In this thesis we focus mainly on special finite differences and finite volume methods and apply them to the pricing of barrier options.The structure of this work is the following: in Chapter 1 we introduce the definitions of options and illustrate some properties of vanilla European options and exotic options.Chapter 2 describes a classical model used in the financial world, the  Black-Scholes model. We derive theBlack-Scholes formula and show how stochastic differential equations model financial instruments prices.The aim of this chapter is also to present the initial boundary value problem and the maximum principle.We discuss boundary conditions such as: the first boundary value problem, also called  Dirichlet problem that occur in pricing ofbarrier options and European options. Some kinds of put options lead to the study of a second boundary value problem (Neumann, Robin problem),while the Cauchy problem is associated with one-factor European and American options.Chapter 3 is about finite differences methods such as theta, explicit, implicit and Crank-Nicolson method, which are used forsolving partial differential equations.The exponentially fitted scheme is presented in Chapter 4. It is one of the new classesof a robust difference scheme that is stable, has good convergence and does not produce spurious oscillations.The stability is also advantage of the box method that is presented in Chapter 5.In the beginning of the Chapter 6 we illustrate barrier options and then we consider a novel finite volume discretization for apricing the above options.Chapter 7 describes discretization of the Black-Scholes equation by the fitted finite volume scheme. In  Chapter 8 we present and describe numerical results obtained by using  the finite difference methods illustrated in the previous chapters.

Since the liberalisation of the energy market in Europe in the early 1990s, much opportunity to trade electricity as a commodity has arisen. One significant consequence of this movement is that market prices have become more volatile instead of its tradition constant rate of supply. Spot price markets have also been introduced, affecting the demand of electricity as companies now have the option to not only produce their own supply but also purchase this commodity from the market. Following the liberalisation of the energy market, hence creating a greater demand for trading of electricity and other types of energy, various types of options related to the sales, storage and transmission of electricity have consequently been introduced. Particularly, swing options are popular in the electricity market. As we know, swing-type derivatives are given in various forms and are mainly traded as over-the-counter (OTC) contracts at energy exchanges. These options offer flexibility with respect to timing and quantity. Traditionally, the Geometric Brownian Motion (GBM) model is a very popular and standard approach for modelling the risk neutral price dynamics of underlyings. However, a limitation of this model is that it has very few degrees of freedom, as it does not capture the complex behaviour of electricity prices. In short the GBM model is inefficient in the pricing of options involving electricity. Other models have subsequently been used to bridge this inadequacy, e.g. spot price models, futures price models, etc. To model risk-neutral commodity prices, there are basically two different methodologies, namely spot and futures or so-called term structure models. As swing options are usually written on spot prices, by which we mean the current price at which a particular commodity can be bought or sold at a specified time and place, it is important for us to examine these models in order to more accurately inculcate their effect on the pricing of swing options. Monte Carlo simulation is also a widely used approach for the pricing of swing options in the electricity market. Theoretically, Monte Carlo valuation relies on risk neutral valuation and the technique used is to simulate as many (random) price paths of the underlying(s) as possible, and then to average the calculated payoff for each path, discounted to today's prices, giving the value of the desired derivative. Monte Carlo methods are particularly useful in the valuation of derivatives with multiple sources of uncertainty or complicated features, like our electricity swing options in question. However, they are generally too slow to be considered a competitive form of valuation, if any analytical techniques of valuation exist. In other words, the Monte Carlo approach is, in a sense, a method of last resort. In this thesis, we aim to examine a numerical method involved in the pricing of swing options in the electricity market. We will consider an existing and widely accepted electricity price process model, use the finite volume method to formulate a numerical scheme in order to calibrate the prices of swing options and make a comparison with numerical solutions obtained using the theta-scheme. Further contributions of this thesis include a comparison of results and also a brief discussion of other possible methods.

An article of Zariphopoulou and Musiela "An example of indifference prices under exponential preferences", was background of our work.

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) and partial differential equations (PDEs) that describe the behaviour of this electrical activity. Generating an efficient numerical solution of these models is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this thesis, we discuss two sets of experiments that test ideas for making the numerical solution process more efficient. In the first set of experiments, we examine the numerical solution of four single cell cardiac electrophysiological models, which consist solely of ODEs. We study the efficiency of using implicit-explicit Runge-Kutta (IMEX-RK) splitting methods to solve these models. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform most methods commonly used in practice for two of the models, and they outperform all methods commonly used in practice for the remaining models. In the second set of experiments, we examine the solution of the bidomain model, a model consisting of both ODEs and PDEs that are typically solved separately. We focus these experiments on numerical methods for the solution of the two PDEs in the bidomain model. The most popular method for this task, the Crank-Nicolson method, produces unphysical oscillations; we propose a method based on a second-order L-stable singly diagonally implicit Runge-Kutta (SDIRK) method to eliminate these oscillations.<p> We find that although the SDIRK method is able to eliminate these unphysical oscillations, it is only more efficient for crude error tolerances.

