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.

An exploration of the positive and negative relationships associated with the development of asthma and atopic disorders in primary school children in Cork

Childhood asthma, allergic rhinitis and eczema are complex heterogenic chronic inflammatory allergic disorders which constitute a major burden to children, their families. The prevalence of childhood allergic disorders is increasing worldwide and merely rudimentary understanding exists regarding causality, or the influence of the environment on disease expression. Phase Three of the International Study of Asthma and Allergy in Childhood (ISAAC) reported that Irish adolescents had the 4th highest eczema and rhinoconjunctivitis prevalence and 3rd highest asthma prevalence in the world. There are no ISAAC data pertaining to young Irish children. In 2002, Sturley reported a high prevalence of current asthma in Cork primary school children aged 6-9 years. This thesis comprises of three cross-sectional studies which examined the prevalence of and associations with childhood allergy and a quasi-retrospective cohort study which observed the natural history of allergy from 6-9 until 11-13 years. Although not part of ISAAC, data was attained by parentally completed ISAAC-based questionnaires, using the ISAAC protocol. The prevalence, natural history and risk factors of childhood allergy in Ireland, as described in this thesis, echo those in worldwide allergy research. The variations of prevalence in different populations worldwide and the recurring themes of associations between childhood allergy and microbial exposures, from farming environments and/or gastrointestinal infections, as shown in this thesis, strengthen the mounting evidence that microbial exposure on GALT may hold the key to the mechanisms of allergy development. In this regard, probiotics may be an area of particular interest in allergy modification. Although their effects in relation to allergy, have been investigated now for several years, our knowledge of their diversity, complex functions and interactions with gut microflora, remain rudimentary. Birth cohort studies which include genomic and microbiomic research are recommended in order to examine the underlying mechanisms and the natural course of allergic diseases.