There are some nonlinear models for pricing financial derivatives which can improve the linear Black-Scholes model introduced by Black, Scholes and Merton. In these models volatility is not constant anymore, but depends on some extra variables. It can be, for example, transaction costs, a risk from a portfolio, preferences of a large trader, etc. In this thesis we focus on these models. In the first chapter we introduce some important theory of financial derivatives. The second chapter is devoted to the volatility models. We derive three models concerning transaction costs (RAPM, Leland's  and Barles-Soner's model) and Frey's model which assumes a large (dominant) trader on the market. In the third and in the forth chapter we derive portfolio and make numerical experiments with a free boundary. We use the first order additive and the second order Strang splitting methods. We also use approximations of Barles-Soner's model using the identity function and introduce an approximation with the logarithm function of Barles-Soner's model. These models we finally compare with models where the volatility includes constant transaction costs.

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.

In this thesis a finite element model of a household washing machine cylinder is built and analysed in ABAQUS 6.9-2. The aim is to help Asko appliances in conducting similar analysis for future manufacturing of high capacity cylinders by reducing experimentation. The analysis is mainly concerned with an evenly distributed load at a constant angular velocity. The load is applied with the help of lead plates instead of clothes. The cylinder is loaded with three thin (2 mm) lead plates weighing 2 kg each. The plates with dimensional 370x240x2 mm are mounted with one strip of double sided foam tape inside the cylinder. To estimate the behavior of the cylinder the strains are measured when the cylinder is rotating at 1620 and 2200 revolution per minute (rpm). To validate the model the numerical analyses are compared with experimental results. The results clearly show that the numerical strain values fit with experimental strain values.

In this thesis we investigate Asian oating strike options. We particu-larly focus on options with early exercise - American options. This typeof options are very lucrative to the end-users of commodities or ener-gies who are tend to be exposed to the average prices over time. Asianoptions are also very popular with corporations, who have ongoing cur-rency exposures. The main idea of the pricing is to examine the freeboundary position on which the value of the option is depending. Wefocus on developing a ecient numerical algorithm for this boundary.In the rst Chapter we give an informative description of the nancialderivatives including Asian options. The second Chapter is devoted tothe analytical derivation of the corresponding partial dierential equa-tion coming from the original Black - Scholes equation. The problemis simplied using transformation methods and dimension reduction. Inthe third and fourth Chapter we describe important numerical methodsand discretize the problem. We use the rst order Lie splitting and thesecond order Strang splitting. Finally, in the fth Chapter we makenumerical experiments with the free boundary and compare the resultwith other known methods.

Asian options are exotic financial derivative products which price must be calculated by numerical evaluation. In this thesis, we study certain ways of solving partial differential equations, which are associated with these derivatives. Since standard numerical techniques for Asian options are often incorrect and impractical, we discuss their variations, which are efficiently applicable for handling frequent numerical instabilities reflected in form of oscillatory solutions. We will show that this crucial problem can be treated and eliminated by adopting flux limiting techniques, which are total variation dimishing.

A meshfree approximation scheme based on the radial basis function methods is presented for the numerical solution of the options pricing model. This thesis deals with the valuation of the European, Barrier, Asian, American options of a single asset and American options of multi assets. The option prices are modeled by the Black-Scholes equation. The θ-method is used to discretize the equation with respect to time. By the next step, the option price is approximated in space with radial basis functions (RBF) with unknown parameters, in particular, we con- sider multiquadric radial basis functions (MQ-RBF). In case of Ameri- can options a penalty method is used, i.e. removing the free boundary is achieved by adding a small and continuous penalty term to the Black- Scholes equation. Finally, a comparison of analytical and finite difference solutions and numerical results from the literature is included.

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In this masters’ thesis report we present a graph theoretical method used for automatic bilingual lexicon acquisition with parallel texts. We analyze the concept of graph similarity and give an interpretation, of the parallel texts, connected to the vector space model. We represent the parallel texts by a directed, tripartite graph and from here use the corresponding adjacency matrix, A, to compute the similarity of the graph. By solving the eigenvalue problem ρS = ASAT + ATSA we obtain the self-similarity matrix S and the Perron root ρ. A rank k approximation of the self-similarity matrix is computed by implementations of the singular value decomposition and the non-negative matrix factorization algorithm GD-CLS. We construct an algorithm in order to extract the bilingual lexicon from the self-similarity matrix and apply a statistical model to estimate the precision, the correctness, of the translations in the bilingual lexicon. The best result is achieved with an application of the vector space model with a precision of about 80 %. This is a good result and can be compared with the precision of about 60 % found in the literature.

<p>Consider a mixture of two immiscible, incompressible fluids e.g. oil and water. Since the fluids do not mix, an interface between the two fluids will form and move in time. The motion of the two fluids can be modelled by the incompressible Navier-Stokes equations for two phase flow with surface tension together with a representation of the moving interface. The parameters in the Navier-Stokes equations will depend on the position and other properties of the interface. The interface should move with the velocity of the flow at the interface. Since the fluids are incompressible, the density of each fluid is constant. Mass conservation then implies that the volume occupied by each of the two fluids should not change with time. The object of this thesis has been to develop a new numerical method to simulate incompressible two phase flow accurately that conserves mass and volume of each fluid correctly.</p><p>Numerical simulations of incompressible two phase flow with surface tension have been a challenge for many years. Several methods have been developed and used prior to the work presented in this thesis. The two most commonly used methods are volume of fluid methods and level set methods. There are advantages and disadvantages of both of the methods.</p><p>In volume of fluid methods the interface is represented by a discontinuity of a globally defined function. Because of the discontinuity it is hard both to move the interface as well as to calculate properties of the interface such as curvature. Specially designed methods have to be used, and all these methods are low order accurate. Volume of fluid methods do however conserve the volumes of the two fluids correctly.</p><p>In level set methods the interface is represented by the zero contour of the globally defined signed distance function. This function is smooth across the interface. Since the function is smooth, standard methods for partial differential equations can be used to advect the interface accurately. A reinitialization is however needed to make sure that the level set function remains a signed distance function. During this process the zero contour might move slightly. Because of this, the volume conservation of the method becomes poor.</p><p>In this thesis we present a new level set method. The method is designed such that the volume of each fluid is conserved, at least approximately. The interface is represented by the 0.5 contour of a regularized characteristic function. As for standard level set methods, the interface is moved first by an advective step, and then reinitialized. Unlike traditional level set methods, we can formulate the reinitialization as a conservation law. Conservative methods can then be used to move and to reinitialize the level set function numerically. Since the level set function is a regularized characteristic function, we can expect good conservation of the volume bounded by the interface.</p><p>The method is discretized using both finite differences and finite elements. Uniform and adaptive grids are used in both two and three space dimensions. Good convergence as well as volume conservation is observed. Theoretical studies are performed to investigate the conservation and the computational time needed for reinitialization.</p>