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.

Effect of remote ischaemic preconditioning in cardiac dysfunction and end-organ injury following cardiac surgery with cardiopulmonary bypass in children: A translational approach investigating clinical outcome and myocardial molecular biology

Congenital heart disease (CHD) is the most common birth defect, causing an important rate of morbidity and mortality. Treatment of CHD requires surgical correction in a significant percentage of cases which exposes patients to cardiac and end organ injury. Cardiac surgical procedures often require the utilisation of cardiopulmonary bypass (CPB), a system that replaces heart and lungs function by diverting circulation into an external circuit. The use of CPB can initiate potent inflammatory responses, in addition a proportion of procedures require a period of aortic cross clamp during which the heart is rendered ischaemic and is exposed to injury. High O2 concentrations are used during cardiac procedures and when circulation is re-established to the heart which had adjusted metabolically to ischaemia, further injury is caused in a process known as ischaemic reperfusion injury (IRI). Several strategies are in place in order to protect the heart during surgery, however injury is still caused, having detrimental effects in patients at short and long term. Remote ischaemic preconditioning (RIPC) is a technique proposed as a potential cardioprotective measure. It consists of exposing a remote tissue bed to brief episodes of ischaemia prior to surgery in order to activate protective pathways that would act during CPB, ischaemia and reperfusion. This study aimed to assess RIPC in paediatric patients requiring CHD surgical correction with a translational approach, integrating clinical outcome, marker analysis, cardiac function parameters and molecular mechanisms within the cardiac tissue. A prospective, single blinded, randomized, controlled trial was conducted applying a RIPC protocol to randomised patients through episodes of limb ischaemia on the day before surgery which was repeated right before the surgery started, after anaesthesia induction. Blood samples were obtained before surgery and at three post-operative time points from venous lines, additional pre and post-bypass blood samples were obtained from the right atrium. Myocardial tissue was resected during the ischaemic period of surgery. Echocardiographic images were obtained before the surgery started after anaesthetic induction and the day after surgery, images were stored for later off line analysis. PICU surveillance data was collected including ventilation parameters, inotrope use, standard laboratory analysis and six hourly blood gas analysis. Pre and post-operative quantitation of markers in blood specimens included cardiac troponin I (cTnI) and B-type natriuretic peptide (BNP), inflammatory mediators including interleukins IL-6, IL-8, IL-10, tumour necrosis factor (TNF-α), and the adhesion molecules ICAM-1 and VCAM-1; the renal marker Cystatin C and the cardiovascular markers asymmetric dymethylarginine (ADMA) and symmetric dymethylarginine (SDMA). Nitric oxide (NO) metabolites and cyclic guanosine monophosphate (cGMP) were measured before and after bypass. Myocardial tissue was processed at baseline and after incubation at hyperoxic concentration during four hours in order to mimic surgical conditions. Expression of genes involved in IRI and RIPC pathways was analysed including heat shock proteins (HSPs), toll like receptors (TLRs), transcription factors nuclear factor κ-B (NF- κ-B) and hypoxia inducible factor 1 (HIF-1). The participation of hydrogen sulfide enzymatic genes, apelin and its receptor were explored. There was no significant difference according to group allocation in any of the echocardiographic parameters. There was a tendency for higher cTnI values and inotropic score in control patients post-operatively, however this was not statistically significant. BNP presented no significant difference according to group allocation. Inflammatory parameters tended to be higher in the control group, however only TNF- α was significantly higher. There was no difference in levels of Cystatin C, NO metabolites, cGMP, ADMA or SDMA. RIPC patients required shorter PICU stay, all other clinical and laboratory analysis presented no difference related to the intervention. Gene expression analysis revealed interesting patterns before and after incubation. HSP-60 presented a lower expression at baseline in tissue corresponding to RIPC patients, no other differences were found. This study provided with valuable descriptive information on previously known and newly explored parameters in the study population. Demographic characteristics and the presence of cyanosis before surgery influenced patterns of activity in several parameters, numerous indicators were linked to the degree of injury suffered by the myocardium. RIPC did not reduce markers of cardiac injury or improved echocardiographic parameters and it did not have an effect on end organ function; some effects were seen in inflammatory responses and gene expression analysis. Nevertheless, an important clinical outcome indicator, PICU length of stay was reduced suggesting benefit from the intervention. Larger studies with more statistical power could determine if the tendency of lower injury and inflammatory markers linked to RIPC is real. The present results mostly support findings of larger multicentre trials which have reported no cardiac benefit from RIPC in paediatric cardiac surgery.