This document presents a numerical solution and comparison of linear Black-Scholes models using finite difference and finite element methods. It begins with an introduction to the Black-Scholes partial differential equation and previous analytical and numerical solutions in the literature. The document then transforms the Black-Scholes equation into a heat equation and presents the finite element formulation and discretization. Numerical results are obtained for the European call and put options and compared between finite difference and finite element methods.
This chapter discusses integration, including defining indefinite integrals, evaluating definite integrals, and techniques for integration. The key topics covered include:
- Defining antiderivatives and indefinite integrals, and using properties like linearity to evaluate integrals.
- Applying integration to solve problems involving rates of change, such as calculating total cost from a marginal cost function.
- Evaluating definite integrals to find the area under a curve over a specified interval.
- Covering techniques for integrating common functions like polynomials, exponentials, and logarithms using rules like power, substitution and integration by parts.
This document contains sample questions from a mathematics exam blueprint and marking scheme for Class 12. It includes:
- A blueprint showing the distribution of questions across different units of the syllabus for very short answer (1 mark), short answer (4 marks) and long answer (6 marks) questions.
- Sample questions from sections A to D with varying marks. The questions cover topics like relations and functions, matrices, calculus, vectors, probability and linear programming.
- A marking scheme providing solutions to the sample questions with marks allocated for each step.
The document discusses the Gram-Schmidt process for finding an orthogonal basis. It presents the Gram-Schmidt algorithm in 3 steps: 1) the first vector in the orthogonal basis is the first vector in the original basis, 2) subsequent vectors are found by removing projections onto previous vectors, 3) repeat step 2 for all other vectors. It also provides an example applying Gram-Schmidt to find an orthogonal basis for the space spanned by three vectors.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIRai University
This document provides an overview of Unit II - Complex Integration from the Engineering Mathematics-IV course at RAI University, Ahmedabad. It covers key topics such as:
1) Complex line integrals and Cauchy's integral theorem which states that the integral of an analytic function around a closed curve is zero.
2) Cauchy's integral formula which can be used to evaluate integrals and find derivatives of analytic functions.
3) Taylor and Laurent series expansions of functions, including their regions of convergence.
4) The residue theorem which can be used to evaluate real integrals involving trigonometric or rational functions.
1. The orthogonal decomposition theorem states that any vector y in Rn can be written uniquely as the sum of a vector ŷ in a subspace W and a vector z orthogonal to W.
2. The vector ŷ is called the orthogonal projection of y onto W. It is the closest vector to y that lies in W.
3. The best approximation theorem states that the orthogonal projection ŷ provides the best or closest approximation of y using only vectors that lie in the subspace W. The distance from y to ŷ is less than the distance from y to any other vector in
Derivation and Application of Six-Point Linear Multistep Numerical Method for...IOSR Journals
A six-step Continuous Block method of order (5, 5, 5, 5, 5, 5) T is proposed for direct solution of the second (2nd) order initial value problems. The main method and additional ones are obtained from the same continuous interpolant derived through interpolation and collocation procedures. The methods are derived by interpolating the continuous interpolant at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 6 and collocating the first and second derivative of the
continuous interpolant at 𝑥𝑛+𝑗 , 𝑗 = 0 and 𝑗 = 2, 3, … 5 respectively. The stability properties of the methods are discussed and the stability region shown. The methods are then applied in block form as simultaneous numerical integrators. Two numerical experiments are given to illustrate the efficiency of the new methods.
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
1. The document discusses properties of vectors and systems of linear equations, including the parallelogram rule for vector addition, properties of linear combinations of vectors, homogeneous and nonhomogeneous systems, and writing solutions to consistent systems in parametric vector form.
2. Key concepts covered include the definition of the zero vector, properties of vector addition and scalar multiplication, the relationship between matrix-vector equations and systems of linear equations, and determining whether a vector can be written as a linear combination of other vectors.
3. Methods are presented for determining if a homogeneous system has nontrivial solutions, describing the solution set of a homogeneous system, and writing the solution set of a consistent nonhomogeneous system in parametric vector form
This document provides information about solving linear simultaneous equations and simultaneous equations involving one linear and one quadratic equation. It discusses several methods for solving these types of equations including: the elimination method, the substitution method, and finding where the graphs of the equations intersect. It also discusses solving linear inequalities and how to check solutions to inequalities.
This document provides an overview of vector integration, including line integrals, surface integrals, and volume integrals. It defines each type of integral and provides examples of evaluating them. For line integrals, it discusses work, circulation, and path independence. Surface integrals are defined as the integral of the normal component of a vector field over an enclosed surface. Volume integrals integrate a vector field over a three-dimensional volume. Worked examples demonstrate evaluating specific line, surface, and volume integrals.
This chapter discusses additional topics in differentiation including:
- Derivatives of logarithmic and exponential functions
- Elasticity of demand
- Implicit differentiation
- Logarithmic differentiation
- Newton's method for approximating roots
- Finding higher-order derivatives directly and implicitly.
Examples are provided for each topic to illustrate the differentiation techniques.
The document discusses functions and graphs in chapter 2 of an introductory mathematical analysis textbook. It introduces key concepts such as functions, domains, ranges, combinations of functions, inverse functions, and graphs in rectangular coordinates. It provides examples of determining equality of functions, finding function values, combining functions, and finding inverses. It also discusses special functions, graphs, symmetry, and intercepts. The chapter aims to define functions and domains, introduce different types of functions and their operations, and familiarize students with graphing equations and basic function shapes.
The document discusses orthogonal bases and the Gram-Schmidt process. It defines the Gram-Schmidt process as an algorithm for finding an orthogonal basis from a given basis by making each new vector orthogonal to the previous ones. It also discusses orthonormal bases, QR factorization, inner products, length and distance, and applying Gram-Schmidt to produce an orthogonal basis for the vector space of polynomials up to degree 2.
This document summarizes Chapter 9 from a textbook on introductory mathematical analysis. Section 9.1 discusses discrete random variables and expected value. Section 9.2 covers the binomial distribution and how it relates to the binomial theorem. Section 9.3 introduces Markov chains and their associated transition matrices. Examples are provided for each topic to illustrate key concepts like calculating expected values, applying the binomial distribution formula, and determining probabilities using Markov chains.
This document appears to be the table of contents and problems from Chapter 0 of a mathematics textbook. The table of contents lists 17 chapters and their corresponding page numbers. The problems cover a range of algebra topics including integers, rational numbers, properties of operations, solving equations, and rational expressions. There are over 70 problems presented without solutions for students to work through.
