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.

The role of transformative thinking and practice in a property entrepreneur's response to the funding crisis since 2008

This portfolio of exploration explores the role of transformative thinking and practice in a property entrepreneur’s response to the financial crisis which swept over Ireland from 2008. The complexity of this challenge and the mental capacity to meet its demands is at the core of the exploration. This inquiry emerged from the challenges the financial crisis presented to my values, beliefs, assumptions, and theories, i.e. the interpretive lens through which I make meaning of my experiences. Given the issue identified, this inquiry is grounded in aspects of theories of constructive developmental psychology, applied developmental science, and philosophy. Integrating and linking these elements to business practice is the applied element of the Portfolio. As the 2008 crisis unfolded I realised I was at the limits of my way of knowing. I came to understand that the underlying structure of a way of knowing is the ‘subject-object relationship’ i.e. what a way of knowing can reflect upon, look at, have perspective on, in other words, make object, as against what is it embedded in, attached to, identified within, or subject to. My goal became enhancing my awareness of how I made meaning and how new insights, which would transform a way of knowing, are created. The focus was on enhancing my practice. This Portfolio is structured into three essays. Essay One reported on my self-reflection and external evaluation out of which emerged my developmental goals. In Essay Two I undertake a reading for change programme in which different meaning making systems were confronted in order to challenge me as a meaning maker. Essay Three reported on my experiment which concerned the question whether it was possible for me as a property entrepreneur, and for others alike, to retain bank finance in the face of the overwhelming objective of the bank to deleverage their balance sheet of property loans. The output of my research can be grouped into General Developmental and Specific Business Implications. Firstly I address those who are interested in a transformational-based response to the challenges of operating in the property sector in Ireland during a crisis. I outline the apparatus of thought that I used to create insight, and thus transform how I thought, these are Awareness, Subject-object separation, Exploring other’s perspectives from the position of incompleteness, Dialectical thinking and Collingwood’s Questioning activity. Secondly I set out my learnings from the crisis and their impact on entrepreneurial behaviour and the business of property development. I identify ten key insights that have emerged from leading a property company through the crisis. Many of these are grounded in common sense, however, in my experience these were, to borrow Shakespeare’s words, “More honor'd in the breach than the observance” in pre Crisis Ireland. Finally I set out a four-step approach for forging a strategy. This requires my peer practitioners to identify (i) what they are subject to, (ii) Assess the Opportunity or challenge in a Systemic Context, (iii) Explore Multiple Perspectives on the opportunity or Challenge with an Orientation to change how you know and (iv) Using the Questioning Activity to create Knowledge. Based on my experience I conclude that transformative thinking and practice is a key enabler for a property entrepreneur, in responding to a major collapse of traditional (bank debt) funding.