Finite Difference Approach to Option Pricing
|
|
- Lewis Ryan
- 8 years ago
- Views:
Transcription
1 Finite Difference Approach to Option Pricing February 998 CS5 Lab Note. Ordinary differential equation An ordinary differential equation, or ODE, is an equation of the form du = fut ( (), t) (.) dt where t is the time variable, u is a real or complex scalar or vector function of t, and f is a function. Initial value problem is to find a differentiable function ut () such that u( ) = u (.) du () t = fut ( (), t) for all t [, T] dt For the solution of ordinary differential equations, one of the most powerful discretization strategies is linear multistep methods. Let k > be a real number, the time step, and let t, t, t,... be defined t n = nk. Our goal is to construct a sequence of values v, v,... such that Let f n be the abbreviation v n ut ( n ) n (.3) = fv ( n, t n ). (.4) A linear multistep method is a formula for calculating each new value v n + from some of the previous values v,..., v n and f,..., f n. The simplest linear multistep method is a one step method : the Euler formula defined by f n + = v n + kf n v n Euler method is an example of an explicit one-step formula. (.5) A related linear multistep formula is the backward Euler, also a one-step formula, defined by + = v n + kf n + v n (.6) v n + To implement an implicit formula, one must employ a scheme to solve for the unknown, and this involves extra work.
2 The advantage of an implicit method is that in some situations it may be stable when an explicit one is catastrophically unstable. Throughout the numerical solution of differential equations, there is a tradeoff between explicit methods, which tend to be easier to implement, and implicit ones, which tend ot be more stable. Example - du dt = uu, ( ) = + = v n + kv n, v = (Euler method) v n + = v n + kv n +, v = (Backward Euler method) v n (.7) (.8) Example - More formulas Trapezoid rule(implicit one-step formula) + = v n + k -- ( f n + f n + ) v n (.9) Midpoint rule(explicit two-step formula) v n + = v n + kf n (.). Partial Differential Equation Partial differential equations fall roughly into three great classes which can be loosely described as follows elliptic -- time-dependent parabolic -- time-dependent and diffusive hyperbolic -- time-dependent and wave like The simplest example of a hyperbolic equation is u t the one-dimensional first-order wave equation, which describes advection of a quantity at the constant velocity. = u x (.) ux (, t)
3 The simplest example of a parabolic equation is u t = (.) the one-dimensional heat equation, which describes diffusion of a quantity such as heat or salinity. Let h > and kbe > a fixed space step and time step, respectively and set x j = and jh t n = nk for any integers j and n. The points ( x j, t n ) define a regular grid or mesh in two dimensions. n The aim of finite difference is to approximate continuous functions ux (, t) by grid functions v j, u x ux ( j, t n ) v n n represents the spatial grid function { for v j, a j fixed Z} value. n v j n (.3) The simplest kind of finite procedure is an s-step finite difference formula, which is a fixed n + formula that prescribes v j as a function of