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.

Regulation of prostate cancer cell function by activators of AMP-activated protein kinase

AMP-activated protein kinase (AMPK) is a key regulator of cell energy homeostasis. More recently, it has become apparent that AMPK regulates cell proliferation, migration and inflammation. Previous evidence has suggested that AMPK may influence proliferation and invasion by regulating the pro-proliferative mitogen-activated protein kinases (MAPKs). However, the mechanisms underlying this crosstalk between AMPK and MAPK signalling are not fully understood. As AMPK activation has been reported to have anti-proliferative effects, there has been increasing interest in AMPK activation as a therapeutic target for tumourigenesis. The aim of this study was to investigate whether AMPK activation influenced prostate cancer (PC) cell line proliferation, migration and signalling. Therefore, different PC cell lines were incubated with two structurally-unrelated molecules that activate AMPK by different mechanisms, AICAR and A769662. Both chemicals activated AMPK in a concentration- and time-dependent manner in PC3, DU145 and LNCaP cell lines. AMPK activity as assessed by AMPK activating phosphorylation as well as phosphorylation of the AMPK substrate ACC increased along with tumour severity in PC biopsies. Furthermore, both activators of AMPK decreased cell proliferation and migration in the androgen-independent PC cell lines PC3 and DU145. Inhibition of proliferation by A769662 was attenuated in AMPK α1-/- AMPK α2-/- knockout (KO) mouse embryonic fibroblasts (MEFs) compared to wild type (WT) MEFs, and the inhibitory effect on migration of AICAR lost significance in PC3 cells infected with adenoviruses expressing a dominant negative AMPK α mutant, indicating these effects are partially mediated by AMPK. Furthermore, long-term activation of AMPK was associated with inhibition of both the phosphatidylinositol 3’-kinase/protein kinase B (PI3K/Akt) signalling pathway in addition to the extracellular signal-regulated kinase 1/2 (ERK1/2) signalling pathway. Indeed, the actions of AMPK activators on PC cell line viability were mimicked by selective inhibitors of Akt and ERK1/2 pathways. In contrast to the effects of prolonged incubation with AMPK activators, short-term incubation with AMPK activators had no effect on epidermal growth factor (EGF)-stimulated ERK1/2 phosphorylation in PC cell lines. In addition, AMPK activation did not influence phosphorylation of the other MAPK family members p38 and JNK. Interestingly, both AICAR and A769662 decreased EGF-stimulated ERK5 phosphorylation in PC3, DU145 and LNCaP cells as assessed with an anti-phospho-ERK5 antibody. Further characterisation of this effect indicated that prior stimulation with the AMPK activators had no effect on ERK5 phosphorylation stimulated by transient transfection with a constitutively active ERK5 kinase (MEK5DD), which represents the only known canonical kinase for ERK5. Intriguingly, the pattern of EGF-stimulated ERK5 phosphorylation was distinct from that mediated by MEK5DD activation of ERK5. This finding indicates that AMPK activation inhibits EGF-stimulated ERK5 phosphorylation at a point at or above the level of MEK5, although why EGF and constitutively active MEK5 stimulate markedly different immunoreactive species recognised by the anti-phospho-ERK5 antibody requires further study. A769662 had a tendency to reduce EGF-stimulated ERK5 phosphorylation in WT MEFs, yet was without effect in MEFs lacking AMPK. These data indicate that AMPK may underlie the effect of A769662 to reduce EGF-stimulated ERK5 phosphorylation. Prolonged stimulation of PC cell lines with AICAR or A769662 inhibited EGF-stimulated Akt Ser473 phosphorylation, whereas only incubation with A769662 rapidly inhibited Akt phosphorylation. This difference in the actions of the different AMPK activators may suggest an AMPK-independent effect of A769662. Furthermore, AICAR increased phosphorylation of Akt in WT MEFs, an effect that was absent in MEFs lacking AMPK, indicating that this effect of AICAR may be AMPK-dependent. Taken together, the data presented in this study suggest that AMPK activators markedly inhibit proliferation and migration of PC cell lines, reduce EGF-stimulated ERK1/2 and Akt phosphorylation after prolonged incubation and rapidly inhibit ERK5 phosphorylation. Both AMPK activators exhibit a number of effects that are likely to be independent of AMPK in PC cell lines, although inhibition of ERK1/2, ERK5 and Akt may underlie the effects of AMPK activators on proliferation, viability and migration. Further studies are required to understand the crosstalk between those signalling pathways and their underlying significance in PC progression.