In the literature, many methods are proposed to value American options. However, due to computational difficulty, there are only approximate solution or numerical method to evaluate American options. It is not easy for general investors either to understand nor to apply. In this thesis, we build up an option pricing and virtual asset model system, which provides a friendly environment for general public to calculate early exercise boundary of an American option. This system modularize the well-handled pricing models to provide the investors an easy way to value American options without learning difficult financial theories. The system consists two parts: the first one is an option pricing system, the other one is an asset model simulation system. The option pricing system provides various option pricing methods to the users; the virtual asset model system generates virtual asset prices for different underlying models.

Contactless electrical impedance imaging technique is based upon the measurement of secondary electromagnetic fields caused by induced currents inside the body. In this study, a circular single-coil is used as a transmitter and a receiver. The purpose of this study is twofold: (1) to solve the induced current density distribution inside the realistic head model resulting from a sinusoidal excitation, (2) to calculate the impedance change of the same coil from the induced current distribution inside the head model. The Finite Difference Method is used to solve the induced current density in the head. The realistic head model is formed by seven tissues with a 1 mm resolution. The electrical properties of the model are assigned as a function of frequency. The quasi-stationary assumptions, especially for head tissues, are explored. It is shown that, numerical solution of only the scalar potential is sufficient to obtain the induced current density in the head below 10 MHz operating frequency. This simplification not only reduce the excessive size of the solution domain, but also reduces the number of unknowns by a factor of 4. For higher frequencies (depending on the application) induction and propagation effects become important. Additionally it is observed that dynamic monitoring of hemorrhage at any frequency seems feasible. It is concluded that the methodology provides useful information about the electrical properties of the human head via contactless measurements and has a potent as a new imaging modality for different clinical applications.

The globalization leads to integration of the economies worldwide. As the firms&#039 / businesses also get integrated with each other, the financing choices of the firms diversify. Among these choices, the popularity and the share of foreign currency borrowing in total borrowing by non-financial firms increase in Turkey similar to the global developments. The main purpose of this thesis is to price the risks of default and financial distress due to foreign currency denominated loans of non-financial firms in Turkey. The valuation model of foreign currency corporate loans is established by two state variable option pricing model based on the study of Cox, Ingersoll and Ross. In our model, the main risk factors are identified as the exchange rate and the interest rate, which are the state variables of the main partial differential equation whose solution gives the value of the asset. The numerical results are tested for different parameters and for different economic environments. The findings show that interest rate fluctuations are more important both for the default and financial distress option values than the fluctuations in exchange rate. However, the effect of upside movements of exchange rate on the financial distress and default values is sharper than the downside movement effect of interest rate. Furthermore, high loan-to-value (LTV) foreign currency loans result in significantly high financial distress values that cannot be disregarded and can lead to default of the firm. To the best of our knowledge, this thesis is the first study that develops a structural model to evaluate foreign currency denominated corporate loans in an option-pricing framework.

This study deals with the use of the generalized finite difference method (GFDM) in perfectly matched layer (PML) analysis of the problems in wave mechanics, in particular, in elastodynamics. It is known that PML plays the role of an absorbing layer, for an unbounded domain, eliminating reflections of waves for all directions of incidence and frequencies. The study is initiated for purpose of detecting any possible advantages of using GFDM in PML analysis: GFDM is a meshless method suitable for any geometry of the domain, handling the boundary conditions properly and having an easy implementation for PML analysis. In the study, first, a bounded 2D fictitious plane strain problem is solved by GFDM to determine its appropriate parameters (weighting function, radius of influence, etc.). Then, a 1D semi-infinite rod on elastic foundation is considered to estimate PML parameters for GFDM. Finally, the proposed procedure, that is, the use of GFDM in PML analysis, is assessed by considering the compliance functions (in frequency domain) of surface and embedded rigid strip foundations. The surface foundation is assumed to be supported by three types of soil medium: rigid strip foundation on half space (HS), on soil layer overlying rigid bedrock, and on soil layer overlying HS. For the embedded rigid strip foundation, the supporting soil medium is taken as HS. In addition of frequency space analyses stated above, the direct time domain analysis is also performed for the reaction forces of rigid strip foundation over HS. The results of GFDM for both frequency and time spaces are compared with those of finite element method (FEM) with PML and boundary element method (BEM), when possible, also with those of other studies. The excellent matches observed in the results show the reliability of the proposed procedure in PML analysis (that is, of using GFDM in PML analysis).

