Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

Integral equation has been one of the essential tools for various area of applied mathematics. In this work, we employed different numerical methods for solving both linear and nonlinear Fredholm integral equations. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. This work, specially, deals with the development of different wavelet methods for solving integral and intgro-differential equations. Wavelets theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms. Wavelets can be separated into two distinct types, orthogonal and semi-orthogonal. The preliminary concept of integral equations and wavelets are first presented in Chapter 1. Classification of integral equations, construction of wavelets and multi-resolution analysis (MRA) have been briefly discussed and provided in this chapter. In Chapter 2, different wavelet methods are constructed and function approximation by these methods with convergence analysis have been presented. In Chapter 3, linear semi-orthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations (both linear and nonlinear) of the second kind and their systems. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Convergence analysis of B-spline method has been discussed in this chapter. Again, in Chapter 4, system of nonlinear Fredholm integral equations have been solved by using hybrid Legendre Block-Pulse functions and xiii Bernstein collocation method. In Chapter 5, two practical problems arising from chemical phenomenon, have been modeled as Fredholm- Hammerstein integral equations and solved numerically by different numerical techniques. First, COSMO-RS model has been solved by Bernstein collocation method, Haar wavelet method and Sinc collocation method. Second, Hammerstein integral equation arising from chemical reactor theory has been solved by B-spline wavelet method. Comparison of results have been demonstrated through illustrative examples. In Chapter 6, Legendre wavelet method and Bernoulli wavelet method have been developed to solve system of integro-differential equations. Legendre wavelets along with their operational matrices are developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. Also, nonlinear Volterra weakly singular integro-differential equations system has been solved by Bernoulli wavelet method. The properties of these wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton's method. Rigorous convergence analysis has been done for these wavelet methods. Illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. In Chapter 7, we have solved the second order Lane-Emden type singular differential equation. First, the second order differential equation is transformed into integro-differential equation and then solved by Legendre multi-wavelet method and Chebyshev wavelet method. Convergence of these wavelet methods have been discussed in this chapter. In Chapter 8, we have developed a efficient collocation technique called Legendre spectral collocation method to solve the Fredholm integro-differential-difference equations with variable coefficients and system of two nonlinear integro-differential equations which arise in biological model. The proposed method is based on the Gauss-Legendre points with the basis functions of Lagrange polynomials. The present method reduces this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by Newton's method. The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. In Chapter 9, we have proposed some numerical techniques for solving fuzzy integral equations and fuzzy integro-differential equations. Fundamentals of fuzzy calculus have been discussed in this chapter. Nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with homotopy analysis method. We have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by Bernoulli wavelet method and then compared with B-spline wavelet method. Finally, fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with homotopy analysis method. In fuzzy case, we have applied two-dimensional numerical methods which are discussed in chapter 2. Convergence analysis and error estimate have been also provided for Bernoulli wavelet method. xiv The study of fractional calculus, fractional differential equations and fractional integral equations has a great importance in the field of science and engineering. Most of the physical phenomenon can be best modeled by using fractional calculus. Applications of fractional differential equations and fractional integral equations create a wide area of research for many researchers. This motivates to work on fractional integral equations, which results in the form of Chapter 10. First, the preliminary definitions and theorems of fractional calculus have been presented in this chapter. The nonlinear fractional mixed Volterra-Fredholm integro-differential equations along with mixed boundary conditions have been solved by Legendre wavelet method. A numerical scheme has been developed by using Petrov-Galerkin method where the trial and test functions are Legendre wavelets basis functions. Also, this method has been applied to solve fractional Volterra integro-differential equations. Uniqueness and existence of the problem have been discussed and the error estimate of the proposed method has been presented in this work. Sinc Galerkin method is developed to approximate the solution of fractional Volterra-Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the Sinc function approximation. Uniqueness and existence of the problem have been discussed and the error analysis of the proposed method have been presented in this chapte

Various Techniques for De-noise Image

Wavelet decomposition has a great role in eliminating noise, the aim of this work on different noise removal techniques by analyzing the color image. Based on the analysis of different image compression techniques, this paper provides a survey of existing research papers. Different types of method for noise are analyzed from the necessary image, where the disturbance was removed using wavelets with basic theories, and the most important details that will be presented in this work, which clarifies the proposed smooth and effective theory in terms of accuracy in our results. By creating new algorithms that explain how to use the proposed theory, some medical applications were used Discrete Wavelet Transform (DWT) where the results were satisfactory obtained, our proposed theory has been proven to be effective, and examples used will demonstrated this method

The Effect of Set Partitioning in Hierarchical Trees with Wavelet Decomposition Levels Algorithm for Image Compression

The transfer and storage of images is done through compression technology using various transformations to find suitable transformations for image analysis and compression, wavelet based images are analyzed with image compression technology, which is necessary for channel image transmission. The purpose of this article is to determine the appropriate wavelengths for compressing images by recording the parameters that are created by compressing and using an appropriate compression method. The compression method Set Partitioning in Hierarchical Trees (SPIHT) is used to obtain a better compression of the image with a high compression ratio using different wavelets and to compare the results that the techniques were implemented in the Matlab program through the results such as basic criteria for the compressed image quality scale

The Continuity of Fuzzy Metric Spaces

In the present paper, the definitions of a fuzzy continuous function and uniformly fuzzy continuous  are introduced. we prove that a function  from a fuzzy metric space ( ) into a fuzzy metric space ( )  is fuzzy continuous if and only if for every fuzzy open subset  of ,   is fuzzy open in . Also the composition function of two uniformly fuzzy continuous functions is proved to be a uniformly fuzzy continuous function

Bounded and Continuous Linear Operators on Linear 2-Normed Space

In this article, at first basic definitions and properties of linear 2-normed space are presented. Then the author defines bounded operator as an introduction to defined norm of an operator. In addition, the definition of continuous operator on a linear 2-normed space is given. This article proved an operator Γ from a linear 2-normed space (U,〖||.||〗_U) into a linear 2-normed space (V, 〖||.||〗_V) is bounded operator if and only if Γ is continuous operator

Collocation Orthonormal Berntein Polynomials method for Solving Integral Equations

In this paper, we use a combination of Orthonormal Bernstein functions on the interval  for degree ,and 6 to produce anew approach implementing Bernstein Operational matrix of derivative as a method for the numerical solution of linear Fredholm integral equations of the second kind and Volterra integral equations. The method converges rapidly to the exact solution and gives very accurate results even by low value of m. Illustrative examples are included to demonstrate the validity and efficiency of the technique and convergence of method to the exact solution. Keywords: Bernstein polynomials, Operational Matrix of Derivative, Linear Fredholm Integral Equations of the Second  Kind and Volterra Integral Equations

Parameters Estimation for Mathematical Model of Solar Cell

This paper presents, a simplified accuracy solar cell mathematical model is suggested depend on the analysis of single-diode PV cell mathematical model, and afford a parameter determination method depend on two methods Newton-Raphson method (NRM). The voltage of the single-diode is measured numerically based on NRM, then the current and power of the diode is predicted with the variable resistance parameter characteristics are tested under different values of load resistance  from (1-5) Ω under room temperature conditions. The results show that the proposed mathematical model (equation) can quickly and accurately for the PV model I-V and P-V characteristics, which have good methods, and supplies strong support for solar cell system related work