B.tech ii unit-5 material vector integrationRai University
This document discusses various vector integration topics:
1. It defines line, surface, and volume integrals and provides examples of evaluating each. Line integrals deal with vector fields along paths, surface integrals deal with vector fields over surfaces, and volume integrals deal with vector fields throughout a volume.
2. Green's theorem, Stokes' theorem, and Gauss's theorem are introduced as relationships between these types of integrals but their proofs are not shown.
3. Examples are provided to demonstrate evaluating line integrals of conservative and non-conservative vector fields, as well as a surface integral over a spherical surface.
This chapter discusses differentiation, including:
- Defining the derivative using the limit definition of the slope of a tangent line.
- Basic differentiation rules for constants, polynomials, sums and differences.
- Interpreting the derivative as an instantaneous rate of change.
- Applying the product rule and quotient rule to differentiate products and quotients.
- Using differentiation to find equations of tangent lines, velocities, marginal costs, and other rates of change.
Lec. 6 the explanation of tipitaka in brief by atthakathacariyaBhik Samādhipuñño
The document discusses the Atthakathacariyas, who composed the early Pali commentaries. It describes how Buddhaghosa built upon their work in the 5th century CE when compiling his famous commentaries. It also mentions other important commentators like Dhammapala, Upasena, and Buddhadatta who further expanded on the commentarial tradition in Sri Lanka. Overall, the document provides historical context about the authors and development of the Pali commentarial literature.
Presented by Carol Roye, EdD, CPNP, RN, Professor of Nursing, Assistant Dean for Research, Hunter College School of Nursing at the 2013 National Chlamydia Coalition Meeting
INTERROGATING THE IMPORTANCE AND RELEVANCE OF ARABIC LANGUAGE TO THE STUDY OF...SCHOLEDGE R&D CENTER
This paper attempts interrogating Arabic language as a language, the importance and relevance to the study of Shari’ah generally. It demonstrates that Arabic, is a medium of communication, not a sacred language as some believe. Prophet Muhammad received his message from God in Arabic and with the rise of Islam, Arabic shifted from a little-known tribal language to the lingua franca for the Muslim world and plays great role in international affairs today. The study found that the Eleventh century marked a period of stagnation for Arabic language but its status as the language of Islam was never threatened. Shari’ah’s language remains Arabic in which it was revealed and which the language of the prophet Muhammad is. Thus, the understanding of the rules of law from the Qur’an and the Sunnah can only be derived if stylistic peculiarities of Arabic language, its lexical meanings and structure are understood. All sources of Shari’ah and contributions of jurists to it have been preserved in Arabic. Prayers and pilgrimage were to be observed with Arabic. The paper discovers that, Classical Arabic has a vocabulary in which the meaning of each root-word is so comprehensive that it is difficult to interpret it in a modern analytical language word for word, or by the use of the same word in all places where the original word occurs in the text. Thus, study of Shari’ah without the least knowledge of Arabic may be as futile as dealing with English law without the knowledge of English language.
The document provides guidance on formulating a research hypothesis for an interdisciplinary paper. It explains that a research hypothesis is a statement that predicts the relationship between two or more variables that will be tested through research. It highlights that a strong hypothesis involves at least two disciplines, including law, and presents sample student hypotheses as examples. The document outlines tips for crafting a hypothesis such as being objective, clearly defining terms, and obtaining approval from a professor.
The document provides instructions for completing an observation procedure. It states that the observer should document where, when, and what they did for the observation, including creating a coding scheme. It notes that a complete procedure should provide enough information for someone else to replicate the observation without asking any questions. It asks if any information has been missed from the procedure.
Chapter 6 the review of related literature and studiesMaria Theresa
Here are the steps to take to write a literature review:
1. Define your research topic. Your literature review should be focused on a specific area related to your research problem or question.
2. Search academic databases and other sources. Use keywords related to your topic to search databases like Google Scholar, ERIC, PsycINFO, and more.
3. Take detailed notes. As you find relevant sources, take thorough notes including the author, year, title, source, key findings and conclusions. Cite sources using APA or other required style.
4. Organize your sources. Group related sources together around important themes, theories, concepts or debates. This will help structure your review.
5
A bibliography is a list of all sources used in an assignment, organized alphabetically by author's last name. If no author is listed, sources are organized alphabetically by title. The bibliography includes full details of books, book chapters, journal articles, newspaper articles, encyclopedia articles, audiovisual materials, webpages, and personal communications. Personal communications are not included in the bibliography but referenced in-text. The document provides examples of how to format different source types in a bibliography using APA referencing style.
51 ways to introduce learning objectivesDavid Didau
The document provides 51 ways to introduce learning objectives to students in an engaging manner, such as through word games, images, movies, music, coding, translating objectives into other languages, and having students determine objectives through problem solving or at the end of a lesson. Some methods encourage guessing objectives or determining success criteria. A few suggestions note that explicitly stating objectives can sometimes limit learning.
This document contains a list of 133 potential MBA project topics. The topics cover a wide range of business subjects including marketing, finance, human resources, operations management, and more. Some of the topics listed include customer satisfaction studies, investment pattern analyses, brand analyses, capital structure analyses, and export/import procedures. The list provides students with many options for choosing an MBA project on an area of business that interests them.
We disclose a simple and straightforward method of solving ordinary or linear partial differential equations of any order and apply it to solve the generalized Euler-Tricomi equation. The method is easier than classical methods and also didactic.
Date: Jan, 10, 202
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
- The document presents a probabilistic algorithm for computing the polynomial greatest common divisor (PGCD) with smaller factors.
- It summarizes previous work on the subresultant algorithm for computing PGCD and discusses its limitations, such as not always correctly determining the variant τ.
- The new algorithm aims to determine τ correctly in most cases when given two polynomials f(x) and g(x). It does so by adding a few steps instead of directly computing the polynomial t(x) in the relation s(x)f(x) + t(x)g(x) = r(x).
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This summary provides the key details from the document in 3 sentences:
The document discusses a finite difference method for solving a nonlocal singularly perturbed problem. It presents properties of the exact solution and establishes a uniformly convergent finite difference scheme on a Bakhvalov mesh. The scheme is proven to be first-order convergent in the discrete maximum norm and a numerical example is used to validate the theoretical convergence results.
This document presents three methods for numerically solving linear Volterra-Fredholm integro-differential equations (LVFIDEs) of the first order: original Lagrange polynomial method, barycentric Lagrange polynomial method, and modified Lagrange polynomial method. It derives the equations for approximating the solution using each method. The document also provides algorithms to implement each method. It includes some test examples and their numerical solutions to validate the accuracy of the techniques.
