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Numerical Methods in Computational Finance. A Partial Differential Equation (PDE/FDM) Approach. Edition No. 1. Wiley Finance

  • Book

  • 544 Pages
  • March 2022
  • John Wiley and Sons Ltd
  • ID: 5837283

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.

Table of Contents

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 Transform 84

5.8 Summary and Conclusions 86

Chapter 6 : Numerical Solutions of Boundary Value Problems 87

6.1 Introduction and Objectives 87

6.2 An Introduction to Numerical Linear Algebra 87

6.2.1 BLAS (Basic Linear Algebra Subprograms) 90

6.3 Direct Methods for Linear Systems 92

6.3.1 LU Decomposition 92

6.3.2 Cholesky Decomposition 94

6.4 Solving Tridiagonal Systems 94

6.4.1 Double Sweep Method 94

6.4.2 Thomas Algorithm 96

6.4.3 Block Tridiagonal Systems 97

6.5 Two-Point Boundary Value Problems 99

6.5.1 Finite Difference Approximation 100

6.5.2 Approximation of Boundary Conditions 102

6.6 Iterative Matrix Solvers 103

6.6.1 Iterative Methods 103

6.6.2 Jacobi Method 104

6.6.3 Gauss-Seidel Method 104

6.6.4 Successive Over-Relaxation (SOR) 105

6.6.5 Other Methods 105

6.7 Example: Iterative Solvers for Elliptic PDEs 106

6.8 Summary and Conclusions 107

Chapter 7 : Black-Scholes Finite Differences for the Impatient 109

7.1 Introduction and Objectives 109

7.2 The Black-Scholes Equation: Fully Implicit and Crank-Nicolson Methods 110

7.2.1 Fully Implicit Method 110

7.2.2 Crank-Nicolson Method 111

7.2.3 Final Remarks 114

7.3 The Black-Scholes Equation: Trinomial Method 115

7.3.1 Comparison with Other Methods 115

7.4 The Heat Equation and Alternating Direction Explicit (ADE) Method 120

7.4.1 Background and Motivation 120

7.5 ADE for Black-Scholes: Some Test Results 121

7.6 Summary and Conclusions 126

Part B : Mathematical Foundation for Two-Factor Problems

Chapter 8 : Classifying and Transforming Partial Differential Equations 129

8.1 Introduction and Objectives 129

8.2 Background and Problem Statement 129

8.3 Introduction to Elliptic Equations 130

8.3.1 What is an Elliptic Operator? 130

8.3.2 Total and Principal Symbols 131

8.3.3 The Adjoint Equation 132

8.3.4 Self-Adjoint Operators and Equations 133

8.3.5 Numerical Approximation of PDEs in Adjoint Form 134

8.3.6 Elliptic Equations with Non-Negative Characteristic Form 135

8.4 Classification of Second-Order Equations 135

8.4.1 Characteristics 136

8.4.2 Model Example 137

8.4.3 Test your Knowledge 138

8.5 Examples of Two-Factor Models from Computational Finance 139

8.5.1 Multi-Asset Options 139

8.5.2 Stochastic Dividend PDE 140

8.6 Summary and Conclusions 141

Chapter 9 : Transforming Partial Differential Equations to a Bounded Domain 143

9.1 Introduction and Objectives 143

9.2 The Domain in Which a PDE Is Defined: Preamble 143

9.2.1 Background and Specific Mappings 144

9.2.2 Initial Examples 146

9.3 Other Examples 147

9.4 Hotspots 148

9.5 What Happened to Domain Truncation? 148

9.6 Another Way to Remove Mixed Derivative Terms 149

9.7 Summary and Conclusions 151

Chapter 10 : Boundary Value Problems for Elliptic and Parabolic Partial Differential Equations 153