a finite number of other grid values at time steps n + s through n(explicit case) or n+ (implicit case). To compute an approximation n { v j } to ux (, t), we shall begin with initial data v,..., v s, and compute values v s, v s +,... in succession by applying finite difference formula. This process is sometimes known as marching with respect to t. Example u u k Finite difference approximations for the heat equation =, with σ = t x h n + n n n n Euler : v j = v j + σ( v j + v j + v j ) n + n n + n + n + Backward Euler : v j = v j + σ( v j + v j + v j ) n + n n n n n + n + n + Crank-Nicholson : v j = v j + -- σ ( v j + v j + v j ) + -- σ ( v j + v j + v j ) (The above two sections are from unpublished manuscript of Lloyd N. Trefethen)
4 3. Option Pricing via Finite Difference Method 3. Implicit method(see lecture note for derivation and notation) Let f i, j denote the value of option price at the ( i, j) point, i.e., when tand = i t S. = j S Implicit method for American put option is a j f i, j b j f i, j c j f i, j + = f i +, for i j =...N,,, and j =...M,,, where a j = -- rj t -- σ j t b j = + σ j t+ r t c j = -- rj t -- σ j t (3.) (3.) MATLAB Implementation function put = imfdamput(smax, ds, T, dt, X, R, SIG); % put = imfdamput(smax, ds, T, dt, X, R, SIG); % Smax : maximum stock price % ds : increment of stock price % T : maturity date % dt : time step % X : exercise price % R : risk free interest rate % % reference : John C. Hull, Options, Futures, and Other Derivatives % 3rd Ed., Chap 5 M = ceil(smax/ds); ds = Smax / M; N = ceil(t/dt); dt = T / N; J = :M-; a =.5*R*dt*J -.5*SIG^*dt*J.^; b = + SIG^*dt*J.^ + R*dt; c = -.5*R*dt*J -.5*SIG^*dt*J.^; A = diag(b) + diag(a(:m-), -) + diag(c(:m-), ); put = zeros(n+, M+); put(n+, :) = max(x - [:ds:smax], ); put(:, ) = X; put(:, M+) = ; for i = N:-:
5 end y = put(i+, :M) ; y() = y() - a()*x; put(i, :M) = [A \ y] ; put(i, :) = max(x - [:ds:smax], put(i,:)); The following figure shows the American put option price when Smax = 5; ds = ; T = ; dt =.; X = 5; R =.; and SIG =.5; Time Stock Price 3. Explicit Method The difference equation is where f i j, = a j f i +, j + b j f i +, j + c j f i +, j + a j = rj t + -- σ j t + r t b j = ( σ j + r t t ) c j = + r t -- rj t + -- σ j t (3.3) (3.4) For the explicit method to be stable, all be positive. --rj t + -- σ j t, σ j t, -- rj t + -- σ j t should
6 MATLAB Implementation function put = exfdamput(smax, ds, T, dt, X, R, SIG); M = ceil(smax/ds); ds = Smax / M; N = ceil(t/dt); dt = T / N; J = :M-; a = (-.5*R*dt*J +.5*SIG^*dt*J.^) / (+R*dt); b = ( - SIG^*dt*J.^) / (+R*dt); c = (.5*R*dt*J +.5*SIG^*dt*J.^) / ( + R*dt); A = diag(b) + diag(a(:m-), -) + diag(c(:m-), ); put = zeros(n+, M+); put(n+, :) = max(x - [:ds:smax], ); put(:, ) = X; put(:, M+) = ; for i = N:-: end y = zeros(, M-); y() = a()*put(i+, ); y(m-) = c(m-)*put(i+,m+); put(i, :M) = put(i+, :M) * A + y; put(i, :) = max(x - [:ds:smax], put(i,:)); The following figure shows the unstability of the explict method with the wrong choice of Smax = ; ds = 5; T = ; dt =.; R =.; SIG =.4; and X = 5. American Put : Explicit FD Time.4. 4 Stock Price 6 8 It s stable when Smax = 5; ds = 5; T = ; dt =.; X = 5; R =.; and SIG =.5.
