Analysing structured products using partial differential equations

In this thesis, the common special features and the main assumptions of valuing and pricing Structured Products are analysed. The method reviewed uses one dimensional modelling in valuing European and American options. In order to accurately price path-dependent equity derivatives, a numerical method using finite differences partial differential equations was applied. The Black-Scholes model has been examined as well as the Constant Elasticity of Variance (CEV) model. The CEV model illustrated and was used to evaluate and price exotic options, which has been used to model and price two Structured Products: Barrier Reverse Convertibles and Capital Protected Product with Double Barrier. The three finite differences methods have been compared when valuing an European option according to accuracy for each method. Then, the best one was used for valuing American options and the CEV model for pricing Structured Products was applied. This thesis also provides important aspects for structuring a Structured Product, and is followed by the pay-off techniques that determined how the return is calculated. Then, there is a discussion about the importance of hedging and second order Greeks of Structured Products in the market. This thesis draws attention to the risks that could affect the value of the structured notes, after which is the methodology used to contribute to the sustainable development of investment products

Evolution and Regularisation of Vacuum Brill Gravitational Waves in Spherical Polar Coordinates

In this thesis the universal collapse of vacuum Brill waves is demonstrated numerically and analytically. This thesis presents the mathematical and numerical methods necessary to regularise and evolve Brill Gravitational Waves in spherical polar coordinates. A Cauchy ADM formulation is used for the time evolution. We find strong evidence that all IVP formulations of pure vacuum Brill gravitational waves collapse to form singularities/black holes, and we do not observe critical black hole mass scaling phenomena in the IVP parameter phase space that has been characterised in non-vacuum systems. A theoretical framework to prove this result analytically is presented. We discuss the meaning of Brill metric variables, the topology of trapped surfaces for various scenarios, and verify other results in the field related to critical values of initial value parameters and black hole formation approaching spatial infinity. The instability of Minkowski (flat) space under Brill wave and more general perturbations is demonstrated. The main numerical tools employed to achieve a stable evolution code are (1) derivation of appropriate regularity conditions on the lapse function and metric function q, (2) the move to a 4th order correct discretisation scheme with appropriate boundary conditions, (3) the use of exponential metric terms, (4) an understanding of the right mix of free versus constrained evolution and (5) the development of appropriate numerical techniques for discretisation and differencing to reduce numerical error, along with a characterisation of condition numbers.Comment: PhD Thesis, 2014, University of Calgary, 368 page

A Theoretical Study of Quantum Ballistic Transport in Semiconductor Ring Structures

Recent developments in microfabrication technology have enabled the manufacture of semiconductor devices in which the carriers scatter very infrequently over typical device lengths. Transport of this kind is termed ballistic, and under such conditions, coherent quantum interference phenomena become an increasingly important part of the conduction process. In particular, the conductors of such devices now assume the role of electron waveguides. Most previous attempts at modelling quantum ballistic transport have been based on one-dimensional models. However, relatively little was known about the true nature of wavepacket propagation in real structures where diffraction from apertures or around obstacles could occur. This thesis presents the first theoretical study of quantum ballistic transport in a two-dimensional quantum waveguide network. The study specifically concentrates on modelling the Aharonov-Bohm effect in ring structures, which is an exclusively quantum-mechanical effect. The method of investigation was to numerically solve the two-dimensional time-dependent Schrodinger equation for an idealised ring structure using a computer algorithm which incorporated several novel techniques. One-dimensional calculations show that one can expect a modulation depth of 100% in the oscillations in the magneto-resistance characteristic of such rings. Present oscillation amplitudes measured experimentally however fall far short of this figure, typically being about 0.1% of the background resistance in metal rings and about 10% in rings formed in the two-dimensional electron gas at a heterojunction interface. Computer simulation of wavepacket propagation in these latter structures clearly show a multi-mode structure in the wavefunction across the conductors of realistically-sized rings. It is shown that it is the transmission of more than one mode at the exit of the ring which is a major factor in reducing the amplitude of the magneto-resistance oscillations. Good agreement between the average magneto-resistance oscillation amplitude in the simulated and experimental characteristics for a ring formed at a heterojunction was obtained. The two-dimensional model can therefore be regarded as a major improvement on earlier one-dimensional models. Evidence suggesting a damping of the magneto-resistance oscillations as a result of the direct action of the magnetic field acting on the conductors is also found. It is estimated that the approximate cut-off field would be about 0.5 Tesla for the particular device modelled, which is consistent with experimental observations of a decline in the oscillation amplitude in the range 0.5-1.0 Tesla. A modification of the basic ring structure to achieve larger magneto-resistance oscillations by constricting the exit of the ring is proposed and computer simulation of wavepacket propagation through this structure shows that a substantial increase in modulation depth can be expected. The techniques developed in this thesis have therefore been able to successfully model existing quantum interference devices and also assess the likely improvement in performance of a hypothetical device. These techniques could a be applied to the modelling of wavepacket propagation in other types of sub-micron quantum-interference devices where transport can be considered to be ballistic

