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.

A study of the regulation of Trib family members expression in normal and malignant haematopoiesis

The Tribbles family of genes consist of three members; TRIB1, TRIB2 and TRIB3. Trib1 and Trib2 have been identified as oncogenes that can induce AML in mice. However little is known about how the expressions of the Tribbles family genes are controlled in the cell during haematopoiesis or leukaemogenesis. To investigate the Tribbles genes in leukaemia a bioinformatics approach was used. TRIB2 expression was found to be elevated in T-ALL and ALL with t(1;19). TRIB1 was found not to be significantly elevated in any leukaemic subtypes. Analyses of the TRIB1 and TRIB2 gene signatures in both leukaemic and normal haematopoietic cells identified pathways and transcription factors associated with these signatures. Pathways enriched for the TRIB1 signature included TLR signalling pathways and NF-κB pathways. Transcription factors enriched for this signature include C/EBP and SRF. Enriched for the TRIB2 signature includes T cell signalling pathways and Notch signalling pathways. Transcription factors enriched for this signature include E2F and ETS. Further investigation in vitro confirmed the finding that E2F1 was as a potential regulator of TRIB2 expression. E2F1 is able to directly bind to the TRIB2 promoter region and induce TRIB2 expression. C/EBPα p42 was found to inhibit E2F1 and the p30 isoform was found to cooperate with E2F1 induced activation of the TRIB2 promoter. Indicating the potential presence of a regulatory loop involved in the regulation of the TRIB2 gene. In conclusion we have investigated the Tribbles gene signatures in both normal haematopoietic and leukaemic cells. This has led to the identification of a number of pathways and transcription factors associated with these genes. We have also identified a family of transcription factors directly responsible for the regulation of TRIB2 expression. This regulatory pathway has the potential to be targeted in the treatment of leukaemia with a high TRIB2 signature.