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.

Two closely coupled lasers, dynamics & applications

The dynamics of two mutually coupled identical single-mode semi-conductor lasers are theoretically investigated. For small separation and large coupling between the lasers, symmetry-broken one-colour states are shown to be stable. In this case the light output of the lasers have significantly different intensities whilst at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable two-colour states, where both single-mode lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. For low coupling but possibly large separation, the frequency of the relaxation oscillations of the freerunning lasers defines the dynamics. Chaotic and quasi-periodic states are identified and shown to be stable. For weak coupling undamped relaxation oscillations dominate where each laser is locked to three or more odd number of colours spaced by the relaxation oscillation frequency. It is shown that the instabilities that lead to these states are directly connected to the two colour mechanism where the change in the number of optical colours due to a change in the plane of oscillation. At initial coupling, in-phase and anti-phase one colour states are shown to emerge from “on” uncoupled lasers using a perturbation method. Similarly symmetry-broken one-colour states come from considering one free-running laser initially “on” and the other laser initially “off”. The mechanism that leads to a bi-stability between in-phase and anti-phase one-colour states is understood. Due to an equivariant phase space symmetry of being able to exchange the identical lasers, a symmetric and symmetry-broken variant of all states mentioned above exists and is shown to be stable. Using a five dimensional model we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetry-broken one-colour, symmetric and symmetry-broken two-colour, symmetric and symmetry-broken undamped relaxation oscillations, symmetric and symmetry-broken quasi-periodic, and symmetric and symmetry-broken chaotic states. As symmetry-broken states always exist in pairs, they naturally give rise to bi-stability. Several of these states show multistabilities between symmetric and symmetry-broken variants and among states. Three memory elements on the basis of bi-stabilities in one and two colour states for two coupled single-mode lasers are proposed. The switching performance of selected designs of optical memory elements is studied numerically.