Numerical methods in computational finance : a partial differential equation (PDE/FDM) approach /: a partial differential equation (PDE/FDM) approach. (2022)

• Record Type: Book
• Further Information: Note: Daniel J. Duffy.
• Authors: Duffy, Daniel J
• Contents: Chapter 1 Real Analysis Foundations for this Book 1 1.1 Introduction and Objectives 1 1.2 Continuous Functions 1 1.3 Differential Calculus 5 1.4 Partial Derivatives 8 1.5 Functions and Implicit Forms 9 1.6 Metric Spaces and Cauchy Sequences 11 1.7 Summary and Conclusions 15 Chapter 2 Ordinary Differential Equations (ODEs), Part 1 17 2.1 Introduction and Objectives 17 2.2 Background and Problem Statement 17 2.3 Discretisation of Initial Value Problems: Fundamentals 20 2.4 Special Schemes 24 2.5 Foundations of Discrete Time Approximations 26 2.6 Stiff ODEs 31 2.7 Intermezzo: Explicit Solutions 33 2.8 Summary and Conclusions 34 Chapter 3 Ordinary Differential Equations (ODEs), Part 2 35 3.1 Introduction and Objectives 35 3.2 Existence and Uniqueness Results 35 3.3 Other Model Examples 37 3.4 Existence Theorems for Stochastic Differential Equations (SDEs) 39 3.5 Numerical Methods for ODEs 42 3.6 The Riccati Equation 45 3.7 Matrix Differential Equations 48 3.8 Summary and Conclusions 50 Chapter 4 An Introduction to Finite Dimensional Vector Spaces 51 4.1 Short Introduction and Objectives 51 4.2 What is a Vector Space? 52 4.3 Subspaces 55 4.4 Linear Independence and Bases 56 4.5 Linear Transformations 57 4.6 Summary and Conclusions 59 Chapter 5 Guide to Matrix Theory and Numerical Linear Algebra 61 5.1 Introduction and Objectives 61 5.2 From Vector Spaces to Matrices 61 5.3 Inner Product Spaces 62 5.4 From Vector Spaces to Matrices 63 5.5 Fundamental Matrix Properties 65 5.6 Essential Matrix Types 67 5.7 The Cayley Transform 71 5.8 Summary and Conclusions 73 Chapter 6 Numerical Solutions of Boundary Value Problems 75 6.1 Introduction and Objectives 75 6.2 An Introduction to Numerical Linear Algebra 75 6.3 Direct Methods for Linear Systems 79 6.4 Solving Tridiagonal Systems 81 6.5 Two-Point Boundary Value Problems 85 6.6 Iterative Matrix Solvers 89 6.7 Example: Iterative Solvers for Elliptic PDEs 92 6.8 Summary and Conclusions 93 Chapter 7 Black Scholes Finite Differences for the Impatient 95 7.1 Introduction and Objectives 95 7.2 The Black Scholes Equation: Fully Implicit and Crank Nicolson Methods 95 7.3 The Black Scholes Equation: Trinomial Method 99 7.4 The Heat Equation and Alternating Direction Explicit (ADE) Method 103 7.5 ADE for Black Scholes: some Test Results 104 7.6 Summary and Conclusions 108 Chapter 8 Classifying and Transforming Partial Differential Equations 109 8.1 Introduction and Objectives 109 8.2 Background and Problem Statement 109 8.3 Introduction to Elliptic Equations 109 8.4 Classification of Second-Order Equations 114 8.5 Examples of Two-Factor Models from Computational Finance 116 8.6 Summary and Conclusions 118 Chapter 9 Transforming Partial Differential Equations to a Bounded Domain 121 9.1 Introduction and Objectives 121 9.2 The Domain in which a PDE is defined: Preamble 121 9.3 Other Examples 124 9.4 Hotspots 125 9.5 What happened to Domain Truncation? 