The document describes using the homotopy perturbation method to solve the Lane-Emden equation. It first provides an overview of the homotopy perturbation method and Lane-Emden equation. It then constructs the homotopy for n=2 and solves for the first three terms of the solution series. The summary provides the key steps and outcomes while keeping the response to 3 sentences.
This document describes a new decomposition method called the Andualem-Khan Decomposition Method (AKDM) for solving nonlinear differential equations. The AKDM combines the AK transform with the Adomian decomposition method.
The AK transform is a new integral transform introduced by Andualem and Khan in 2022. Properties of the AK transform and how it can be used to solve differential equations are presented.
The document then explains how the AKDM works by applying the AK transform to a general nonlinear differential equation, representing the solution as an infinite series, and decomposing the nonlinearity into Adomian polynomials to generate a recurrence relation for obtaining successive terms of the series solution.
Examples of applying the AKDM to
The study is concerned with a different perspective which the numerical solution of the singularly
perturbed nonlinear boundary value problem with integral boundary condition using finite difference method on
Bakhvalov mesh. So, we show some properties of the exact solution. We establish uniformly convergent finite
difference scheme on Bakhvalov mesh. The error analysis for the difference scheme is performed. The numerical
experiment implies that the method is the first order convergent in the discrete maximum norm, independently of
휀- singular perturbation parameter with effective and efficient iterative algorithm. The numerical results are
shown in table and graphs.
A high accuracy approximation for half - space problems with anisotropic scat...IOSR Journals
An approximate model, which is developed previously, is extended to solve the half – space problems
in the case of extremely anisotropic scattering kernels. The scattering kernel is assumed to be a combination of
isotropic plus a forward and backward leak. The transport equation is transformed into an equivalent fictitious
one involving only multiple isotropic scattering, therefore permitting the application of the previously developed
method for treating isotropic scattering. It has been shown that the method solves the albedo half – space
problem in a concise manner and leads to fast converging numerical results as shown in the Tables. For pure
scattering and weakly absorbing medium the computations can be performed by hand with a pocket calculator
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
A Family Of Extragradient Methods For Solving Equilibrium ProblemsYasmine Anino
The document discusses using variational inequalities and bilevel programming models to analyze the optimal pollution emission price problem. Specifically, it presents a continuous-time central planning model where the government chooses the optimal price of pollution emissions considering how manufacturers in a supply chain will respond to the price. The lower-level problem involves the manufacturers determining their optimal production levels given the emission price, while the upper-level problem involves the government selecting the price to maximize social welfare. Existence of solutions is analyzed using variational inequality theory.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
A Two Grid Discretization Method For Decoupling Time-Harmonic Maxwell’s Equat...IOSR Journals
This document summarizes a two grid discretization method for decoupling time-harmonic Maxwell's equations. The method discretizes the coupled partial differential equations using continuous 1H-conforming finite elements on two grids - a fine grid and a coarser grid. This allows solving the coupled equations on the coarse grid along with decoupled equations on the fine grid, reducing computational costs compared to solving on just the fine grid. The document reviews relevant fundamentals, formulates the variational problem, and proves a regularity result stating the solution has higher regularity if the right hand side is more regular.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
The Scientific journal “Norwegian Journal of development of the International Science” is issued 24 times a year and is a scientific publication on topical problems of science.
On The Distribution of Non - Zero Zeros of Generalized Mittag – Leffler Funct...IJERA Editor
In this work, we derive some theorems involving distribution of non – zero zeros of generalized Mittag – Leffler functions of one and two variables. Mathematics Subject Classification 2010: Primary; 33E12. Secondary; 33C65, 26A33, 44A20
This document provides an overview of solving partial differential equations using the homotopy perturbation method and separation of variables. Key points:
- The document introduces the Laplace, wave, and heat equations and outlines methods to solve them, including homotopy perturbation and separation of variables.
- Homotopy perturbation method involves constructing a homotopy equation with an embedding parameter and expanding the solution as a power series in this parameter.
- Separation of variables involves assuming the solution can be written as a product of functions involving only one variable, leading to ordinary differential equations that can be solved.
- Examples are provided of applying these methods to solve the Laplace equation and estimating the error compared to other methods.
The Weak Solution of Black-Scholes Option Pricing Model with Transaction Costmathsjournal
This document summarizes a research paper that examines the Black-Scholes option pricing model when transaction costs are included. It establishes the existence, uniqueness, and continuity of the weak solution to the Black-Scholes PDE with a transaction cost term added. The paper reviews previous literature on this topic and presents lemmas demonstrating that the weak solution exists and is unique. It then defines approximation solutions and establishes existence of approximation solutions to the PDE, laying the groundwork for demonstrating existence of the true weak solution.
THE WEAK SOLUTION OF BLACK-SCHOLE’S OPTION PRICING MODEL WITH TRANSACTION COSTmathsjournal
The
existence, uniqueness and continuous dependence of the weak solution of the Black-Scholes model with
transaction cost are established.The continuity of weak solution of the parameters was discussed and
similar solution as in literature obtained.
THE WEAK SOLUTION OF BLACK-SCHOLE’S OPTION PRICING MODEL WITH TRANSACTION COSTmathsjournal
The
existence, uniqueness and continuous dependence of the weak solution of the Black-Scholes model with
transaction cost are established.The continuity of weak solution of the parameters was discussed and
similar solution as in literature obtained.
This document proposes methods for generating electricity from speed breakers. It discusses 5 classifications of speed breaker power generators that use different mechanisms: 1) a chain drive mechanism, 2) a rack and pinion system, 3) direct use of the load through a reciprocating device, 4) a translator and stator topology, and 5) a pressure lever mechanism. The document also outlines the advantages of using speed breakers for power generation such as low cost and maintenance and being a renewable source. Some challenges are also noted such as selecting a suitable generator and dealing with rain damage.
Cassava waste water was used as an admixture to replace distilled water in ratios of 5%, 10%, 15%, and 20% for producing sandcrete blocks. 60 sandcrete blocks of size 450mm x 150mm x 225mm were produced with different admixture ratios and a control with 0% admixture. The blocks were cured for 7, 14, 21, and 28 days and then tested for moisture content, specific gravity, water absorption, and compressive strength. Test results showed that blocks with 20% cassava waste water admixture met the minimum compressive strength requirement of 3.30 N/mm2 set by Nigerian standards, indicating the potential of cassava waste water to improve sandcrete block quality and
The document presents a theorem on random fixed points in metric spaces. It begins with introductions to fixed point theory, random fixed point theory, and relevant definitions. The main result is Theorem 3.1, which proves that if a self-mapping E on a complete metric space X satisfies certain contraction conditions involving parameters between 0 and 1, then E has a unique fixed point. The proof constructs a Cauchy sequence that converges to the unique fixed point. The document contributes to the study of random equations and random fixed point theory, which has applications in nonlinear analysis, probability theory, and other fields.