The thesis analyzes the third-order one-dimensional pseudoparabolic equations with two types of nonlocal conditions. The stability of difference schemes for this problem was studied using the analysis of the spectrum structure of a difference operator with nonlocal conditions. The analysis of the increased accuracy difference schemes for third-order one-dimensional and two-dimensional pseudoparabolic equations with integral conditions has been made. The thesis considers a two-dimensional pseudoparabolic equation with nonlocal integral conditions in one coordinate direction. This problem was solved by a locally one-dimensional method. The stability of a difference scheme has been investigated based on the spectrum structure. The doctoral disertation investigates three-layer difference schemes for one-dimensional pseudoparabolic equations with various, including nonlocal, conditions. Also, the conditions for the stability of three-layer explicit difference schemes have been explored. / Disertacijoje išnagrinėta trečiosios eilės vienmatė pseudoparabolinė lygtis su dviejų tipų nelokaliosiomis sąlygomis. Šiems uždaviniams spręsti sudarytos skirtuminės schemos, kurių stabilumas tiriamas, taikant skirtuminių operatorių su nelokaliosiomis sąlygomis spektro struktūrą. Trečiosios eilės vienmatėms ir dvimatėms pseudoparabolinėms lygtims su integralinėmis sąlygomis sudarytos ir išnagrinėtos padidinto tikslumo skirtuminės schemos. Išnagrinėta dvimatė pseudoparabolinė lygtis su nelokaliosiomis integralinėmis sąlygomis viena koordinačių kryptimi. Tokiam uždaviniui spręsti pritaikytas ir išnagrinėtas lokaliai vienmatis metodas, ištirtos šio metodo stabilumo sąlygos. Taip pat išnagrinėtos: trisluoksnės skirtuminės schemos vienmatei pseudoparabolinei lygčiai su įvairiomis, taip pat ir nelokaliosiomis, sąlygomis; trisluoksnių išreikštinių skirtuminių schemų stabilumo sąlygos.

Darbo tikslas - išnagrinėti dvimatės parabolinio tipo lygties su nelokaliąja integraline sąlyga sprendimą baigtinių skirtumų metodu. Išnagrinėtas kintamųjų krypčių metodo algoritmas tokiam uždaviniui spręsti. Išnagrinėtas dvimatės parabolinės lygties su keliomis nelokaliosiomis integralinėmis kraštinėmis sąlygomis sprendimas kintamųjų krypčių metodu. Uždavinio sprendinys randamas papildomai išsprendžiant neaukštos eilės algebrinę tiesinių lygčių sistemą, kuri sudaroma panaudojant nelokaliąsias integralines sąlygas. Išanalizuota skirtuminio operatoriaus su nelokaliosiomis sąlygomis spektro struktūra. Spektro struktūra išanalizuota tuo tikslu, kad galima būtų išnagrinėti dvimačio parabolinio uždavinio su viena nelokaliąja integraline sąlyga sprendžiamo kintamųjų krypčių ar lokaliai vienmačiu metodu, stabilumą. Nustatyta nelokaliosios sąlygos įtaka spektro struktūrai. Sudarytas elipsinio uždavinio su papildoma nelokaliąja sąlyga sprendimo algoritmas. / The aim of the work is to analyze the finite difference method for solving two-dimensional parabolic equation with an integral boundary condition. The alternating direction method for solving the problem of this kind is analyzed. This method is applied the alternating direction method for solving two-dimensional parabolic equation with two nonlocal integral condition is analyzed. Solution of the problem is found by resolving an additional linear system of equations of lower order . Structure of the spectrum for difference operator with nonlocal condition is analyzed. In order to analyze stability of two-dimensional parabolic equation with one integral condition the structure of spectrum is analyzed. Influence of nonlocal condition for structure of the spectrum is determined. The finite difference method for elliptic problem is constructed.

