall, the findings of this thesis demonstrate for the first time that the actions of Ang-(1-9) in cardiac pathology are dependent on its time of administration and that Ang-(1-9) can reverse Ang II-induced cardiac contractile dysfunction by acting as a positive inotrope. Furthermore, this thesis demonstrates evidence for an involvement of EndMT and MV signalling as novel pathways contributing to Ang II-induced cardiac fibrosis and hypertrophy, respectively. These findings provide incentive to further investigate the therapeutic potential of Ang-(1-9) in the treatment of cardiac contractile dysfunction in heart disease, establish the importance of novel pathways in Ang II-mediated cardiac remodelling and evaluate the significance of the presence of Ang II in plasma-derived MVs.