John Wiley & Sons Numerical Methods in Computational Finance Cover This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary an.. Product #: 978-1-119-71967-0 Regular price: $80.28 $80.28 In Stock

Numerical Methods in Computational Finance

A Partial Differential Equation (PDE/FDM) Approach

Duffy, Daniel J.

Wiley Finance Editions

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1. Edition March 2022
544 Pages, Hardcover
Wiley & Sons Ltd

ISBN: 978-1-119-71967-0
John Wiley & Sons

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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.

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 269

18.3 Notation, Prerequisites and Model Problems 270

18.4 Motivation: Two-Dimensional Heat Equation 273

18.5 Other Related Schemes for the Heat Equation 277

18.6 Boundary Conditions 281

18.7 Two-Dimensional Convection PDEs 282

18.8 Three-Dimensional Problems 284

18.9 The Hopscotch Method 285

18.10 Software Design and Implementation Guidelines 286

18.11 The Future: Convection-Diffusion Equations 287

18.12 Summary and Conclusions 287

Chapter 19 The Alternating Direction Explicit (ADE) Method 289

19.1 Introduction and Objectives 289

19.2 Background and Problem Statement 290

19.3 Global Overview and Applicability of ADE 290

19.4 Motivating Examples: One-Dimensional and Two-Dimensional Diffusion Equations 291

19.5 ADE for Convection (Advection) Equation 294

19.6 Convection-Diffusion PDEs 295

19.7 Attention Points with ADE 299

19.8 Summary and Conclusions 300

Chapter 20 The Method of Lines (MOL), Splitting and the Matrix Exponential 303

20.1 Introduction and Objectives 303

20.2 Notation and Prerequisites: The Exponential Function 303

20.3 The Exponential of a Matrix: Advanced Topics 305

20.4 Motivation: One-dimensional Heat Equation 307

20.5 Semilinear Problems 309

20.6 Test Case: Double-Barrier Options 311

20.7 Summary and Conclusions 318

Chapter 21 Free and Moving Boundary Value Problems 321

21.1 Introduction and Objectives 321

21.2 Background, Problem Statement and Formulations 321

21.3 Notation and Prerequisites 321

21.4 Some Initial Examples of Free and Moving Boundary Value Problems 322

21.5 An Introduction to Parabolic Variational Inequalities 325

21.6 An Introduction to Front-Fixing 332

21.7 Python Code Example: ADE for American Option Pricing 333

21.8 Summary and Conclusions 336

Chapter 22 Splitting Methods, Part II 337

22.1 Introduction and Objectives 337

22.2 Background and Problem Statement: the Essence of Sequential Splitting 337

22.3 Notation and Mathematical Formulation 338

22.4 Mathematical Foundations of Splitting Methods 340

22.5 Some Popular Splitting Methods 343

22.6 Applications and Relationships to Computational Finance 345

22.7 Software Design and Implementation Guidelines 346

22.8 Experience Report: Comparing ADI and Splitting 347

22.9 Summary and Conclusions 348

Chapter 23 Multi-Asset Options 349

23.1 Introduction and Objectives 349

23.2 Background and Goals 349

23.3 The Bivariate Normal Distribution (BVN) and its Applications 351

23.4 PDE Models for Multi-Asset Option Problems: Requirements and Design 356

23.5 An Overview of Finite Difference Schemes for Multi-Asset Option Problems 357

23.6 American Spread Options 361

23.7 Appendices 362

23.8 Summary and Conclusions 364

Chapter 24 Asian (Average Value) Options 365

24.1 Introduction and Objectives 365

24.2 Background and Problem Statement 365

24.3 Prototype PDE Model 367

24.4 The Many Ways to handle the Convective Term 369

24.5 ADE for Asian Options 371

24.6 ADI for Asian Options 372

24.7 Summary and Conclusions 374

Chapter 25 Interest Rate Models 375

25.1 Introduction and Objectives 375

25.2 Main Use Cases 375

25.3 The CIR Model 376

25.4 Well-Posedness of the CIRPDE Model 379

25.5 Finite Difference Methods for the CIR Model 381

25.6 Heston Model and the Feller Condition 383

25.7 Summary and Conclusion 386

Chapter 26 Epilogue Models Follow-up Chapters 1 to 25 387

26.1 Introduction and Objectives 387

26.2 Mixed Derivatives and Monotone Schemes 387

26.3 The Complex Step Method (CSM) Revisited 392

26.4 Extending the Hull-White: Possible Projects 398

26.5 Summary and Conclusions 400
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.

D. J. Duffy, Datasim Education BV