1. The document discusses applying multi-curve reconstruction technology to seismic inversion to improve accuracy and reliability. It focuses on reconstructing SP and RMN curves from well logs that are affected by various distortions.
2. The process of reconstructing the curves involves removing baseline drift, standardizing values, applying linear filtering, and fitting the curves. This removes interference and retains valid lithological information.
3. Reconstructing high quality curves improves the resolution and credibility of seismic inversion results. The method is shown to effectively predict sand distribution with little error.
This document compares the performance of a Minimum-Mean-Square-Error (MMSE) adaptive receiver and a conventional Rake receiver for receiving Ultra-Wideband (UWB) signals over a multipath fading channel. It first describes the UWB pulse shapes and channel model used, including the 6th derivative of the Gaussian pulse and the IEEE 802.15.3a modified Saleh-Valenzuela channel model. It then discusses the Direct-Sequence and Time-Hopping transmission and multiple access schemes for UWB. The document presents the receiver structures for the MMSE adaptive receiver and Rake receiver and compares their performance using MATLAB simulations.
This document summarizes a study on establishing logging interpretation models for reservoir parameters like porosity, permeability, oil saturation, and gas saturation in the Gaotaizi Reservoir of the L Oilfield. Models were developed using core data from 4 wells and include:
1) A porosity model relating acoustic travel time to porosity with an error of 0.92%
2) A permeability model relating permeability to porosity with an error of 0.31%
3) An oil saturation model using resistivity data with empirically determined parameters
4) A method to determine original gas saturation from mercury injection data.
Application of the models improved interpretation precision and allowed recalculation of oil and gas reserves for the
This document discusses predicting spam videos on social media platforms using machine learning. It proposes using attributes like number of likes, comments, and view count to classify videos as spam or not spam. A predictive algorithm is developed that uses threshold values for attributes and natural language processing of comments to classify videos. Testing of the algorithm on a dataset achieved a spam prediction precision of 93.6%. Issues with small datasets decreasing accuracy are also discussed, along with continuing work to address this issue.
1) The study experimentally evaluated the compatibility relationship between polymer solutions and oil layers through core flooding tests with different permeability cores.
2) The results showed that injection rate decreased with increasing polymer concentration and molecular weight, and increased with permeability.
3) Based on the results, boundaries for injection capability were established and a compatibility chart was proposed to guide polymer solution selection for different sedimentary microfacies in the field based on permeability and pore size.
1. The document discusses the identification of lithologic traps in the D3 Member of the Gaonan Region using seismic attribute analysis, acoustic impedance inversion, and sedimentary microfacies analysis.
2. Several lithologic traps were identified in the I and II oil groups of the D3 Member, with the largest trap located between wells G46 and G146X1 covering an area of about 2.35 km2.
3. Impedance inversion, seismic attribute analysis, and sedimentary microfacies characterization using 3D seismic data helped determine the location and development of effective lithologic traps in the thin sandstone-shale interbeds of the target stratum.
This document examines using coal ash as a partial replacement for cement in concrete. Coal ash was substituted for cement at rates of 5%, 10%, and 15% by weight. Testing found that concrete with a 5% substitution of coal ash exhibited only a slight decrease in compressive strength of 2% at 28 days while gaining improved workability. Higher substitution rates of 10% and 15% coal ash led to greater decreases in compressive and tensile strength. The study concludes that a 5% substitution of coal ash for cement provides benefits of reduced cost and improved workability with minimal strength impacts, representing an effective use of a waste material that addresses sustainability.
Accounting professional judgment involves handling accounting events and compiling financial reports according to regulations and standards. However, professional judgment is sometimes manipulated to distort accounting information. The document discusses three ways manipulation occurs: 1) abandoning accounting principles, 2) optional changes to accounting policies, and 3) abuse of accounting estimates. The causes of manipulation include distorted motivations from corporate governance issues and catering to various stakeholder interests. Strengthening supervision and improving the accounting system are proposed to manage manipulation of professional judgment.
The document discusses research on the distribution of oil and water in the eastern block of the Chao202-2 area in China. It establishes standards for identifying oil, poor oil, dry, and water layers using well logging data. Analysis shows structural reservoirs are dominant and fault and sand body configuration control oil-water distribution. Oil-water distribution varies between fault blocks from "up oil, bottom water" to "up water, bottom oil" depending on structure and sand body development.
The document describes an intelligent fault diagnosis system for reciprocating pumps that uses pressure and flow signals as inputs. It consists of hardware for data acquisition and a software system for signal processing, feature extraction, and fault diagnosis using wavelet neural networks. The system was able to accurately diagnose three main fault types - seal ring faults, valve damage, and spring faults - based on differences observed in the pressure curves. Testing on over 12 samples of each fault type achieved a correct diagnosis rate of over 94%. The system provides a fast and effective means of remotely monitoring reciprocating pumps and identifying faults.
This document discusses the application of meta-learning algorithms in banking sector data mining for fraud detection. It proposes using Classification and Regression Tree (CART), AdaBoost, LogitBoost, Bagging and Dagging algorithms for classification of banking transaction data. The experimental results show that Bagging algorithm has the best performance with the lowest misclassification rate, making it effective for banking fraud detection through data mining. Data mining can help banks detect patterns for applications like credit scoring, payment default prediction, fraud detection and risk management by analyzing customer transaction history and loan details.
This document presents a numerical solution for unsteady heat and mass transfer flow past an infinite vertical plate with variable thermal conductivity, taking into account Dufour number and heat source effects. The governing equations are non-linear and coupled, and were solved numerically using an implicit finite difference scheme. Various parameters, including Dufour number and heat source, were found to influence the velocity, temperature, and concentration profiles. Skin friction, Nusselt number, and Sherwood number were also calculated.
The document discusses methods for obtaining a background image using depth information from a depth camera to more accurately extract foreground objects. It finds that accumulating depth images and taking the median value at each pixel provides the most accurate background image. The accuracy of three methods - average, median, and mode - are evaluated using simulated depth data of a captured plane. The median method provides the best results, followed by average, while mode performs worst. More accumulated images provide a more accurate background image across all methods.
