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 emerging strategic partnership between the European Union and the African Union

Many among the emerging generation of political elites in Africa see the role the European Union (EU) plays in the maintenance of an unprecedented period of peace in Western Europe as an inspirational example of the manner in which the African Union (AU) can contribute to peace and stability in Africa. This doctoral thesis examines security cooperation between the EU and the AU, with a particular focus on the nature and substance of that cooperation. It suggests that despite the establishment of various EU–AU institutions and ties with a role in security policy and cooperation, such security cooperation is limited in substance. This study argues that EU–AU security cooperation is especially constrained by the emergence of alternative partners, most notably China, and by failures of implementation and follow-through. Two case studies, the first dealing with EU–AU cooperation in peacekeeping, and the second addressing the silent water crisis along with the link between water and security, have been analysed in detail to determine the effectiveness and sustainability of the EU–AU partnership. A number of important lessons for regionalism, interregionalism and multilateralism are drawn from the bond between the EU and the AU. This doctoral thesis will prove that, despite an emphasis on the problematic term ‘strategic’ by both EU and AU policymakers, EU–AU cooperation is limited and somewhat lacking in strategic direction. The cooperation between the EU and the AU focuses mainly on EU financial support for AU peacekeeping and specific projects in Africa (e.g. in the water sector), as well as on a limited political dialogue. Nonetheless, the EU–AU link represents the most comprehensive partnership the AU has with any non-African actor. This study will furthermore demonstrate that the United Nations (UN) is an indispensable third-party to their relationship and it is therefore more appropriate to speak of the AU–EU–UN nexus. This doctoral thesis concludes that the AU–EU–UN nexus is an important example of interregionalism in a global context and that such interregionalism is an important emerging part of global governance.