7 American Put : Explicit FD Time.4. Stock Price Crank-Nicholson Method The Crank-Nicholson scheme is an average of the explicit and implicit methods. It s given by and g i, j = f i j, a j f i, j b j f i, j c j f i, j + (3.5) g i j, = a j f i, j + b j f i, j + c j f i, j + f i, j The implementation of the Crank-Nicholson method is similar to that of the implicit method. (3.6) MATLAB Implementation function put = cnfdamput(smax, ds, T, dt, X, R, SIG); M = ceil(smax/ds); ds = Smax / M; N = ceil(t/dt); dt = T / N; J = :M-; a = (-.5*R*dt*J +.5*SIG^*dt*J.^) / (+R*dt); b = ( - SIG^*dt*J.^) / (+R*dt); c = (.5*R*dt*J +.5*SIG^*dt*J.^) / ( + R*dt); A = diag(-b) + diag(-a(:m-), -) + diag(-c(:m-), ); aa =.5*R*dt*J -.5*SIG^*dt*J.^; bb = + SIG^*dt*J.^ + R*dt; cc = -.5*R*dt*J -.5*SIG^*dt*J.^; B = diag(bb-) + diag(aa(:m-), -) + diag(cc(:m-), );
8 g = zeros(, M-); put = zeros(n+, M+); put(n+, :) = max(x - [:ds:smax], ); put(:, ) = X; put(:, M+) = ; for i = N:-: end g = put(i+, :M) * A ; g() = g() - a()*x - aa()*x; put(i, :M) = [B \ g ] ; put(i, :) = max(x - [:ds:smax], put(i,:)); Using finite difference methods, we can get the boundary between the regions where it s optimal to exercise an American option or not. The following figure shows different boundaries for SIG =.:.:.9. The first curve from the right is when SIG =.. Free boundary : American Put Option Time Stock Price
CS 294-73 Software Engineering for Scientific Computing. http://www.cs.berkeley.edu/~colella/cs294fall2013. Lecture 16: Particle Methods; Homework #4
CS 294-73 Software Engineering for Scientific Computing http://www.cs.berkeley.edu/~colella/cs294fall2013 Lecture 16: Particle Methods; Homework #4 Discretizing Time-Dependent Problems From here on in,
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationSIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I
Lennart Edsberg, Nada, KTH Autumn 2008 SIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I Parameter values and functions occurring in the questions belowwill be exchanged
More information440 Geophysics: Heat flow with finite differences
440 Geophysics: Heat flow with finite differences Thorsten Becker, University of Southern California, 03/2005 Some physical problems, such as heat flow, can be tricky or impossible to solve analytically
More information5.4 The Heat Equation and Convection-Diffusion
5.4. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and Convection-Diffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The
More informationCollege of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions
College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More informationMEL 807 Computational Heat Transfer (2-0-4) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi
MEL 807 Computational Heat Transfer (2-0-4) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi Time and Venue Course Coordinator: Dr. Prabal Talukdar Room No: III, 357
More informationOPTION PRICING WITH PADÉ APPROXIMATIONS
C om m unfacsciu niva nkseries A 1 Volum e 61, N um b er, Pages 45 50 (01) ISSN 1303 5991 OPTION PRICING WITH PADÉ APPROXIMATIONS CANAN KÖROĞLU A In this paper, Padé approximations are applied Black-Scholes
More informationN 1. (q k+1 q k ) 2 + α 3. k=0
Teoretisk Fysik Hand-in problem B, SI1142, Spring 2010 In 1955 Fermi, Pasta and Ulam 1 numerically studied a simple model for a one dimensional chain of non-linear oscillators to see how the energy distribution
More informationOn computer algebra-aided stability analysis of dierence schemes generated by means of Gr obner bases
On computer algebra-aided stability analysis of dierence schemes generated by means of Gr obner bases Vladimir Gerdt 1 Yuri Blinkov 2 1 Laboratory of Information Technologies Joint Institute for Nuclear
More informationNotes for AA214, Chapter 7. T. H. Pulliam Stanford University
Notes for AA214, Chapter 7 T. H. Pulliam Stanford University 1 Stability of Linear Systems Stability will be defined in terms of ODE s and O E s ODE: Couples System O E : Matrix form from applying Eq.
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 1: Initial value problems in ODEs Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,
More informationAdvanced CFD Methods 1
Advanced CFD Methods 1 Prof. Patrick Jenny, FS 2014 Date: 15.08.14, Time: 13:00, Student: Federico Danieli Summary The exam took place in Prof. Jenny s office, with his assistant taking notes on the answers.
