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.

Visualising ribosome profiling and using it for reading frame detection and exploration of eukaryotic translation initiation

Ribosome profiling (ribo-seq) is a recently developed technique that provides genomewide information on protein synthesis (GWIPS) in vivo. The high resolution of ribo-seq is one of the exciting properties of this technique. In Chapter 2, I present a computational method that utilises the sub-codon precision and triplet periodicity of ribosome profiling data to detect transitions in the translated reading frame. Application of this method to ribosome profiling data generated for human HeLa cells allowed us to detect several human genes where the same genomic segment is translated in more than one reading frame. Since the initial publication of the ribosome profiling technique in 2009, there has been a proliferation of studies that have used the technique to explore various questions with respect to translation. A review of the many uses and adaptations of the technique is provided in Chapter 1. Indeed, owing to the increasing popularity of the technique and the growing number of published ribosome profiling datasets, we have developed GWIPS-viz (http://gwips.ucc.ie), a ribo-seq dedicated genome browser. Details on the development of the browser and its usage are provided in Chapter 3. One of the surprising findings of ribosome profiling of initiating ribosomes carried out in 3 independent studies, was the widespread use of non-AUG codons as translation initiation start sites in mammals. Although initiation at non-AUG codons in mammals has been documented for some time, the extent of non-AUG initiation reported by these ribo-seq studies was unexpected. In Chapter 4, I present an approach for estimating the strength of initiating codons based on the leaky scanning model of translation initiation. Application of this approach to ribo-seq data illustrates that initiation at non-AUG codons is inefficient compared to initiation at AUG codons. In addition, our approach provides a probability of initiation score for each start site that allows its strength of initiation to be evaluated.