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.

The consequences of phenotypic plasticity on postglacial fishes

Phenotypic differences within a species significantly contribute to the variation we see among plants and animals. Plasticity as a concept helps us to understand some of this variation. Phenotypic plasticity plays a significant role in multiple ecological and evolutionary processes. Because plasticity can be driven by the environment it is more likely to produce beneficial alternative phenotypes than rare and often deleterious genetic mutations. Furthermore, differences in phenotypes that arise in response to the environment can affect multiple individuals from the same population (or entire populations) simultaneously and are therefore of greater evolutionary significance. This allows similar, beneficial alternative phenotypes to increase quickly within a single generation and allow new environments to produce and select for new phenotypes instantly. The direction of the present thesis is to increase our understanding of how phenotypic plasticity, coupled with contrasting environmental conditions, can produce alternative phenotypes within a population. Plasticity provides a source of variation for natural selection to act upon, and may lead to genetic isolation as a by-product. For example, there are multiple cases of polymorphic populations of fish, where groups belonging to multiple isolated gene pools, have arisen in sympatry. Here it is shown that although plasticity is important in sympatric speciation events, plasticity alone is not responsible for the frequency in which sympatric polymorphic populations occur. The most frequently observed differences among sympatric polymorphic populations are morphological differences associated with parts of the anatomy used in the detection, handling and capture of prey. Moreover, it is shown here that there are physiological effects associated with foraging on alternative prey that may significantly contribute towards ecological speciation. It is also shown in this study that anthropogenic abiotic factors can disrupt developmental processes during early ontogeny, significantly influencing morphology, and therefore having ecological consequences. Phenotypic structuring in postglacial fish is most frequently based around a divergence towards either pelagic or littoral benthic foraging specialisms. Divergences that deviate from this pattern are of greater scientific interest as they increase our understanding of how evolutionary processes and selection pressures work. Here we describe a rare divergence not based around the typical pelagic/littoral benthic foraging specialisms. Finally, in this study, the effectiveness of local level conservation policy shows that species of fish which are highly variable in their life history strategies are harder to effectively manage and often poorly represented at a local level.