En este trabajo se realiza un estudio de varias extensiones del método de las diferencias finitas en el dominio del tiempo (FDTD) que permiten la simulación electromagnética de estructuras híbridas de microondas, incluyendo circuitos activos y pasivos. En primer lugar, se revisan los métodos lumped-element (LE) -FDTD y lumped-network (LN) -FDTD, los cuales permiten la incorporación de circuitos concentrados de dos terminales dentro del formalismo FDTD. En el caso del método LN-FDTD, se realiza también un estudio de sus propiedades numéricas. A continuación se presenta el método two-port (TP) -LN-FDTD, el cual permite incorporar circuitos lineales concentrados de dos puertas en las estructuras híbridas estudiadas. Este método parte de una descripción del cuadripolo en términos de su matriz admitancia expresada en el dominio de Laplace. La discretización se realiza con la ayuda de la técnica de la transformación de Moebius. Por último, una vez validado, este método se combina con otras técnicas para la simulación distintos tipos de circuitos híbridos de microondas. / In this work, a study of several extensions of the conventional finite difference time domain (FDTD) method is been carried out. These extensions enable the electromagnetic simulation of microwave hybrid structures, including passive and active circuits. First, an exhaustive revision of both the lumped-element (LE) -FDTD and the lumped-network (LN) -FDTD methods is performed. These methods allow us to incorporate two-terminal lumped circuits into the FDTD. In addition, the numerical properties of the LN-FDTD method are studied for the first time. Second, the two-port (TP)-LN-FDTD is presented. This method enables the incorporation of linear two-port lumped circuits into the studied hybrid structures. This technique basically consists of describing a TP-LN by means of its admittance matrix in the Laplace domain. Then, by applying the Mobius transformation technique, we obtain the discretized admittance matrix. Finally, this method is combined with other existing techniques to allow the simulation of several microwave hybrid circuits.

For centuries physicians, scientists, engineers, mathematicians, and many others have been asking: 'what are the forces that drive tissues in an embryo to their final geometric forms?' At the tissue and whole embryo level, a multitude of very different morphogenetic processes, such as gastrulation and neurulation are involved. However, at the cellular level, virtually all of these processes are evidently driven by a relatively small number of internal structures all of whose forces can be resolved into equivalent interfacial tensions γ. Measuring the cell-level forces that drive specific morphogenetic events remains one of the great unsolved problems of biomechanics. Here I present a novel approach that allows these forces to be estimated from time lapse images. In this approach, the motions of all visible triple junctions formed between trios of cells adjacent to each other in epithelia (2D cell sheets) are tracked in time-lapse images. An existing cell-based Finite Element (FE) model is then used to calculate the viscous forces needed to deform each cell in the observed way. A recursive least squares technique with variable forgetting factors is then used to estimate the interfacial tensions that would have to be present along each cell-cell interface to provide those forces, along with the attendant pressures in each cell. The algorithm is tested extensively using synthetic data from an FE model. Emphasis is placed on features likely to be encountered in data from live tissues during morphogenesis and wound healing. Those features include algorithm stability and tracking despite input noise, interfacial tensions that could change slowly or suddenly, and complications from imaging small regions of a larger epithelial tissue (the frayed boundary problem). Although the basic algorithm is highly sensitive to input noise due to the ill-conditioned nature of the system of equations that must be solved to obtain the interfacial tensions, methods are introduced to improve the resulting force and pressure estimates. The final algorithm returns very good estimates for interfacial tensions and intracellular cellular pressures when used with synthetic data, and it holds great promise for calculating the forces that remodel live tissue.

