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.

Role of substrate quality on the performance of semipolar (11 2 - 2) InGaN light-emitting diodes

We compare the optical properties and device performance of unpackaged InGaN/GaN multiple-quantum-well light-emitting diodes (LEDs) emitting at ∼430 nm grown simultaneously on a high-cost small-size bulk semipolar (11 2 - 2) GaN substrate (Bulk-GaN) and a low-cost large-size (11 2 - 2) GaN template created on patterned (10 1 - 2) r-plane sapphire substrate (PSS-GaN). The Bulk-GaN substrate has the threading dislocation density (TDD) of ∼ and basal-plane stacking fault (BSF) density of 0 cm-1, while the PSS-GaN substrate has the TDD of ∼2 × 108cm-2 and BSF density of ∼1 × 103cm-1. Despite an enhanced light extraction efficiency, the LED grown on PSS-GaN has two-times lower internal quantum efficiency than the LED grown on Bulk-GaN as determined by photoluminescence measurements. The LED grown on PSS-GaN substrate also has about two-times lower output power compared to the LED grown on Bulk-GaN substrate. This lower output power was attributed to the higher TDD and BSF density.