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.

Vascular calcification and mineral bone disorder in chronic kidney disease

Chronic Kidney Disease (CKD), osteoporosis and mild hyponatremia are all prevalent chronic conditions that may coexist and are often under-recognized. Mineral-Bone Disorder begins early in the natural history of CKD and results in complex abnormalities of bone which ultimately confers a well-established increased risk of fragility fractures in End Stage Kidney Disease. Hyponatremia is a novel, usually renal mediated metabolic perturbation, that most commonly occurs independently of the stage of renal dysfunction but which may also predispose to increased fracture risk. The extent -if any- to which either early stages of renal dysfunction or the presence of hyponatremia contribute to fracture occurrence in the general population, independently of osteoporosis, is unclear. Renal transplantation is the treatment of choice for ESKD and although it restores endogenous renal function it typically fails to normalize either the long term cardiovascular or fracture risk. One potential mechanism contributing to these elevated long-term risks and to diminished Health Related Quality of Life is persistent, post-transplant hyperparathyroidism. In this study we retrospectively examine the association of renal function and serum sodium with Bone Mineral Density and fracture occurrence in a retrospective cohort of 1930 female members of the general population who underwent routine DXA scan. We then prospectively recruited a cohort of 90 renal transplant recipients in order to examine the association of post transplant parathyroid hormone (PTH) level with measures of CKD Mineral Bone Disorder, including, DXA Bone Mineral Density, Vascular Calcification (assessed using both abdominal radiography and CT techniques, as well as indirectly by carotid-femoral Pulse Wave Velocity) and Quality of Life (using the Short Form-12 and a PTH specific symptom score). In the retrospective DXA cohort, moderate CKD (eGFR 30-59ml/min/1.73m2) and hyponatremia (<135mmol/L) were associated with fracture occurrence, independently of BMD, with an adjusted Odds Ratio (95% Confidence Interval), of 1.37 (1.0, 1.89) and 2.25 (1.24, 4.09) respectively. In the renal transplant study, PTH was independently associated with the presence of osteoporosis, adjusted Odds Ratio (95% Confidence Interval), 1.15 (per 10ng/ml increment), (1.04, 1.26). The presence of osteoporosis but not PTH was independently associated with measures of vascular calcification, adjusted ß (95% Confidence Interval), 12.45, (1.16, 23.75). Of the eight quality-of-life domains examined, post-transplant PTH (per 10ng/ml increment), was only significantly and independently associated with reduced Physical Functioning, (95% Confidence Interval), 1.12 (1.01, 1.23). CKD and hyponatremia are both common health problems that may contribute to fracture occurrence in the general population, a major on-going public health concern. PTH and decreased Bone Mineral Density may signal sub-optimal long-term outcomes post renal transplantation, influencing bone and vascular health and to a limited extent long term Health Related Quality of Life