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.

The application of computer modelling to develop a methodology for ‘existing building’ upgrades to move towards carbon-neutral buildings

The retrofitting of existing buildings for decreased energy usage, through increased energy efficiency and for minimum carbon dioxide emissions throughout their remaining lifetime is a major area of research. This research area requires development to provide building professionals with more efficient building retrofit solution determination tools. The overarching objective of this research is to develop a tool for this purpose through the implementation of a prescribed methodology. This has been achieved in three distinct steps. Firstly, the concept of using the degree-days modelling method as an adequate means of basing retrofit decision upon was analysed and the results illustrated that the concept had merit. Secondly, the concept of combining the degree-days modelling method and the Genetic Algorithms optimisation method is investigated as a method of determining optimal thermal energy retrofit solutions. Thirdly, the combination of the degree-days modelling method and the Genetic Algorithms optimisation method were packaged into a building retrofit decision-support tool and named BRaSS (Building Retrofit Support Software). The results demonstrate clearly that, fundamental building information, simplified occupancy profiles and weather data used in a static simulation modelling method is a sufficient and adequate means to base retrofitting decisions upon. The results also show that basing retrofit decisions upon energy analysis results are the best means to guide a retrofit project and also to achieve results which are optimum for a particular building. The results also indicate that the building retrofit decision-support tool, BRaSS, is an effective method to determine optimum thermal energy retrofit solutions.