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.

Developing executive capability — banking and beyond

The transition to becoming a leader is perhaps the least understood and most difficult in business. This Portfolio of Exploration examines the development of conscious awareness and meaning complexity as key transformational requirements to operate competently at leadership level and to succeed in a work environment characterised by change and complexity. It recognises that developing executive leadership capability is not just an issue of personality increasing what we know or expertise. It requires development of complexity in terms of how we know ourselves, relate to others, construe leadership and organisation, problem solve in business and understand the world as a whole. The exploration is grounded in the theory of adult mental development as outlined by Robert Kegan (1982, 1994) and in his collaborations with Lisa Laskow Lahey (2001, 2009). The theory points to levels of consciousness which impact on how we make meaning of and experience the world around us and respond to it. Critically it also points to transformational processes which enable us to evolve how we make meaning of our world as a means to close the mismatch between the demands of this world and our ability to cope. The exploration is laid out in three stages. Using Kegan’s (1982, 1994) theory as a framework it begins with a reflection of my career to surface how I made meaning of banking, management and subsequently leadership. In stage two I engage with a range of source thinkers in the areas of leadership, decision making, business, organisation, growth and complexity in a transformational process of developing greater conscious and complex understanding of organisational leadership (also recognising ever increasing complexity in the world). Finally, in stage three, I explore how qualitative changes as a result of this transformational effort have benefitted my professional, leadership and organisational capabilities.