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.

Moral shielding: A grounded theory of integrity maintenance in multidisciplinary teams

Using a classic grounded theory methodology (CGT), this study explores the phenomenon of moral shielding within mental health multidisciplinary teams (MDTS). The study was located within three catchment areas engaged in acute mental health service practice. The main concern identified was the maintenance of a sense of personal integrity during situational binds. Through theoretical sampling thirty two practitioners, including; doctors, nurses, social workers, occupational therapists, counsellors and psychologists, where interviewed face to face. In addition, emergent concepts were identified through observation of MDTs in clinical and research practice. Following a classic grounded theory methodology, data collection and analysis occurred simultaneously. A constant comparative approach was adopted and resulted in the immergence of three sub- core categories; moral abdication, moral hinting and pseudo-compliance. Moral abdication seeks to re-position within an event in order to avoid or deflect the initial obligation to act, it is a strategy used to remove or reduce moral ownership. Moral gauging represents the monitoring of an event with the goal of judging the congruence of personal principles and commitments with that of other practitioners. This strategy is enacted in a bid to seek allies for the support of a given moral position. Pseudo-compliance represents behaviour that hides desired principles and commitments in order to shield them from challenge. This strategy portrays agreement with the dominant position within the MDT, whilst holding a contrary position. It seeks to preserve a reservoir of emotional energy required to maintain a sense of personal integrity. Practitioners who were successful in enacting moral shielding were found to not experience significant emotional distress associated with the phenomenon of moral distress; suggesting that these practitioners had found mechanisms to manage situational binds that threatened their sense of personal integrity.