The Heston model under stochastic interest rates

Tese de mestrado, Matemática Financeira, Faculdade de Ciências, Universidade de Lisboa,2008In this dissertation the Heston (1993) model is considered, but using, instead of a constant interest rate, stochastic interest rates according to Vasicek (1977) and to Cox, Ingersoll and Ross (1985) models. Under this framework, a closed-form solution is determined for the price of European standard calls, which, by using a manipulation implemented by Attari (2004), only require the evaluation of one characteristic function. For forward-start European calls, starting from the result for standard calls and using analytic characteristic functions, it is determined a closed-form solution that only requires one numerical integration. In the end, the results of these closedform solutions are compared with the results presented by Monte Carlo simulations for the considered models.Nesta dissertação é considerado o modelo de Heston (1993), mas em vez de utilizar uma taxa de juro constante, considera-se taxas de juro estocásticas segundo os modelos de Vasicek (1977) e de Cox, Ingersoll e Ross (1985). Neste contexto, é determinada uma solução fechada para a avaliação de standard calls Europeias, que, por ter sido usada uma manipulação implementada por Attari (2004), apenas necessitará da avaliação de uma função característica. Para calls forward-start Europeias, partindo do resultado apresentado para standard calls e utilizando funções característica analíticas, é determinada uma solução fechada que também recorrerá a apenas uma integração numérica. No final, os resultados destas fórmulas fechadas são comparados com os resultantes de simulações de Monte Carlo para os modelos considerados

Clustering of Gaussian Random Vector Fields in Multiple Trajectory Modelling

This paper concerns the estimation of multiple dynamical models from a set of observed trajectories. It proposes vector valued gaussian random fields, representing dynamical models and their vector fields, combined with a modified k- means clustering algorithm to assign observed trajectories to models. The assignment is done according to a likelihood function obtained from applying the random field associated to a cluster, to the data. The algorithm is shown to have several advantages when compared with others: 1) it does not depend on a grid, region of interest, grid resolution or interpolation method; 2) the estimated vector fields has an associated uncertainty which is given by the algorithm and taken into account. The paper presents results obtained on synthetic trajectories that illustrate the performance of the proposed algorithm

The planar-to-tubular structural transition in boron clusters from optical absorption

The optical response of the lowest energy isomers of the B_20 family is calculated using time-dependent density functional theory within a real-space, real-time scheme. Significant differences are found among the absorption spectra of the clusters studied. We show that these differences can be easily related to changes in the overall geometry. Optical spectroscopy is thus an efficient tool to characterize the planar to tubular structural transition, known to be present in these boron based systems

Recursive bayesian identification of nonlinear autonomous systems

This paper concerns the recursive identification of nonlinear discrete-time systems for which the original equations of motion are not known. Since the true model structure is not available, we replace it with a generic nonlinear model. This generic model discretizes the state space into a finite grid and associates a set of velocity vectors to the nodes of the grid. The velocity vectors are then interpolated to define a vector field on the complete state space. The proposed method follows a Bayesian framework where the identified velocity vectors are selected by the maximum a posteriori (MAP) criterion. The resulting algorithms allow a recursive update of the velocity vectors as new data is obtained. Simulation examples using the recursive algorithm are presented

On the instability of Reissner-Nordstrom black holes in de Sitter backgrounds

Recent numerical investigations have uncovered a surprising result: Reissner-Nordstrom-de Sitter black holes are unstable for spacetime dimensions larger than 6. Here we prove the existence of such instability analytically, and we compute the timescale in the near-extremal limit. We find very good agreement with the previous numerical results. Our results may me helpful in shedding some light on the nature of the instability.Comment: Published in Phys.Rev.

Magnetic response of carbon nanotubes from ab initio calculations

We present {\it ab initio} calculations of the magnetic susceptibility and of the $^{13}$C chemical shift for carbon nanotubes, both isolated and in bundles. These calculations are performed using the recently proposed gauge-including projector augmented-wave approach for the calculation of magnetic response in periodic insulating systems. We have focused on the semiconducting zigzag nanotubes with diameters ranging from 0.6 to 1.6 nm. Both the susceptibility and the isotropic shift exhibit a dependence with the diameter (D) and the chirality of the tube (although this dependence is stronger for the susceptibility). The isotropic shift behaves asymptotically as $\alpha/D + 116.0$, where $\alpha$ is a different constant for each family of nanotubes. For a tube diameter of around 1.2 nm, a value normally found in experimental samples, our results are in excellent agreement with experiments. Moreover, we calculated the chemical shift of a double-wall tube. We found a diamagnetic shift of the isotropic lines corresponding to the atoms of the inner tube due to the effect of the outer tube. This shift is in good agreement with recent experiments, and can be easily explained by demagnetizing currents circulating the outer tube.Comment: 7 pages, 4 figure

Optimization of the ionization time of an atom with tailored laser pulses: a theoretical study

How fast can a laser pulse ionize an atom? We address this question by considering pulses that carry a fixed time-integrated energy per-area, and finding those that achieve the double requirement of maximizing the ionization that they induce, while having the shortest duration. We formulate this double-objective quantum optimal control problem by making use of the Pareto approach to multi-objetive optimization, and the differential evolution genetic algorithm. The goal is to find out how much a precise time-profiling of ultra-fast, large-bandwidth pulses may speed up the ionization process with respect to simple-shape pulses. We work on a simple one-dimensional model of hydrogen-like atoms (the P\"oschl-Teller potential), that allows to tune the number of bound states that play a role in the ionization dynamics. We show how the detailed shape of the pulse accelerates the ionization process, and how the presence or absence of bound states influences the velocity of the process

Alignment of velocity fields for video surveillance

Velocity fields play an important role in surveillance since they describe typical motion behaviors of video objects (e.g., pedestrians) in the scene. This paper presents an algorithm for the alignment of velocity fields acquired by different cameras, at different time intervals, from different viewpoints. Velocity fields are aligned using a warping function which maps corresponding points and vectors in both fields. The warping parameters are estimated by minimizing a non-linear least squares energy. Experimental tests show that the proposed model is able to compensate significant misalignments, including translation, rotation and scaling

Benchmark calculations for reduced density-matrix functional theory

Reduced density-matrix functional theory (RDMFT) is a promising alternative approach to the problem of electron correlation. Like standard density functional theory, it contains an unknown exchange-correlation functional, for which several approximations have been proposed in the last years. In this article, we benchmark some of these functionals in an extended set of molecules with respect to total and atomization energies. Our results show that the most recent RDMFT functionals give very satisfactory results compared to more involved quantum chemistry and density functional approaches.Comment: 17 pages, 1 figur