More informationNumerical Methods for Engineers
Steven C. Chapra Berger Chair in Computing and Engineering Tufts University RaymondP. Canale Professor Emeritus of Civil Engineering University of Michigan Numerical Methods for Engineers With Software
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationDoes Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationHow To Price A Call Option
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 5
Advanced Computational Fluid Dynamics AA5A Lecture 5 Antony Jameson Winter Quarter, 0, Stanford, CA Abstract Lecture 5 shock capturing schemes for scalar conservation laws Contents Shock Capturing Schemes
More informationIntroduction to the Finite Element Method
Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross
More informationChapter 9 Partial Differential Equations
363 One must learn by doing the thing; though you think you know it, you have no certainty until you try. Sophocles (495-406)BCE Chapter 9 Partial Differential Equations A linear second order partial differential
More information1 Finite difference example: 1D implicit heat equation
1 Finite difference example: 1D implicit heat equation 1.1 Boundary conditions Neumann and Dirichlet We solve the transient heat equation ρc p t = ( k ) (1) on the domain L/2 x L/2 subject to the following
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationFinite Differences Schemes for Pricing of European and American Options
Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the Black-Scholes
More informationFeature Commercial codes In-house codes
A simple finite element solver for thermo-mechanical problems Keywords: Scilab, Open source software, thermo-elasticity Introduction In this paper we would like to show how it is possible to develop a
More informationDomain Decomposition Methods. Partial Differential Equations
Domain Decomposition Methods for Partial Differential Equations ALFIO QUARTERONI Professor ofnumericalanalysis, Politecnico di Milano, Italy, and Ecole Polytechnique Federale de Lausanne, Switzerland ALBERTO
More informationThe Black-Scholes-Merton Approach to Pricing Options
he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More informationChapter 5. Methods for ordinary differential equations. 5.1 Initial-value problems
Chapter 5 Methods for ordinary differential equations 5.1 Initial-value problems Initial-value problems (IVP) are those for which the solution is entirely known at some time, say t = 0, and the question
More informationPricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation
Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Yoon W. Kwon CIMS 1, Math. Finance Suzanne A. Lewis CIMS, Math. Finance May 9, 000 1 Courant Institue of Mathematical Science,
More informationFigure 1 - Unsteady-State Heat Conduction in a One-dimensional Slab
The Numerical Method of Lines for Partial Differential Equations by Michael B. Cutlip, University of Connecticut and Mordechai Shacham, Ben-Gurion University of the Negev The method of lines is a general
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationNumerical Analysis An Introduction
Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver. Finite Difference Methods for Partial Differential Equations As you are well aware, most differential equations are much too complicated to be solved by
More informationDiffusion: Diffusive initial value problems and how to solve them
84 Chapter 6 Diffusion: Diffusive initial value problems and how to solve them Selected Reading Numerical Recipes, 2nd edition: Chapter 19 This section will consider the physics and solution of the simplest
More informationHow To Calculate Energy From Water
A bi-projection method for Bingham type flows Laurent Chupin, Thierry Dubois Laboratoire de Mathématiques Université Blaise Pascal, Clermont-Ferrand Ecoulements Gravitaires et RIsques Naturels Juin 2015
More informationNumerical Methods for Ordinary Differential Equations 30.06.2013.
Numerical Methods for Ordinary Differential Equations István Faragó 30.06.2013. Contents Introduction, motivation 1 I Numerical methods for initial value problems 5 1 Basics of the theory of initial value
More information- momentum conservation equation ρ = ρf. These are equivalent to four scalar equations with four unknowns: - pressure p - velocity components
J. Szantyr Lecture No. 14 The closed system of equations of the fluid mechanics The above presented equations form the closed system of the fluid mechanics equations, which may be employed for description
More informationTo define concepts such as distance, displacement, speed, velocity, and acceleration.