10.1 Introduction and Objectives 153

10.2 Notation and Prerequisites 154

10.3 The Laplace Equation 154

10.3.1 Harmonic Functions and the Cauchy-Riemann Equations 154

10.4 Properties of The Laplace Equation 156

10.4.1 Maximum-Minimum Principle for Laplace’s Equation 158

10.5 Some Elliptic Boundary Value Problems 159

10.5.1 Some Motivating Examples 159

10.6 Extended Maximum-Minimum Principles 159

10.6.1 An Example 161

10.7 Summary and Conclusions 162

Chapter 11 : Fichera Theory, Energy Inequalities and Integral Relations 163

11.1 Introduction and Objectives 163

11.2 Background and Problem Statement 163

11.2.1 The ‘Big Bang’: Cauchy-Euler Equation 163

11.3 Well-Posed Problems and Energy Estimates 165

11.3.1 Time to Reflect: What Have We Achieved and What’s Next? 167

11.4 The Fichera Theory: Overview 168

11.5 The Fichera Theory: The Core Business 168

11.6 The Fichera Theory: Further Examples and Applications 171

11.6.1 Cox-Ingersoll-Ross (CIR) 171

11.6.2 Heston Model Fundamenals 172

11.6.3 Heston Model by Fichera Theory 176

11.6.4 First-Order Hyperbolic PDE in One and Two Space Variables 177

11.7 Some Useful Theorems 178

11.7.1 Divergence (Gauss-Ostrogradsky) Theorem 179

11.7.2 Green’s Theorem/Formula 180

11.7.3 Green’s First and Second Identities 180

11.8 Summary and Conclusions 180

Chapter 12 : An Introduction to Time-Dependent Partial Differential Equations 181

12.1 Introduction and Objectives 181

12.2 Notation and Prerequisites 181

12.3 Preamble: Separation of Variables for the Heat Equation 182

12.4 Well-Posed Problems 184

12.4.1 Examples of an ill-posed Problem 185

12.4.2 The Importance of Proving that Problems Are Well-Posed 187

12.5 Variations on Initial Boundary Value Problem for the Heat Equation 188

12.5.1 Smoothness and Compatibility Conditions 188

12.6 Maximum-Minimum Principles for Parabolic PDEs 189

12.7 Parabolic Equations with Time-Dependent Boundaries 190

12.8 Uniqueness Theorems for Boundary Value Problems in Two Dimensions 192

12.8.1 Laplace Equation 192

12.8.2 Heat Equation 193

12.9 Summary and Conclusions 193

Chapter 13 : Stochastics Representations of PDEs and Applications 195

13.1 Introduction and Objectives 195

13.2 Background, Requirements and Problem Statement 196

13.3 An Overview of Stochastic Differential Equations (SDEs) 196

13.4 An Introduction to One-Dimensional Random Processes 196

13.5 An Introduction to the Numerical Approximation of SDEs 199

13.5.1 Euler-Maruyama Method 199

13.5.2 Milstein Method 201

13.5.3 Predictor-Corrector Method 201

13.5.4 Drift-Adjusted Predictor-Corrector Method 202

13.6 Path Evolution and Monte Carlo Option Pricing 203

13.6.1 Monte Carlo Option Pricing 204

13.6.2 Some C++ Code 205

13.7 Two-Factor Problems 209

13.7.1 Spread Options with Stochastic Volatility 209

13.7.2 Heston Stochastic Volatility Model 211

13.8 The Ito Formula 215

13.9 Stochastics Meets PDEs 215

13.9.1 A Statistics Refresher 215

13.9.2 The Feynman-Kac Formula 217

13.9.3 Kolmogorov Equations 218

13.9.4 Kolmogorov Forward (Fokker-Planck (FPE)) Equation 218

13.9.5 Multi-Dimensional Problems and Boundary Conditions 219

13.9.6 Kolmogorov Backward Equation (KBE) 220

13.10 First Exit-Time Problems 221

13.11 Summary and Conclusions 222

Part C : The Foundations of the Finite Difference Method (FDM)

Chapter 14 : Mathematical and Numerical Foundations of the Finite Difference Method, Part I 225

14.1 Introduction and Objectives 225

14.2 Notation and Prerequisites 226

14.3 What Is the Finite Difference Method, Really? 227

14.4 Fourier Analysis of Linear PDEs 227

14.4.1 Fourier Transform for Advection Equation 229

14.4.2 Fourier Transform for Diffusion Equation 230

14.5 Discrete Fourier Transform 232

14.5.1 Finite and Infinite Dimensional Sequences and Their Norms 232

14.5.2 Discrete Fourier Transform (DFT) 233

14.5.3 Discrete von Neumann Stability Criterion 235

14.5.4 Some More Examples 235

14.6 Theoretical Considerations 237

14.6.1 Consistency 237

14.6.2 Stability 238

14.6.3 Convergence 239

14.7 First-Order Partial Differential Equations 239

14.7.1 Why First-Order Equations are Different: Essential Difficulties 242

14.7.2 A Simple Explicit Scheme 243

14.7.3 Some Common Schemes for Initial Value Problems 245

14.7.4 Some Other Schemes 246

14.7.5 General Linear Problems 248

14.8 Summary and Conclusions 248

Chapter 15: Mathematical and Numerical Foundations of the Finite Difference Method, Part II 249