This document presents a mathematical model for determining the minimum overtaking sight distance (OSDm) required for an ascending vehicle to safely pass another slower vehicle on a single lane highway with an incline. It defines sight distance, stopping sight distance, perception-reaction time and derives equations to calculate the reaction distance (d1), overtaking distance (d2), vehicle travel distance during overtaking (d3), and total minimum OSDm based on vehicle characteristics, road geometry, and coefficients of friction. The safe overtaking zone is defined as 3 times the minimum OSDm. The model accounts for effects of slope angle and aims to satisfy laws of mechanics for overtaking maneuvers on inclined two-way single lane highways.
This document discusses a novel technique for better analysis of ice properties using Kalman filtering. It summarizes previous research on sea ice segmentation using SAR imagery and dual polarization techniques. It proposes using an automated SAR algorithm along with Kalman filtering to more accurately detect sea ice properties from RADARSAT1 and RADARSAT2 imagery data. The document reviews techniques for image segmentation, dual polarization, PMA detection, and related work on sea ice classification using statistical ice properties, edge preserving region models, and object extraction methods.
This document summarizes a study on the bioaccumulation of heavy metals in bass fish (Morone Saxatilis) caught at Rodoni Cape in the Adriatic Sea in Albania. Samples of bass fish were collected from five sites and analyzed for mercury, lead, and cadmium levels in their muscles. The concentrations of heavy metals varied between fish and sites but were below international limits for human consumption. While the fish were found to be safe for eating, the study recommends continuous monitoring of metal levels in fish from the area due to various factors that can influence metal uptake over time.
This document discusses optimal maintenance policies for repairable systems with linearly increasing hazard rates. It considers a system with a constant repair rate and predetermined availability requirement. There are two maintenance policies: corrective maintenance only, and preventive maintenance at set time intervals. The goal is to determine the preventive maintenance interval that guarantees the availability requirement at minimum cost. Equations are developed to calculate the availability under each policy and the optimal preventive maintenance interval based on both availability and cost. A numerical example is provided to demonstrate the decision process in determining the optimal policy.
Discovery Series - Zero to Hero - Task Mining Session 1DianaGray10
This session is focused on providing you with an introduction to task mining. We will go over different types of task mining and provide you with a real-world demo on each type of task mining in detail.
Redefining Cybersecurity with AI CapabilitiesPriyanka Aash
In this comprehensive overview of Cisco's latest innovations in cybersecurity, the focus is squarely on resilience and adaptation in the face of evolving threats. The discussion covers the imperative of tackling Mal information, the increasing sophistication of insider attacks, and the expanding attack surfaces in a hybrid work environment. Emphasizing a shift towards integrated platforms over fragmented tools, Cisco introduces its Security Cloud, designed to provide end-to-end visibility and robust protection across user interactions, cloud environments, and breaches. AI emerges as a pivotal tool, from enhancing user experiences to predicting and defending against cyber threats. The blog underscores Cisco's commitment to simplifying security stacks while ensuring efficacy and economic feasibility, making a compelling case for their platform approach in safeguarding digital landscapes.
Welcome to Cyberbiosecurity. Because regular cybersecurity wasn't complicated...Snarky Security
How wonderful it is that in our modern age, every bit of our biological data can be digitized, stored, and potentially pilfered by cyber thieves! Isn't it just splendid to think that while scientists are busy pushing the boundaries of biotechnology, hackers could be plotting the next big bio-data heist? This delightful scenario is brought to you by the ever-expanding digital landscape of biology and biotechnology, where the integration of computer science, engineering, and data science transforms our understanding and manipulation of biological systems.
While the fusion of technology and biology offers immense benefits, it also necessitates a careful consideration of the ethical, security, and associated social implications. But let's be honest, in the grand scheme of things, what's a little risk compared to potential scientific achievements? After all, progress in biotechnology waits for no one, and we're just along for the ride in this thrilling, slightly terrifying, adventure.
So, as we continue to navigate this complex landscape, let's not forget the importance of robust data protection measures and collaborative international efforts to safeguard sensitive biological information. After all, what could possibly go wrong?
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This document provides a comprehensive analysis of the security implications biological data use. The analysis explores various aspects of biological data security, including the vulnerabilities associated with data access, the potential for misuse by state and non-state actors, and the implications for national and transnational security. Key aspects considered include the impact of technological advancements on data security, the role of international policies in data governance, and the strategies for mitigating risks associated with unauthorized data access.
This view offers valuable insights for security professionals, policymakers, and industry leaders across various sectors, highlighting the importance of robust data protection measures and collaborative international efforts to safeguard sensitive biological information. The analysis serves as a crucial resource for understanding the complex dynamics at the intersection of biotechnology and security, providing actionable recommendations to enhance biosecurity in an digital and interconnected world.
The evolving landscape of biology and biotechnology, significantly influenced by advancements in computer science, engineering, and data science, is reshaping our understanding and manipulation of biological systems. The integration of these disciplines has led to the development of fields such as computational biology and synthetic biology, which utilize computational power and engineering principles to solve complex biological problems and innovate new biotechnological applications. This interdisciplinary approach has not only accelerated research and development but also introduced new capabilities such as gene editing and biomanufact
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G05834551
1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 05, Issue 08 (August. 2015), ||V3|| PP 45-51
International organization of Scientific Research 45 | P a g e
Numerical Solution of a Linear Black-Scholes Models: A
Comparative Overview
Md. Kazi Salah Uddin, Md. Noor-A-Alam Siddiki, Md. Anowar Hossain
Department of Natural Science, Stamford University Bangladesh, Dhaka-1209, Bangladesh.
Abstract: - Black-Scholes equation is a well known partial differential equation in financial mathematics. In
this paper we try to solve the European options (Call and Put) using different numerical methods as well as
analytical methods. We approximate the model using a Finite Element Method (FEM) followed by weighted
average method using different weights for numerical approximations. We present the numerical result of semi-
discrete and full discrete schemes for European Call option and Put option by Finite Difference Method and
Finite Element Method. We also present the difference of these two methods. Finally, we investigate some
linear algebra solvers to verify the superiority of the solvers.
Keywords: Black-Scholes model; call and put options; exact solution; finite difference schemes, Finite Element
Methods.
I. INTRODUCTION
A powerful tool for valuation of equity options is the Black-Scholes model[12,15]. This model is used for
finding the prices of stocks.
R. Company, A.L. Gonzalez, L. Jodar [14] solved the modified Black-Scholes equation pricing option with
discrete dividend.