125 9.6 Another Way to remove Mixed Derivative Terms 126 9.7 Summary and Conclusions 128 Chapter 10 Boundary Value Problems for Elliptic and Parabolic Partial Differential Equations 129 10.1 Introduction and Objectives 129 10.2 Notation and Prerequisites 129 10.3 The Laplace Equation 129 10.4 Properties of The Laplace Equation 131 10.5 Some Elliptic Boundary Value Problems 134 10.6 Extended Maximum-Minimum Principles 134 10.7 Summary and Conclusions 136 Chapter 11 Fichera Theory, Energy Inequalities and Integral Relations 137 11.1 Introduction and Objectives 137 11.2 Background and Problem Statement 137 11.3 Well-posed Problems and Energy Estimates 139 11.4 The Fichera Theory: Overview 140 11.5 The Fichera Theory: The Core Business 141 11.6 The Fichera Theory: Further Examples and Applications 143 11.7 Some Useful Theorems 149 11.8 Summary and Conclusions 151 Chapter 12 An Introduction to Time-dependent Partial Differential Equations 153 12.1 Introduction and Objectives 153 12.2 Notation and Prerequisites 153 12.3 Preamble: Separation of Variables for the Heat Equation 153 12.4 Well-posed Problems 155 12.5 Variations on Initial Boundary Value Problem for the Heat Equation 159 12.6 Maximum-Minimum Principles for Parabolic PDEs 160 12.7 Parabolic Equations with Time-Dependent Boundaries 160 12.8 Uniqueness Theorems for Boundary Value Problems in Two Dimensions 162 12.9 Summary and Conclusions 164 Chapter 13 Stochastics Representations of PDEs and Applications 165 13.1 Introduction and Objectives 165 13.2 Background, Requirements and Problem Statement 165 13.3 An Overview of Stochastic Differential Equations (SDEs) 165 13.4 An Introduction to One-Dimensional Random Processes 166 13.5 An Introduction to the Numerical Approximation of SDEs 168 13.6 Path Evolution and Monte Carlo Option Pricing 172 13.7 Two-Factor Problems 177 13.8 The Ito Formula 181 13.9 Stochastics meets PDEs 182 13.10 First Exit-Time Problems 187 13.11 Summary and Conclusions 188 Chapter 14 Mathematical and Numerical Foundations of the Finite Difference Method, Part I 189 14.1 Introduction and Objectives 189 14.2 Notation and Prerequisites 189 14.3 What is the Finite Difference Method, really? 190 14.4 Fourier Analysis of Linear PDEs 190 14.5 Discrete Fourier Transform 194 14.6 Theoretical Considerations 199 4.7 First-Order Partial Differential Equations 201 14.8 Summary and Conclusions 208 Chapter 15 Mathematical and Numerical Foundations of the Finite Difference Method, Part II 209 15.1 Introduction and Objectives 209 15.2 A Short History of Numerical Methods for CDR Equations 210 15.3 Exponential Fitting and Time-dependent Convection-Diffusion 216 15.4 Stability and Convergence Analysis 217 15.5 Special limiting Cases 218 15.6 Stability for Initial Boundary Value Problems 218 15.7 Semi-Discretisation for Convection-Diffusion Problems 221 15.8 Padé Matrix Approximation 226 15.9 Time-Dependent Convection-Diffusion Equations 231 15.10 Summary and Conclusions 232 Chapter 16 Sensitivity Analysis, Option Greeks and Parameter Optimisation, Part I 233 16.1 Introduction and Objectives 233 16.2 Helicopter View of Sensitivity Analysis 233 16.3 Black-Scholes-Merton Greeks 234 16.4 Divided Differences 236 16.5 Cubic Spline Interpolation 240 16.6 Some Complex Function Theory 245 16.7 The Complex Step Method (CSM) 251 16.8 Summary and Conclusions 254 Chapter 17 Advanced Topics in Sensitivity Analysis 255 17.1 Introduction and Objectives 255 17.2 Examples of CSE 255 17.3 CSE and Black Scholes PDE 259 17.4 Using Operator Calculus to compute Greeks 262 17.5 An Introduction to Automatic Differentiation (AD) 263 17.6 Dual Numbers 265 17.7 Automatic Differentiation in C++ 266 17.8 Summary and Conclusions 267 Chapter 18 Splitting Methods, Part I 269 18.1 Introduction and Objectives 269 18.2 Background and History 2
• Edition: 1st
• Publisher Details: Hoboken : John Wiley & Sons, Inc
• Publication Date: 2022
• Extent: 1 online resource
• Subjects: 658.15; Financial engineering; Differential equations, Partial
• Languages: English
• ISBNs: 9781119719724
• Notes: Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
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