Chapter 7 Kinematics of a particle Overview In kinematics we are concerned with describing a particle s motion without analysing what causes or changes that motion (forces). In this chapter we look at
More informationValuation of American Options
Valuation of American Options Among the seminal contributions to the mathematics of finance is the paper F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political
More informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More informationNumerical Solution of Differential Equations
Numerical Solution of Differential Equations Dr. Alvaro Islas Applications of Calculus I Spring 2008 We live in a world in constant change We live in a world in constant change We live in a world in constant
More informationSimulating Stochastic Differential Equations
Monte Carlo Simulation: IEOR E473 Fall 24 c 24 by Martin Haugh Simulating Stochastic Differential Equations 1 Brief Review of Stochastic Calculus and Itô s Lemma Let S t be the time t price of a particular
More informationReaction diffusion systems and pattern formation
Chapter 5 Reaction diffusion systems and pattern formation 5.1 Reaction diffusion systems from biology In ecological problems, different species interact with each other, and in chemical reactions, different
More informationValuation, Pricing of Options / Use of MATLAB
CS-5 Computational Tools and Methods in Finance Tom Coleman Valuation, Pricing of Options / Use of MATLAB 1.0 Put-Call Parity (review) Given a European option with no dividends, let t current time T exercise
More informationNUMERICAL ANALYSIS OF OPEN CHANNEL STEADY GRADUALLY VARIED FLOW USING THE SIMPLIFIED SAINT-VENANT EQUATIONS
TASK QUARTERLY 15 No 3 4, 317 328 NUMERICAL ANALYSIS OF OPEN CHANNEL STEADY GRADUALLY VARIED FLOW USING THE SIMPLIFIED SAINT-VENANT EQUATIONS WOJCIECH ARTICHOWICZ Department of Hydraulic Engineering, Faculty
More informationParametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationDiscrete mechanics, optimal control and formation flying spacecraft
Discrete mechanics, optimal control and formation flying spacecraft Oliver Junge Center for Mathematics Munich University of Technology joint work with Jerrold E. Marsden and Sina Ober-Blöbaum partially
More informationPricing European and American bond option under the Hull White extended Vasicek model
1 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Pricing European and American bond option under the Hull White extended Vasicek model Eva Maria Rapoo 1, Mukendi Mpanda 2
More informationInteractive simulation of an ash cloud of the volcano Grímsvötn
Interactive simulation of an ash cloud of the volcano Grímsvötn 1 MATHEMATICAL BACKGROUND Simulating flows in the atmosphere, being part of CFD, is on of the research areas considered in the working group
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More informationINTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS
INTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS I. Dolezel Czech Technical University, Praha, Czech Republic P. Karban University of West Bohemia, Plzeft, Czech Republic P. Solin University of Nevada,
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point
More informationNumerical Resolution Of The Schrödinger Equation
École Normale Supérieure de Lyon Master Sciences de la Matière 2011 Numerical Analysis Project Numerical Resolution Of The Schrödinger Equation Loren Jørgensen, David Lopes Cardozo, Etienne Thibierge Abstract
More informationFast solver for the three-factor Heston-Hull/White problem. F.H.C. Naber Floris.Naber@INGbank.com tw1108735
Fast solver for the three-factor Heston-Hull/White problem F.H.C. Naber Floris.Naber@INGbank.com tw8735 Amsterdam march 27 Contents Introduction 2. Stochastic Models..........................................
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More information1 The Collocation Method
CS410 Assignment 7 Due: 1/5/14 (Fri) at 6pm You must wor eiter on your own or wit one partner. You may discuss bacground issues and general solution strategies wit oters, but te solutions you submit must
More informationPricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching
Pricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching Kees Oosterlee Numerical analysis group, Delft University of Technology Joint work with Coen Leentvaar, Ariel
More informationExample SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross
CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal
More information3. Reaction Diffusion Equations Consider the following ODE model for population growth
3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent
More information1D Numerical Methods With Finite Volumes
1D Numerical Methods With Finite Volumes Guillaume Riflet MARETEC IST 1 The advection-diffusion equation The original concept, applied to a property within a control volume V, from which is derived the
More informationPart II: Finite Difference/Volume Discretisation for CFD
Part II: Finite Difference/Volume Discretisation for CFD Finite Volume Metod of te Advection-Diffusion Equation A Finite Difference/Volume Metod for te Incompressible Navier-Stokes Equations Marker-and-Cell
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More informationLecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows
Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3.- 1 Basics: equations of continuum mechanics - balance equations for mass and momentum - balance equations for the energy and the chemical
More informationOptimization of Supply Chain Networks
Optimization of Supply Chain Networks M. Herty TU Kaiserslautern September 2006 (2006) 1 / 41 Contents 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal
More informationTo give it a definition, an implicit function of x and y is simply any relationship that takes the form:
2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationHeavy Parallelization of Alternating Direction Schemes in Multi-Factor Option Valuation Models. Cris Doloc, Ph.D.