A delta-defining sequence of generalized Dirac-Delta function and the Mellin transformation are used toobtain
an integral formula. Finally numerical quadrature approximation is used to approximate the solution.
In some papers like [13] Mellin transformation is used. They were required neither variable transformation nor
solving diffusion equation.
R. Company, L. Jodar, G. Rubio, R.J. Villanueva [13] found the solution of BS equation with a wide class of
payoff functions that contains not only the Dirac delta type functions but also the ordinary payoff functions with
discontinuities of their derivatives.
Julia Ankudiova, Matthias Ehrhardt [20] solved non linear Black-Scholes equations numerically. They focused
on various models relevant with the Black-Scholes equations with volatility depending on several factors.
They also worked on the European Call option and American Call option analytically using transformation into
a convection -diffusion equation with non-linear term and the free boundary problem respectively.
In our previous paper [7] we discussed about the analytical solution of Black-Scholes equation using Fourier
Transformation method for European options. We formulated the Finite Difference Scheme and found the
solutions of them.
In this paper we discuss the solution with Finite Element Method and compare the result with the result obtained
by Finite Difference Schemes.
II. MODEL EQUATION
The linear Black-Scholes equation [12,15] developed by Fischer Black and Myron Scholes in 1973 is
𝑉𝑡 + 𝑟𝑆𝑉𝑆 +
1
2
𝜎2
𝑆2
𝑉𝑆𝑆 − 𝑟𝑉 = 0 … … … … … … … … … … … … … … … … … … … … (1)
where
𝑉 = 𝑉 𝑆, 𝑡 , 𝑡ℎ𝑒 𝑝𝑎𝑦 − 𝑜𝑓𝑓 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑆 = 𝑆(𝑡), 𝑡ℎ𝑒 𝑠𝑡𝑜𝑐𝑘 𝑝𝑟𝑖𝑐𝑒, 𝑤𝑖𝑡ℎ 𝑆 = 𝑆(𝑡) ≥ 0,
𝑡 = 𝑡𝑖𝑚𝑒,
𝑟 = 𝑅𝑖𝑠𝑘 − 𝐹𝑟𝑒𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒,
𝜎 = 𝑉𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛
and also 𝑡 ∈ (0, 𝑇).
where T is time of maturity.
The terminal and boundary conditions [16] for both the European Call and Put options stated below.
2. Numerical Solution of a Linear Black-Scholes Models: A Comparative Overview
International organization of Scientific Research 46 | P a g e
European Call Option [16]
The solution to the Black-Scholes equation (1) is the value 𝑉(𝑆, 𝑡) of the European Call option on $0 ≤ 𝑆 <
∞, 0 ≤ 𝑡 ≤ 𝑇. The boundary and terminal conditions are as follows
𝑉 0, 𝑡 = 0 𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 𝑇,
𝑉 𝑆, 𝑡 ∼ 𝑆 − 𝐾𝑒−𝑟 𝑇−𝑡
𝑎𝑠 𝑆 → ∞, … … … … … … … … … … … … … … … … … … … … (2)
𝑉 𝑆, 𝑇 = 𝑆 − 𝐾 +
𝑓𝑜𝑟 0 ≤ 𝑆 < ∞.
European Put Option[16]
European Put option is the reciprocal of the European Call option and the boundary and terminal conditions are
𝑉 0, 𝑡 = 𝐾𝑒−𝑟 𝑇−𝑡
𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 𝑇,
𝑉 𝑆, 𝑡 → 0 𝑎𝑠 𝑆 → ∞, … … … … … … … … … … … … … … … … … … … … … … … … … (3)
𝑉 𝑆, 𝑇 = 𝐾 − 𝑆 +
𝑓𝑜𝑟 0 ≤ 𝑆 < ∞.
III. TRANSFORMATION
The model problem stated in (1) is a backward type. This type is little bit difficult to solve. To solve the problem
in (1) with the conditions stated in (2) and (3) we need to make the model in forward type. In this regard, we
have the following transformations.
Let
𝑆 = 𝐾𝑒 𝑥
𝑡 = 𝑇 −
𝜏
𝜎2 /2
And
𝑣(𝑥, 𝜏) =
1
𝐾
𝑉(𝑆, 𝑡)
𝜕𝑉
𝜕 𝑡
=
𝜕𝑉
𝜕𝜏
𝜕𝜏
𝜕𝑡
+
𝜕𝑉
𝜕𝑆
𝜕𝑆
𝜕𝑡
= −
𝜎2
2
𝐾
𝜕𝑣
𝜕𝜏
𝜕𝑉
𝜕𝑆
= 𝐾
𝜕𝑣
𝜕𝑆
=
𝐾
𝑆
𝜕𝑣
𝜕𝑥
𝜕2
𝑉
𝜕𝑆2
= 𝐾
𝜕2
𝑣
𝜕𝑆2
=
𝐾
𝑆2
𝜕2
𝑣
𝜕𝑥2
−
𝜕𝑣
𝜕𝑥
inserting these derivatives in equation (1) we have
−
𝜎2
2
𝐾
𝜕𝑣
𝜕𝜏
+
𝜎2
2
𝐾
𝜕2
𝑣
𝜕𝑥2
−
𝜕𝑣
𝜕𝑥
+ 𝑟𝐾
𝜕𝑣
𝜕𝑥
− 𝑟𝐾𝑣 = 0.
implies
𝜕𝑣
𝜕𝜏
=
𝜕2
𝑣
𝜕𝑥2
+
𝑟
𝜎2
2
− 1
𝜕𝑣
𝜕𝑥
−
𝑟
𝜎2
2
𝑣 … … … … … … … … … … … … … … … … … (4)
Let
𝑟
𝜎2
2
= 𝜃
∴ (4) implies
𝜕𝑣
𝜕𝜏
=
𝜕2
𝑣
𝜕𝑥2
+ 𝜃 − 1
𝜕𝑣
𝜕𝑥
− 𝜃𝑣. … … … … … … … … … … … … … … … … … … . . (5)
Now let
𝜆 =
1
2
(𝜃 − 1), 𝜈 =
1
2
(𝜃 + 1) = 𝜆 + 1
so that
𝜈2
= 𝜆2
+ 𝜃
𝑣(𝑥, 𝜏) = 𝑒−𝜆𝑥 −𝜈2 𝜏
𝑢(𝑥, 𝜏).
