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.

Becoming a father of a child with an intellectual disability- a narrative inquiry of Irish fathers’ stories

Background: This thesis explored men’s experiences of becoming a father of a child with an intellectual disability in the early years. In Ireland, it is estimated that there are almost 97% (n= 9,914) children with intellectual disabilities living at home in the care of parents, siblings, relatives or foster parents. While mothers and fathers are the primary caregivers, mothers’ experiences are well documented in comparison to the dearth of reports on fathers’ experiences. This descriptive narrative study aims to redress this gap in knowledge and understanding of men’s experiences of becoming a father of a child with an intellectual disability in the early years. Method: Narrative inquiry was employed for this study as it allows stories told by fathers to be collected as a means of exploring men’s transition to becoming a father of a child with an intellectual disability. A sample of 10 fathers of children with intellectual disabilities aged between thirteen months and five years of age were recruited from a large intellectual disability Health Service Provider (HSP) in the South of Ireland. Data were collected through semi-structured interviews which were audio-recorded, transcribed, and analysed using a narrative thematic approach. Findings: Findings are presented in four themes: i) ‘becoming a father’, ii) ‘something wrong with my child’, iii) ‘entering the world of disability’ and iv) ‘living a different life’. For all 10 fathers the time of being told that their child had an intellectual disability was laden with negative emotional responses irrespective of whether the diagnosis was at birth or more gradual over the child’s early developmental period. When fathers found out that ‘something was wrong’ they spoke of ‘moving on’ and entering the world of disability. In their narratives, becoming the father of a child with an intellectual disability had changed their lives and would inevitably change their futures. Fathers’ positivity was clearly evident with many fathers identifying that the diagnosis of their child with an intellectual disability was not a life ending event but rather a life changing event. Conclusions: Healthcare professionals have a critical role in supporting fathers during the transition to becoming a father of a child with an intellectual disability. Factors which require consideration include recognising that each father’s experience is unique; that fathers require support; and that fathers achieve personal growth because of their experiences of their transition to becoming a father of a child with an intellectual disability in the early years.

Community workers' understandings and practices of resistance

This research is focused on Community Workers located in Southern Ireland, and their understandings and practices of resistance. It is an attempt to explore the ways in which community workers’ understandings and practices of resistance are formed and, in turn, inform their sense of identity and their responses to the wider context of community development work in Ireland today. This study is specifically located but also has wider application and relevance because of the extended international reach of neo-liberal and managerial rationalities, and their implications for politics, policy and practice. The study considers resistance in a number of inter-related ways: as a collective oppositional position (with negative and positive dimensions); a personal and/or professional value (associated with the ‘expansion of contention’); a strategy for negotiating unequal power relations (in a range of levels and spaces of power); an identity (in relation to the sustaining of ‘reflexive subjectivities’); a set of practices, (which take into account the interplay between economic, political and cultural influences); and an educational process through which practitioners assess and enact personal and professional agency. Critical theorisations of community development and of the Irish state over time, trace the ways in which neo-liberalism and managerialism has inflected community development practice and the positions of community workers and communities in that process. The study draws on James C. Scott, Gramsci, Barnes and Prior, among others, which enabled the interrogation of resistance in relation to everyday practices through engaging with ‘hidden transcripts’ and spaces. The method chosen was focus group discussions with three groups of community workers located in different counties in Southern Ireland. This method facilitated a deep discourse analysis of practitioners’ encounters with resistance in the field of community work. Key findings relate to the various interpretations of the role of resistance, practices of resistance (including current restrictions), the value of resistance work and the conditions that may be conducive to practising resistance.