15.1 Introduction and Objectives 249

15.2 A Short History of Numerical Methods for CDR Equations 250

15.2.1 Temporal and Spatial Stability 251

15.2.2 Motivating Exponential Fitting Methods 253

15.2.3 Eliminating Temporal and Spatial Stability Problems 254

15.3 Exponential Fitting and Time-Dependent Convection-Diffusion 257

15.4 Stability and Convergence Analysis 258

15.5 Special Limiting Cases 260

15.6 Stability for Initial Boundary Value Problems 260

15.6.1 Gerschgorin’s Circle Theorem 261

15.7 Semi-Discretisation for Convection-Diffusion Problems 264

15.7.1 Essentially Positive Matrices 265

15.7.2 Fully Discrete Schemes 267

15.8 Padé Matrix Approximation 269

15.8.1 Padé Matrix Approximations 270

15.9 Time-Dependent Convection-Diffusion Equations 275

15.9.1 Fully Discrete Schemes 275

15.10 Summary and Conclusions 276

Chapter 16 Sensitivity Analysis, Option Greeks and Parameter Optimisation, Part I 277

16.1 Introduction and Objectives 277

16.2 Helicopter View of Sensitivity Analysis 278

16.3 Black-Scholes-Merton Greeks 279

16.3.1 Higher-Order and Mixed Greeks 282

16.4 Divided Differences 282

16.4.1 Approximation to First and Second Derivatives 282

16.4.2 Black-Scholes Numeric Greeks and Divided Differences 285

16.5 Cubic Spline Interpolation 286

16.5.1 Caveat: Cubic Splines with Sparse Input Data 289

16.5.2 Cubic Splines for Option Greeks 290

16.5.3 Boundary Conditions 291

16.6 Some Complex Function Theory 292

16.6.1 Curves and Regions 293

16.6.2 Taylor’s Theorem and Series 294

16.6.3 Laurent’s Theorem and Series 295

16.6.4 Cauchy-Goursat Theorem 296

16.6.5 Cauchy’s Integral Formula 297

16.6.6 Cauchy’s Residue Theorem 298

16.6.7 Gauss’s Mean Value Theorem 299

16.7 The Complex Step Method (CSM) 299

16.7.1 Caveats 302

16.8 Summary and Conclusions 302

Chapter 17 Advanced Topics in Sensitivity Analysis 305

17.1 Introduction and Objectives 305

17.2 Examples of CSE 305

17.2.1 Simple Initial Value Problem 306

17.2.2 Population Dynamics 307

17.2.3 Comparing CSE and Complex Step Method (CSM) 310

17.3 CSE and Black-Scholes PDE 310

17.3.1 Black-Scholes Greeks: Algorithms and Design 311

17.3.2 Some Specific Black-Scholes Greeks 312

17.4 Using Operator Calculus to Compute Greeks 313

17.5 An Introduction to Automatic Differentiation (AD) for the Impatient 314

17.5.1 What Is Automatic Differentiation: The Details 316

17.6 Dual Numbers 317

17.7 Automatic Differentiation in C++ 318

17.8 Summary and Conclusions 319

Part D : Advanced Finite Difference Schemes for Two-Factor Problems

Chapter 18 : Splitting Methods, Part I 323

18.1 Introduction and Objectives 323

18.2 Background and History 324

18.3 Notation, Prerequisites and Model Problems 325

18.4 Motivation: Two-Dimensional Heat Equation 328

18.4.1 Alternating Direction Implicit (ADI) Method 328

18.4.2 Soviet (Operator) Splitting 330

18.4.3 Mixed Derivative and Yanenko Scheme 331

18.5 Other Related Schemes for the Heat Equation 333

18.5.1 D’Yakonov Method 333

18.5.2 Approximate Factorisation of Operators 334

18.5.3 Predictor-Corrector Methods 337

18.5.4 Partial Integro Differential Equations (PIDEs) 338

18.6 Boundary Conditions 339

18.7 Two-Dimensional Convection PDEs 341

18.8 Three-Dimensional Problems 343

18.9 The Hopscotch Method 344

18.10 Software Design and Implementation Guidelines 346

18.11 The Future: Convection-Diffusion Equations 346

18.12 Summary and Conclusions 347