𝜕𝑣
𝜕𝜏
= 𝑒−𝜆𝑥 −𝜈2 𝜏
−𝜈2
𝑢 𝑥, 𝜏 +
𝜕𝑢
𝜕𝜏
𝑒−𝜆𝑥 −𝜈2 𝜏
= 𝑒−𝜆𝑥 −𝜈2 𝜏
[−𝜈2
𝑢 +
𝜕𝑢
𝜕𝜏
],
𝜕𝑣
𝜕𝑥
= 𝑒−𝜆𝑥 −𝜈2 𝜏
[−𝜆𝑢 +
𝜕𝑢
𝜕𝑥
],
3. Numerical Solution of a Linear Black-Scholes Models: A Comparative Overview
International organization of Scientific Research 47 | P a g e
𝜕2
𝑣
𝜕𝑥2
= 𝑒−𝜆𝑥 −𝜈2 𝜏
[𝜆2
𝑢 − 2𝜆
𝜕𝑢
𝜕𝑥
+
𝜕2
𝑢
𝜕𝑥2
].
inserting these into equation(5) and dividing by 𝑒−𝜆𝑥 −𝜈2 𝜏
we get
−𝜈2
𝑢 +
𝜕𝑢
𝜕𝜏
= [𝜆2
𝑢 − 2𝜆
𝜕𝑢
𝜕𝑥
+ (𝜕2
𝑢)/(𝜕𝑥2
)] + (𝜃 − 1)[−𝜆𝑢 +
𝜕𝑢
𝜕𝑥
] − 𝜃𝑢
implies
𝑢 𝜏 = 𝑢 𝑥𝑥 + −2𝜆 + 𝜃 − 1 𝑢 𝑥 + (𝜆2
+ 𝜈2
− 𝜆(𝜃 − 1))𝑢
= 𝑢 𝑥𝑥 .
∴ 𝑢 𝜏 = 𝑢 𝑥𝑥 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (6)
And the initial & boundary conditions for the European Call and Put options are respectively
𝑢 𝑥, 0 = 𝑒 𝜆+1 𝑥
− 𝑒 𝜆𝑥 +
𝑎𝑠 𝑥 ∈ ℝ
𝑢 𝑥, 𝜏 = 0 𝑎𝑠 𝑥 → −∞
𝑢 𝑥, 𝜏 = 𝑒 𝜆+1 𝑥+𝜈2 𝜏
− 𝑒 𝜆𝑥+𝜆2 𝜏
𝑎𝑠 𝑥 → ∞
… … … … … … … … … . . … … … … … … … . (7)
and
𝑢 𝑥, 0 = 𝑒 𝜆𝑥
− 𝑒 𝜆+1 𝑥 +
𝑎𝑠 𝑥 ∈ ℝ
𝑢 𝑥, 𝜏 = 𝑒 𝜆𝑥 +𝜆2 𝜏
𝑎𝑠 𝑥 → −∞ … … … … … … … … . . … … … … … … … … … . (8)
𝑢 𝑥, 𝜏 = 0 𝑎𝑠 𝑥 → ∞
Thus the Black-Scholes equation reduced to a heat diffusion equation.
IV. NUMERICAL APPROXIMATION OF TRANSFORMED LINEAR BLACK-
SCHOLES MODEL
Now we solve the problems numerically. We use the Finite Element Method (FEM) to solve the problems
related to the differential equation (6). Finally back substitution of the coordinate transformation gives the
solution of the problems related to the differential equation (1).
We have the model
𝑢 𝜏 = 𝑢 𝑥𝑥 , 𝑥 ∈ ℝ, 0 ≤ 𝜏 ≤ 𝑇. … … … … … … … … … … … … … … … … … … … … … … … … (9)
and the initial and boundary conditions for call option are
𝑢 𝑥, 0 = 𝑒 𝜆+1 𝑥
− 𝑒 𝜆𝑥 +
𝑎𝑠 𝑥 ∈ ℝ,
𝑢 𝑥, 𝜏 = 0 𝑎𝑠 𝑥 → −∞ … … … … … … … … … … … … … … . . (10)
𝑢 𝑥, 𝜏 = 𝑒 𝜆+1 𝑥+𝜈2 𝜏
− 𝑒 𝜆𝑥+𝜆2 𝜏
𝑎𝑠 𝑥 → ∞
The weak form of the governing equation is
𝜕𝑢
𝜕𝜏ℝ
𝑣(𝑥)𝑑𝑥 +
𝜕𝑢
𝜕𝑥
𝜕𝑣
𝜕𝑥
𝑑𝑥ℝ
= 0 … … … … … … … … … … … … … … … … … … … . (11)
Since 𝑣 𝑥 → 0 as 𝑥 → ±∞.
Discretizing 𝑢(𝑥, 𝜏) spatially, we have
𝑢 𝑥, 𝜏 = 𝜙𝑖 𝜏 𝑁𝑖 𝑥
𝑛
−𝑛
… … … … … … … … … … … … … … … … … … … … … . (12)
where 𝑁𝑖(𝑥) are given shape functions, and 𝜙𝑖(𝜏) are unknown, and 𝑛 is the ordinal number of nodes.
Substituting (12) into (11), we get the weak semidiscretized equation
𝜙𝑖
′
𝜏 𝑁𝑖 𝑥 𝑁𝑗 (𝑥)𝑑𝑥
1
0
𝑛
−𝑛
+ 𝜙𝑖 𝜏 𝑁′𝑖 𝑥 𝑁′𝑗 (𝑥)𝑑𝑥
1
0
𝑛
−𝑛
= 0 … … … … … . . (13)
Let 𝑄, 𝑀 ∈ ℝ 2𝑛−1 × 2𝑛−1
denote the so-called mass and stiffness matrices, respectively, defined by:
𝑀𝑖𝑗 = 𝑁′𝑖 𝑥 𝑁′𝑗 (𝑥)𝑑𝑥
1
0
… … … … … … … … … … … … … … … . . (14)
𝑄𝑖𝑗 = 𝑁𝑖 𝑥 𝑁𝑗 (𝑥)𝑑𝑥
1
0
… … … … … … … … … … … … … . . … … . . (15)
Then (13) can be expressed as:
𝑄𝛷′
+ 𝑀𝛷 = 0 … … … … … … … … … … … … … … . (16)
where Φ ∈ ℝ2𝑛−1
is a vector function with the components 𝜙𝑖.
After performing the integral in (14) and (15) for the linear shape functions, the mass and the stiffness matrices
have the following form
4. Numerical Solution of a Linear Black-Scholes Models: A Comparative Overview
International organization of Scientific Research 48 | P a g e
𝑀 =
1
ℎ
−1 2 −1 … 0
⋮ ⋱ ⋱ ⋱ ⋮
0
0
…
…
−1
0
2 −1
−1 1
; 𝑄 =
6
ℎ
1 4 1 … 0
⋮ ⋱ ⋱ ⋱ ⋮
0
0
…
…
1
0
4 1
1 2
where ℎ is the length of the spatial approximation.
