Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

Synthesis, characterisation and applications of group IV nanocrystals

Group IV materials such as silicon nanocrystals (Si NCs) and carbon quantum dots (CQDs) have received great attention as new functional materials with unique physical/chemical properties that are not found in the bulk material. This thesis reports the synthesis and characterisation of both types of nanocrystal and their application as fluorescence probes for the detection of metal ions. In chapter 2, a simple method is described for the size controlled synthesis of Si NCs within inverse micelles having well defined core diameters ranging from 2 to 6 nm using inert atmospheric synthetic methods. In addition, ligands with different molecular structures were utilised to reduce inter-nanocrystal attraction forces and improve the stability of the NC dispersions in water and a variety of organic solvents. Regulation of the Si NCs size is achieved by variation of the surfactants and addition rates, resulting high quality NCs with standard deviations (σ = Δd/d) of less than 10 %. Large scale production of highly mondisperse Si NC was also successfully demonstrated. In chapter 3, a simple solution phase synthesis of size monodisperse carbon quantum dots (CQDs) using a room temperature microemulsion strategy is demonstrated. The CQDs are synthesized in reverse micelles via the reduction of carbon tetrachloride using a hydride reducing agent. CQDs may be functionalised with covalently attached alkyl or amine monolayers, rendering the CQDs dispersible in wide range of polar or non-polar solvents. Regulation of the CQDs size was achieved by utilizing hydride reducing agents of different strengths. The CQDs possess a high photoluminescence quantum yield in the visible region and exhibit excellent photostability. In chapter 4, a simple and rapid assay for detection of Fe3+ ions was developed, based on quenching of the strong blue-green Si NC photoluminescence. The detection method showed a high selectivity, with only Fe3+ resulting in strong quenching of the fluorescence signal. No quenching of the fluorescence signal was induced by Fe2+ ions, allowing for solution phase discrimination between the same ion in different charge states. The optimised sensor system showed a sensitive detection range from 25- 900 μM and a limit of detection of 20.8 μM

Synthesis, functionalisation and characterisation of germanium nanocrystals & their applications

In the last two decades, semiconductor nanocrystals have been the focus of intense research due to their size dependant optical and electrical properties. Much is now known about how to control their size, shape, composition and surface chemistry, allowing fine control of their photophysical and electronic properties. However, genuine concerns have been raised regarding the heavy metal content of these materials, which is toxic even at relatively low concentrations and may limit their wide scale use. These concerns have driven the development of heavy metal free alternatives. In recent years, germanium nanocrystals (Ge NCs) have emerged as environmentally friendlier alternatives to II-VI and IV-VI semiconductor materials as they are nontoxic, biocompatible and electrochemically stable. This thesis reports the synthesis and characterisation of Ge NCs and their application as fluorescence probes for the detection of metal ions. A room-temperature method for the synthesis of size monodisperse Ge NCs within inverse micelles is reported, with well-defined core diameters that may be tuned from 3.5 to 4.5 nm. The Ge NCs are chemically passivated with amine ligands, minimising surface oxidation while rendering the NCs dispersible in a range of polar solvents. Regulation of the Ge NCs size is achieved by variation of the ammonium salts used to form the micelles. A maximum quantum yield of 20% is shown for the nanocrystals, and a transition from primarily blue to green emission is observed as the NC diameter increases from 3.5 to 4.5 nm. A polydisperse sample with a mixed emission profile is prepared and separated by centrifugation into individual sized NCs which each showed blue and green emission only, with total suppression of other emission colours. A new, efficient one step synthesis of Ge NCs with in situ passivation and straightforward purification steps is also reported. Ge NCs are formed by co-reduction of a mixture of GeCl4 and n-butyltrichlorogermane; the latter is used both as a capping ligand and a germanium source. The surface-bound layer of butyl chains both chemically passivates and stabilises the Ge NCs. Optical spectroscopy confirmed that these NCs are in the strong quantum confinement regime, with significant involvement of surface species in exciton recombination processes. The PL QY is determined to be 37 %, one of the highest values reported for organically terminated Ge NCs. A synthetic method is developed to produce size monodisperse Ge NCs with modified surface chemistries bearing carboxylic acid, acetate, amine and epoxy functional groups. The effect of these different surface terminations on the optical properties of the NCs is also studied. Comparison of the emission properties of these Ge NCs showed that the wavelength position of the PL maxima could be moved from the UV to the blue/green by choice of the appropriate surface group. We also report the application of water-soluble Ge NCs as a fluorescent sensing platform for the fast, highly selective and sensitive detection of Fe3+ ions. The luminescence quenching mechanism is confirmed by lifetime and absorbance spectroscopies, while the applicability of this assay for detection of Fe3+ in real water samples is investigated and found to satisfy the US Environmental Protection Agency requirements for Fe3+ levels in drinkable water supplies.

The occurrence and significance of triploidy in the liver fluke, Fasciola hepatica.

Karyotyping of Fasciola hepatica samples from Britain and Ireland has identified a triploid isolate which is effectively aspermic, rendering it necessarily asexually reproducing. Considering the extensive presence of asexually reproducing diploid and triploid Fasciola in Asia it is suggested that facultative gynogenesis is widespread in this parasite. This has important implications for the population genetics and evolution of Fasciola, especially in relation to the development and spread of drug resistance, and must be considered in the mathematical modelling of this process.