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.

Renal failure and the neural control of the kidney

Renal failure (RF) is associated with an over activation of the sympathetic nervous system. The aim of this thesis was to investigate the hypothesis that as the kidney progresses into RF there is an inappropriate and sustained activation of renal afferent nerves which results in a dysregulation of basal RSNA and reflexly controlled RSNA by the high and low pressure baroreceptors. Baroreflex gain curves for both RSNA and HR were generated in control and RF rats. This study clearly showed a blunted high-pressure baroreflex in RF rats, an impairment which was almost completely corrected by bilateral renal denervation. The integrity of the low-pressure cardiopulmonary receptors to inhibit RSNA was investigated using acute saline volume. Again, a blunted reflex sympatho-inhibition of RSNA was observed, which was corrected by renal denervation. Finally a functional study to examine how the renal excretory response to volume expansion differed in RF was carried out. This study revealed an impairment of the low-pressure baroreflex control of the sympathetic outflow. The result of these studies suggest that cisplatin induced RF initiates a neural signal from within the kidney, which over rides the normal reflex regulation of RSNA by the high and low – pressure baroreceptors and that this impairment in function can be normalised by renal denervation. This raises further questions as to the mechanisms involved in the afferent over activation arising from the diseased kidneys.