Daniel J. Duffy

Numerical Methods in Computational Finance (eBook, PDF)

A Partial Differential Equation (PDE/FDM) Approach

• Format: PDF

Produktbeschreibung

This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance. The book is structured so that it can be read by beginners, novices and expert users. Part A Mathematical Foundation for One-Factor Problems Chapters 1 to 7 introduce the mathematical and numerical analysis concepts that are needed to understand the finite difference method and its application to computational finance. Part B Mathematical Foundation for Two-Factor Problems Chapters 8 to 13 discuss a number of rigorous mathematical techniques relating to elliptic and parabolic partial differential equations in two space variables. In particular, we develop strategies to preprocess and modify a PDE before we approximate it by the finite difference method, thus avoiding ad-hoc and heuristic tricks. Part C The Foundations of the Finite Difference Method (FDM) Chapters 14 to 17 introduce the mathematical background to the finite difference method for initial boundary value problems for parabolic PDEs. It encapsulates all the background information to construct stable and accurate finite difference schemes. Part D Advanced Finite Difference Schemes for Two-Factor Problems Chapters 18 to 22 introduce a number of modern finite difference methods to approximate the solution of two factor partial differential equations. This is the only book we know of that discusses these methods in any detail. Part E Test Cases in Computational Finance Chapters 23 to 26 are concerned with applications based on previous chapters. We discuss finite difference schemes for a wide range of one-factor and two-factor problems. This book is suitable as an entry-level introduction as well as a detailed treatment of modern methods as used by industry quants and MSc/MFE students in finance. The topics have applications to numerical analysis, science and engineering. More on computational finance and the author's online courses, see www.datasim.nl.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 544
• Erscheinungstermin: 14. März 2022
• Englisch
• ISBN-13: 9781119719694
• Artikelnr.: 63617978

Autorenporträt

DANIEL DUFFY, PhD, has BA (Mod), MSc and PhD degrees in pure, applied and numerical mathematics (University of Dublin, Trinity College) and he is active in promoting partial differential equations (PDE) and the Finite Difference Method (FDM) for applications in computational finance. He was responsible for the introduction of the Fractional Step (Soviet Splitting) method and the Alternating Direction Explicit (ADE) method in computational finance. He is the originator of the exponential fitting method for convection-dominated PDEs.

Inhaltsangabe

Preface xix

Who Should Read this Book? xxiii

Part A : Mathematical Foundation for One-Factor Problems

Chapter 1 : Real Analysis Foundations for this Book 3

1.1 Introduction and Objectives 3

1.2 Continuous Functions 4

1.2.1 Formal Definition of Continuity 5

1.2.2 An Example 6

1.2.3 Uniform Continuity 6

1.2.4 Classes of Discontinuous Functions 7

1.3 Differential Calculus 8

1.3.1 Taylor's Theorem 9

1.3.2 Big O and Little o Notation 10

1.4 Partial Derivatives 11

1.5 Functions and Implicit Forms 13

1.6 Metric Spaces and Cauchy Sequences 14

1.6.1 Metric Spaces 15

1.6.2 Cauchy Sequences 16

1.6.3 Lipschitz Continuous Functions 17

1.7 Summary and Conclusions 19

Chapter 2 : Ordinary Differential Equations (ODEs), Part 1 21

2.1 Introduction and Objectives 21

2.2 Background and Problem Statement 22

2.2.1 Qualitative Properties of the Solution and Maximum Principle 22

2.2.2 Rationale and Generalisations 24

2.3 Discretisation of Initial Value Problems: Fundamentals 25

2.3.1 Common Schemes 26

2.3.2 Discrete Maximum Principle 28

2.4 Special Schemes 29

2.4.1 Exponential Fitting 29

2.4.2 Scalar Non-Linear Problems and Predictor-Corrector Method 31

2.4.3 Extrapolation 31

2.5 Foundations of Discrete Time Approximations 32

2.6 Stiff ODEs 37

2.7 Intermezzo: Explicit Solutions 39

2.8 Summary and Conclusions 41

Chapter 3 : Ordinary Differential Equations (ODEs), Part 2 43

3.1 Introduction and Objectives 43

3.2 Existence and Uniqueness Results 43

3.2.1 An Example 45

3.3 Other Model Examples 45

3.3.1 Bernoulli ODE 45

3.3.2 Riccati ODE 46

3.3.3 Predator-Prey Models 47

3.3.4 Logistic Function 48

3.4 Existence Theorems for Stochastic Differential Equations (SDEs) 48

3.4.1 Stochastic Differential Equations (SDEs) 49

3.5 Numerical Methods for ODEs 51

3.5.1 Code Samples in Python 52

3.6 The Riccati Equation 55

3.6.1 Finite Difference Schemes 57

3.7 Matrix Differential Equations 59

3.7.1 Transition Rate Matrices and Continuous Time Markov Chains 61

3.8 Summary and Conclusions 62

Chapter 4 : An Introduction to Finite Dimensional Vector Spaces 63

4.1 Short Introduction and Objectives 63

4.1.1 Notation 64

4.2 What Is a Vector Space? 65

4.3 Subspaces 67

4.4 Linear Independence and Bases 68

4.5 Linear Transformations 69

4.5.1 Invariant Subspaces 70

4.5.2 Rank and Nullity 71

4.6 Summary and Conclusions 72

Chapter 5 : Guide to Matrix Theory and Numerical Linear Algebra 73

5.1 Introduction and Objectives 73

5.2 From Vector Spaces to Matrices 73

5.2.1 Sums and Scalar Products of Linear Transformations 73

5.3 Inner Product Spaces 74

5.3.1 Orthonormal Basis 75

5.4 From Vector Spaces to Matrices 76

5.4.1 Some Examples 76

5.5 Fundamental Matrix Properties 77

5.6 Essential Matrix Types 80

5.6.1 Nilpotent and Related Matrices 80

5.6.2 Normal Matrices 81

5.6.3 Unitary and Orthogonal Matrices 82

5.6.4 Positive Definite Matrices 82

5.6.5 Non-Negative Matrices 83

5.6.6 Irreducible Matrices 83

5.6.7 Other Kinds of Matrices 84

5.7 The Cayley Trans