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.

Silica bonded stationary phases: Innovative functionalization methods and novel chromatographic separations based on molecular recognition

The research work included in this thesis examines the synthesis, characterization and chromatographic evaluation of novel bonded silica stationary phases. Innovative methods of preparation of silica hydride intermediates and octadecylsilica using a “green chemistry” approach eliminate the use of toxic organic solvents and exploit the solvating power and enhanced diffusivity of supercritical carbon dioxide to produce phases with a surface coverage of bonded ligands which is comparable to, or exceeds, that achieved using traditional organic solvent-based methods. A new stationary phase is also discussed which displays chromatographic selectivity based on molecular recognition. Chapter 1 introduces the chemistry of silica stationary phases, the retention mechanisms and theories on which reversed-phase liquid chromatography and hydrophilic interaction chromatograpy are based, the art and science of achieving a well packed liquid chromatography column, the properties of supercritical carbon dioxide and molecular recognition chemistry. Chapter 2 compares the properties of silica hydride materials prepared using supercritical carbon dioxide as the reaction medium with those synthesized in an organic solvent. A higher coverage of hydride groups on the silica surface is seen when a monofunctional silane is reacted in supercritical carbon dioxide while trifunctional silanes result in a phase which exhibits different properties depending on the reaction medium used. The differing chromatographic behaviour of these silica hydride materials prepared using supercritical carbon dioxide and using organic solvent are explored in chapter 3. Chapter 4 focusses on the preparation of octadecylsilica using mono-, di- and trifunctional alkoxysilanes in supercritical carbon dioxide and in anhydrous toluene. The surface coverage of octadecyl groups, as calculated using thermogravimetric analysis and elemental analysis, is highest when a trifunctional alkoxysilane is reacted with silica in supercritical carbon dioxide. A novel silica stationary phase is discussed in chapter 5 which displays selectivity for analytes based on their hydrogen bonding capabilities. The phase is also highly selective for barbituric acid and may have a future application in the solid phase extraction of barbiturates from biological samples.