Now we would like to discrete the equation (16) with respect to time. One may start with a simple scheme.
One of the trivial choice is to use the forward Euler scheme. Firstly we discrete (16) explicitly and we have
𝑄Φ′ + 𝑀Φ = 0,
Φ′ + 𝑄−1
𝑀Φ = 0,
Φ 𝑚+1 − Φ 𝑚
Δ𝜏
+ 𝑄−1
𝑀Φ 𝑚 = 0,
Φ 𝑚+1 − Φ 𝑚 + Δ𝜏𝑄−1
𝑀Φ 𝑚 = 0,
Φ 𝑚+1 − 𝐼 − Δ𝜏𝑄−1
𝑀 Φ 𝑚 = 0,
Φ 𝑚+1 = 𝐼 − Δ𝜏𝑄−1
𝑀 Φ 𝑚 . … … … … … … … … … … … … … … … … . (17)
The difficulty of using the scheme is that it needs very little step size to converge , as a result the scheme is a
slow one, and is not of interest in this advance study.
We want a fast and efficient scheme, so we want larger time stepping, and interested in using implicit
techniques. We discrete (16) implicitly and have
Φ 𝑚+1 − Φ 𝑚
Δ𝜏
+ 𝑄−1
𝑀Φ 𝑚+1 = 0,
Φ 𝑚+1 − Φ 𝑚 + Δ𝜏𝑄−1
𝑀Φ_(𝑚 + 1) = 0,
𝐼 + Δ𝜏𝑄−1
𝑀 Φ 𝑚+1 = Φ 𝑚 ,
Φ 𝑚+1 = 𝐼 + Δ𝜏𝑄−1
𝑀 −1
Φ 𝑚 . … … … … … … … … … … … … … … … … … … (18)
which is a system of linear equations with unknowns Φ 𝑚+1. The advantage of using (18)
is that the scheme is unconditionally stable . Equation (18) accuracy of order 𝑂(𝑘). It is faster than the explicit
Euler scheme since (18) allows us to use large time steps.
We use an weighted average method to discrete (16) with weight 𝛿 and we have
Φ 𝑚+1 − Φ 𝑚
Δ𝜏
+ 𝑄−1
𝑀(𝛿Φ 𝑚+1 + 1 − 𝛿 Φ 𝑚 ) = 0,
Φ 𝑚+1 − Φ 𝑚 + Δ𝜏𝑄−1
𝑀(𝛿Φ 𝑚+1 + 1 − 𝛿 Φ 𝑚 ) = 0,
𝐼 + Δ𝜏𝑄−1
𝑀𝛿 Φ 𝑚+1 = 𝐼 − Δ𝜏𝑄−1
𝑀 1 − 𝛿 Φ 𝑚 ,
Φ 𝑚+1 = 𝐼 + Δ𝜏𝑄−1
𝑀𝛿 −1
𝐼 − Δ𝜏𝑄−1
𝑀 1 − 𝛿 Φ 𝑚 . … … … … … … … . . (19)
This system is also a linear one with unknowns Φ 𝑚+1, where 𝛿 varies from 0 to 1. This
method turns to the explicit method when 𝛿 = 0 i.e., equations (17) and (19) are same and implicit method
when 𝛿 = 1, i.e., equations (18) and (19)are same. For 0 ≤ 𝛿 ≤
1
2
, the scheme is conditionally stable and
unconditionally stable for
1
2
≤ 𝛿 ≤ 1.
The order of the accuracy of the scheme is 𝑂(𝑘).
V. RESULTS, DISCUSSION AND CONCLSTION
In this section we have presented the results by various methods. We have solved the model
analytically [7] by the method of Fourier Transformation. In Figure fig. 1 we placed the analytic solution of two
options (Call Option and Put Option). To solve the model numerically we have applied [7] Finite difference
methods (FDM) and have shown the result of the two options in Figure 2. Our interest in this paper was in the
methods of Finite Elements (FEM) [1]. Firstly, we have discretized the model (6) spatially in the section (4).
Then we have used various one step Euler’s time integrations to discretize the system of linear equations
obtained by semi-discretization. The results have been presented in the Figure 3. We have tried to show
comparison between the methods (FDM and FEM) in Figure 4.
5. Numerical Solution of a Linear Black-Scholes Models: A Comparative Overview
International organization of Scientific Research 49 | P a g e
(a) Call Optin (b) Put Option
Figure 1: Analytic solutions
(a) Call Option (b) Put Option
Figure 2: Numerical Solutions by Finite Difference Method
(a) Call Option (b) Put Option
Figure 3: Numerical Solutions by Finite Element Method
6. Numerical Solution of a Linear Black-Scholes Models: A Comparative Overview
International organization of Scientific Research 50 | P a g e
Figure 4: Comparison of Finite Difference Method and Finite Element Method
The system of linear equations (19) generated by the discretization of the Black-Scholes model can be
solved by many conventional processes. For a large scale linear system, scientists rarely use direct methods as
they are computationally costly. Here, in this section, it is our motivation to solve the system of equation (19)
using various iterative techniques. Here we first investigate which linear solver converges swiftly. To that end,
we consider Jacobi iterative method, Gauss-Seidel iterative method and successive over relaxation method to
start with. In terms of matrices, the Jacobi method can be expressed as
x(k)
= D−1
L + U x(k−1)
+ D−1
b,
Gauss-Seidel method
x(k)
= D − L −1
(Ux k−1
+ b),
and the SOR algorithm can be written as
x k
= D − ωL −1
ωU + 1 − ω D x k−1
+ ω D − ωL −1
b,
where in each case the matrices D, −L, and −U represent the diagonal, strictly lower triangular, and strictly
upper triangular parts of A, respectively.
Figure 5: Time comparison of different linear algebra solvers
We investigate Preconditioned Conjugate Gradient (PCG) Method and Generalized Minimal Residual
(GMRES) Method with a diagonal preconditioning [6]. Here for all computations we consider 𝐾 = 100, 𝜎 =
.2, 𝑟 = .1, 𝑇 = 1𝑦𝑒𝑎𝑟, Δ𝑡 = 0.001. The results are presented with different weights 𝛿. Observing Figure (5),
we notice that Preconditioned Conjugate Gradient (PCG) Method performs the best.
7. Numerical Solution of a Linear Black-Scholes Models: A Comparative Overview
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