A Review of Computational Methods in Materials Science: Examples from Shock-Wave and Polymer Physics

This review discusses several computational methods used on different length and time scales for the simulation of material behavior. First, the importance of physical modeling and its relation to computer simulation on multiscales is discussed. Then, computational methods used on different scales are shortly reviewed, before we focus on the molecular dynamics (MD) method. Here we survey in a tutorial-like fashion some key issues including several MD optimization techniques. Thereafter, computational examples for the capabilities of numerical simulations in materials research are discussed. We focus on recent results of shock wave simulations of a solid which are based on two different modeling approaches and we discuss their respective assets and drawbacks with a view to their application on multiscales. Then, the prospects of computer simulations on the molecular length scale using coarse-grained MD methods are covered by means of examples pertaining to complex topological polymer structures including star-polymers, biomacromolecules such as polyelectrolytes and polymers with intrinsic stiffness. This review ends by highlighting new emerging interdisciplinary applications of computational methods in the field of medical engineering where the application of concepts of polymer physics and of shock waves to biological systems holds a lot of promise for improving medical applications such as extracorporeal shock wave lithotripsy or tumor treatment

Applications of Laplace transform for evaluating occupation time options and other derivatives

The present thesis provides an analysis of possible applications of the Laplace Transform (LT) technique to several pricing problems. In Finance this technique has received very little attention and for this reason, in the first chapter we illustrate with several examples why the use of the LT can considerably simplify the pricing problem. Observed that the analytical inversion is very often difficult or requires the computation of very complicated expressions, we illustrate also how the numerical inversion is remarkably easy to understand and perform and can be done with high accuracy and at very low computational cost. In the second and third chapters we investigate the problem of pricing corridor derivatives, i.e. exotic contracts for which the payoff at maturity depends on the time of permanence of an index inside a band (corridor) or below a given level (hurdle). The index is usually an exchange or interest rate. This kind of bond has evidenced a good popularity in recent years as alternative instruments to common bonds for short term investment and as opportunity for investors believing in stable markets (corridor bonds) or in non appreciating markets (hurdle bonds). In the second chapter, assuming a Geometric Brownian dynamics for the underlying asset and solving the relevant Feynman-Kac equation, we obtain an expression for the Laplace transform of the characteristic function of the occupation time. We then show how to use a multidimensional numerical inversion for obtaining the density function. In the third chapter, we investigate the effect of discrete monitoring on the price of corridor derivatives and, as already observed in the literature for barrier options and for lookback options, we observe substantial differences between discrete and continuous monitoring. The pricing problem with discrete monitoring is based on an appropriate numerical scheme of the system of PDE's. In the fourth chapter we propose a new approximation for pricing Asian options based on the logarithmic moments of the price average

Numerical evaluation of aerodynamic roughness of the built environment and complex terrain

Aerodynamic drag in the atmospheric boundary layer (ABL) is affected by the structure and density of obstacles (surface roughness) and nature of the terrain (topography). In building codes and standards, average roughness is usually determined somewhat subjectively by examination of aerial photographs. For detailed wind mapping, boundary layer wind tunnel (BLWT) testing is usually recommended. This may not be cost effective for many projects, in which case numerical studies become good alternatives. This thesis examines Computational Fluid Dynamics (CFD) for evaluation of aerodynamic roughness of the built environment and complex terrain. The present study started from development of an in-house CFD software tailored for ABL simulations. A three-dimensional finite-volume code was developed using flexible polyhedral elements as building blocks. The program is parallelized using MPI to run on clusters of processors so that micro-scale simulations can be conducted quickly. The program can also utilize the power of latest technology in high performance computing, namely GPUs. Various turbulence models including mixing-length, RANS, and LES models are implemented, and their suitability for ABL simulations assessed. Then the effect of surface roughness alone on wind profiles is assessed using CFD. Cases with various levels of complexity are considered including simplified models with roughness blocks of different arrangement, multiple roughness patches, semi-idealized urban model, and real built environment. Comparison with BLWT data for the first three cases showed good agreement thereby justifying explicit three-dimensional numerical approach. Due to lack of validation data, the real built environment case served only to demonstrate use of CFD for such purposes. Finally, the effect of topographic features on wind profiles was investigated using CFD. This work extends prior work done by the research team on multiple idealized two-dimensional topographic features to more elaborate three-dimensional simulations. It is found that two-dimensional simulations overestimate speed up over crests of hills and also show larger recirculation zones. The current study also emphasized turbulence characterization behind hills. Finally a real complex terrain case of the well-known Askervein hill was simulated and the results validated against published field observations. In general the results obtained from the current simulations compared well with those reported in literature