For centuries physicians, scientists, engineers, mathematicians, and many others have been asking: 'what are the forces that drive tissues in an embryo to their final geometric forms?' At the tissue and whole embryo level, a multitude of very different morphogenetic processes, such as gastrulation and neurulation are involved. However, at the cellular level, virtually all of these processes are evidently driven by a relatively small number of internal structures all of whose forces can be resolved into equivalent interfacial tensions γ. Measuring the cell-level forces that drive specific morphogenetic events remains one of the great unsolved problems of biomechanics. Here I present a novel approach that allows these forces to be estimated from time lapse images. In this approach, the motions of all visible triple junctions formed between trios of cells adjacent to each other in epithelia (2D cell sheets) are tracked in time-lapse images. An existing cell-based Finite Element (FE) model is then used to calculate the viscous forces needed to deform each cell in the observed way. A recursive least squares technique with variable forgetting factors is then used to estimate the interfacial tensions that would have to be present along each cell-cell interface to provide those forces, along with the attendant pressures in each cell. The algorithm is tested extensively using synthetic data from an FE model. Emphasis is placed on features likely to be encountered in data from live tissues during morphogenesis and wound healing. Those features include algorithm stability and tracking despite input noise, interfacial tensions that could change slowly or suddenly, and complications from imaging small regions of a larger epithelial tissue (the frayed boundary problem). Although the basic algorithm is highly sensitive to input noise due to the ill-conditioned nature of the system of equations that must be solved to obtain the interfacial tensions, methods are introduced to improve the resulting force and pressure estimates. The final algorithm returns very good estimates for interfacial tensions and intracellular cellular pressures when used with synthetic data, and it holds great promise for calculating the forces that remodel live tissue.

The dynamics of the world's oceans occur at a vast range of length scales. Although there are theories that aid in understanding the dynamics at planetary scales and microscales, the motions in between are still not yet well understood. This work discusses a numerical model to study barotropic wind-driven gyre flow that is capable of resolving dynamics at the synoptic, O(1000 km), mesoscale, O(100 km) and submesoscales O(10 km). The Quasi-Geostrophic (QG) model has been used predominantly to study ocean circulations but it is limited as it can only describe motions at synoptic scales and mesoscales. The Rotating Shallow Water (SW) model that can describe dynamics at a wider range of horizontal length scales and can better describe motions at the submesoscales. Numerical methods that are capable of high-resolution simulations are discussed for both QG and SW models and the numerical results are compared. To achieve high accuracy and resolve an optimal range of length scales, spectral methods are applied to solve the governing equations and a third-order Adams-Bashforth method is used for the temporal discretization. Several simulations of both models are computed by varying the strength of dissipation. The simulations either tend to a laminar steady state, or a turbulent flow with dynamics occurring at a wide range of length and time scales. The laminar results show similar behaviours in both models, thus QG and SW tend to agree when describing slow, large-scale flows. The turbulent simulations begin to differ as QG breaks down when faster and smaller scale motions occur. Essential differences in the underlying assumptions between the QG and SW models are highlighted using the results from the numerical simulations.

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) and partial differential equations (PDEs) that describe the behaviour of this electrical activity. Generating an efficient numerical solution of these models is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this thesis, we discuss two sets of experiments that test ideas for making the numerical solution process more efficient. In the first set of experiments, we examine the numerical solution of four single cell cardiac electrophysiological models, which consist solely of ODEs. We study the efficiency of using implicit-explicit Runge-Kutta (IMEX-RK) splitting methods to solve these models. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform most methods commonly used in practice for two of the models, and they outperform all methods commonly used in practice for the remaining models. In the second set of experiments, we examine the solution of the bidomain model, a model consisting of both ODEs and PDEs that are typically solved separately. We focus these experiments on numerical methods for the solution of the two PDEs in the bidomain model. The most popular method for this task, the Crank-Nicolson method, produces unphysical oscillations; we propose a method based on a second-order L-stable singly diagonally implicit Runge-Kutta (SDIRK) method to eliminate these oscillations.<p> We find that although the SDIRK method is able to eliminate these unphysical oscillations, it is only more efficient for crude error tolerances.

Optical amplifiers and lasers continue to play its crucial role and they have become an indispensable part of the every fiber optic communication systems being installed from optical network to ultra-long haul systems. It seems that they will keep on to be a promising future technology for high speed, long-distance fiber optic transmission systems. The numerical simulations of the model equations have been already commercialized by the photonic system designers to meet the future challenges. One of the challenging problems for designing Raman amplifiers or lasers is to develop a numerical method that meets all the requirements such as accuracy, robustness and speed. In the last few years, there have been much effort towards solving the coupled differential equations of Raman model with high accuracy and stability. The techniques applied in literature for solving propagation equations are mainly based on the finite differences, shooting or in some cases relaxation methods. We have described a new method to solve the nonlinear equations such as Newton-Krylov iteration and performed numerical simulations using spectral methods. A novel algorithm implementing spectral method (pseuodspectral) for solving the two-point boundary value problem of propagation equations is proposed, for the first time to the authors&#039 / knowledge in this thesis. Numerical results demonstrate that in a few iterations great accuracy is obtained using fewer grid points.