Heavy Parallelization of Alternating Direction Schemes in Multi-Factor Option Valuation Models Cris Doloc, Ph.D. WHO INTRO Ex-physicist Ph.D. in Computational Physics - Applied TN Plasma (10 yrs) Working
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationFourth-Order Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions
Fourth-Order Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions Jennifer Zhao, 1 Weizhong Dai, Tianchan Niu 1 Department of Mathematics and Statistics, University of Michigan-Dearborn,
More informationNumerical Methods for Differential Equations
1 Numerical Methods for Differential Equations 1 2 NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS Introduction Differential equations can describe nearly all systems undergoing change. They are ubiquitous
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationPricing participating policies with rate guarantees and bonuses
Pricing participating policies with rate guarantees and bonuses Chi Chiu Chu and Yue Kuen Kwok Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
More informationVectors. Objectives. Assessment. Assessment. Equations. Physics terms 5/15/14. State the definition and give examples of vector and scalar variables.
Vectors Objectives State the definition and give examples of vector and scalar variables. Analyze and describe position and movement in two dimensions using graphs and Cartesian coordinates. Organize and
More informationTime domain modeling
Time domain modeling Equationof motion of a WEC Frequency domain: Ok if all effects/forces are linear M+ A ω X && % ω = F% ω K + K X% ω B ω + B X% & ω ( ) H PTO PTO + others Time domain: Must be linear
More informationBlack-Scholes Option Pricing Model
Black-Scholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationIterative Solvers for Linear Systems
9th SimLab Course on Parallel Numerical Simulation, 4.10 8.10.2010 Iterative Solvers for Linear Systems Bernhard Gatzhammer Chair of Scientific Computing in Computer Science Technische Universität München
More informationMathematical test criteria for filtering complex systems: Plentiful observations
Available online at www.sciencedirect.com Journal of Computational Physics 7 (8) 678 7 www.elsevier.com/locate/jcp Mathematical test criteria for filtering complex systems: Plentiful ervations E. Castronovo,
More informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationLecture Notes on the Mathematics of Finance
Lecture Notes on the Mathematics of Finance Jerry Alan Veeh February 20, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects
More informationME6130 An introduction to CFD 1-1
ME6130 An introduction to CFD 1-1 What is CFD? Computational fluid dynamics (CFD) is the science of predicting fluid flow, heat and mass transfer, chemical reactions, and related phenomena by solving numerically
More informationNumerical Methods For Image Restoration
Numerical Methods For Image Restoration CIRAM Alessandro Lanza University of Bologna, Italy Faculty of Engineering CIRAM Outline 1. Image Restoration as an inverse problem 2. Image degradation models:
More informationFluid Dynamics and the Navier-Stokes Equation
Fluid Dynamics and the Navier-Stokes Equation CMSC498A: Spring 12 Semester By: Steven Dobek 5/17/2012 Introduction I began this project through a desire to simulate smoke and fire through the use of programming
More informationIntroduction to CFD Basics
Introduction to CFD Basics Rajesh Bhaskaran Lance Collins This is a quick-and-dirty introduction to the basic concepts underlying CFD. The concepts are illustrated by applying them to simple 1D model problems.
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationGrid adaptivity for systems of conservation laws
Grid adaptivity for systems of conservation laws M. Semplice 1 G. Puppo 2 1 Dipartimento di Matematica Università di Torino 2 Dipartimento di Scienze Matematiche Politecnico di Torino Numerical Aspects
More information