The aim of the work is to analyze the finite difference method for solving two-dimensional parabolic equation with an integral boundary condition. The alternating direction method for solving the problem of this kind is analyzed. This method is applied the alternating direction method for solving two-dimensional parabolic equation with two nonlocal integral condition is analyzed. Solution of the problem is found by resolving an additional linear system of equations of lower order . Structure of the spectrum for difference operator with nonlocal condition is analyzed. In order to analyze stability of two-dimensional parabolic equation with one integral condition the structure of spectrum is analyzed. Influence of nonlocal condition for structure of the spectrum is determined. The finite difference method for elliptic problem is constructed. / Darbo tikslas - išnagrinėti dvimatės parabolinio tipo lygties su nelokaliąja integraline sąlyga sprendimą baigtinių skirtumų metodu. Išnagrinėtas kintamųjų krypčių metodo algoritmas tokiam uždaviniui spręsti. Išnagrinėtas dvimatės parabolinės lygties su keliomis nelokaliosiomis integralinėmis kraštinėmis sąlygomis sprendimas kintamųjų krypčių metodu. Uždavinio sprendinys randamas papildomai išsprendžiant neaukštos eilės algebrinę tiesinių lygčių sistemą, kuri sudaroma panaudojant nelokaliąsias integralines sąlygas. Išanalizuota skirtuminio operatoriaus su nelokaliosiomis sąlygomis spektro struktūra. Spektro struktūra išanalizuota tuo tikslu, kad galima būtų išnagrinėti dvimačio parabolinio uždavinio su viena nelokaliąja integraline sąlyga sprendžiamo kintamųjų krypčių ar lokaliai vienmačiu metodu, stabilumą. Nustatyta nelokaliosios sąlygos įtaka spektro struktūrai. Sudarytas elipsinio uždavinio su papildoma nelokaliąja sąlyga sprendimo algoritmas.

The objectives of this study are to investigate the dynamic behavior of an apple slab subjected to drying at constant external conditions and under changing in the drying temperatures and to determine the effects of temperature and time combinations at different steps during drying on the process dynamics parameters, time constant and process gain of the system. For this purpose, a semi-batch dryer system was simulated by using integral method of analysis. Initially, the dynamic behavior of the drying temperature was investigated by using first order system dynamic model. Process dynamic parameters, time constant and process gain of the system, for change in drying temperature were determined. Secondly, investigation of the drying kinetics of the apple slab was carried out under constant external conditions in a semi-batch dryer. A mathematical model for diffusion mechanism assumed in one dimensional transient analysis of moisture distribution was solved by using explicit finite difference method of analysis. Thirdly, investigation of the drying kinetics of the apple slab was carried out under change in drying temperature at different time steps during drying. Inverse response system model was used for the representation of the dynamic behavior of drying. Process dynamic parameters, time constant and process gain of the system were determined. Model predicted results for apple slab drying under constant external condition and under step change in the drying temperature were compared with the experimental data.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Chih-Hang Wu / The fast advancement in computer technologies in the recent years has made the use of simulation to estimate stock/equity performances and pricing possible; however, determining the optimal exercise time and prices of American options using Monte-Carlo simulation is still a computationally challenging task due to the involved computer memory and computational complexity requirements. At each time step, the investor must decide whether to exercise the option to get the immediate payoff, or hold on to the option until a later time. Traditionally, the stock options are simulated using Monte-Carlo methods and all stock prices along the path are stored, and then the optimal exercise time is determined starting at the final time period and continuing backward in time. Also, as the number of paths simulated increases, the number of simultaneous equations that need to be solved at each time step grow proportionally. Currently, two theoretical methods have emerged in determining the optimal exercise problem. The first method uses the concept of least-squares approach in linear regression to estimate the value of continuing to hold on to the option via a set of randomly generated future stock prices. Then, the value of continuing can be compared to the payoff at current time from exercising the option and a decision can be reached, which gives the investor a higher value. The second method uses the finite difference approach to establish an exercise boundary for the American option via an artificially generated mesh on both possible stock prices and decision times. Then, the stock price is simulated and the method checks to see if it is inside the exercise boundary. In this research, these two solution approaches are evaluated and compared using discrete event simulation. This allows complex methods to be simulated with minimal coding efforts. Finally, the results from each method are compared. Although a more conservative method cannot be determined, the least-squares method is faster, more concise, easier to implement, and requires less memory than the mesh method. The motivation for this research stems from interest in simulating and evaluating complicated solution methods to the optimal exercise problem, yet requiring little programming effort to produce accurate and efficient estimation results.

Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.