Chapter 19 : The Alternating Direction Explicit (ADE) Method 349

19.1 Introduction and Objectives 349

19.2 Background and Problem Statement 351

19.3 Global Overview and Applicability of ADE 351

19.4 Motivating Examples: One-Dimensional and Two-Dimensional Diffusion Equations 352

19.4.1 Barakat and Clark (B&C) Method 353

19.4.2 Saul’yev Method 354

19.4.3 Larkin Method 355

19.4.4 Two-Dimensional Diffusion Problems 355

19.5 ADE for Convection (Advection) Equation 356

19.6 Convection-Diffusion PDEs 358

19.6.1 Example: Black-Scholes PDE 359

19.6.2 Boundary Conditions 360

19.6.3 Spatial Amplification Errors 361

19.7 Attention Points with ADE 362

The Consequences of Conditional Consistency 362

Call Pay-Off Behaviour at the Far Field 362

19.7.1 General Formulation of the ADE Method 362

19.8 Summary and Conclusions 364

Chapter 20 : The Method of Lines (MOL), Splitting and the Matrix Exponential 365

20.1 Introduction and Objectives 365

20.2 Notation and Prerequisites: The Exponential Function 366

20.2.1 Initial Results 367

20.2.2 The Exponential of a Matrix 367

20.3 The Exponential of a Matrix: Advanced Topics 368

20.3.1 Fundamental Theorem for Linear Systems 368

Proof of Theorem 20.1. 369

20.3.2 An Example 369

20.4 Motivation: One-Dimensional Heat Equation 370

20.5 Semi-Linear Problems 373

20.6 Test Case: Double-Barrier Options 375

20.6.1 PDE Formulation 376

20.6.2 Using Exponential Fitting of Barrier Options 377

20.6.3 Performing MOL with Boost C++ odeint 378

20.6.4 Computing Sensitivities 381

20.6.5 American Options 384

20.7 Summary and Conclusions 384

Chapter 21 : Free and Moving Boundary Value Problems 387

21.1 Introduction and Objectives 387

21.2 Background, Problem Statement and Formulations 388

21.3 Notation and Prerequisites 388

21.4 Some Initial Examples of Free and Moving Boundary Value Problems 389

21.4.1 Single-Phase Melting Ice 389

21.4.2 Oxygen Diffusion 390

21.4.3 American Option Pricing 391

21.4.4 Two-Phase Melting Ice 392

21.5 An Introduction to Parabolic Variational Inequalities 392

21.5.1 Formulation of Problem: Test Case 392

21.5.2 Examples of Initial Boundary Value Problems 395

21.6 An Introduction to Front-Fixing 399

21.6.1 Front-Fixing for the Heat Equation 399

21.7 Python Code Example: ADE for American Option Pricing 400

21.8 Summary and Conclusions 405

Chapter 22 : Splitting Methods, Part II 407

22.1 Introduction and Objectives 407

22.2 Background and Problem Statement: The Essence of Sequential Splitting 408

22.3 Notation and Mathematical Formulation 408

22.3.1 C0 Semigroups 408

22.3.2 Abstract Cauchy Problem 409

22.3.3 Examples 410

22.4 Mathematical Foundations of Splitting Methods 411

22.4.1 Lie (Trotter) Product Formula 411

22.4.2 Splitting Error 411

22.4.3 Component Splitting and Operator Splitting 413

22.4.4 Splitting as a Discretisation Method 413

22.5 Some Popular Splitting Methods 414

22.5.1 First-Order (Lie-Trotter) Splitting 415

22.5.2 Predictor-Corrector Splitting 415

22.5.3 Marchuk’s Two-Cycle (1-2-2-1) Method 416

22.5.4 Strang Splitting 417

22.6 Applications and Relationships to Computational Finance 417

22.7 Software Design and Implementation Guidelines 418

22.8 Experience Report: Comparing ADI and Splitting 419

22.9 Summary and Conclusions 421

Part E : Test Cases in Computational Finance

Chapter 23 : Multi-Asset Options 425

23.1 Introduction and Objectives 425

23.2 Background and Goals 426

23.3 The Bivariate Normal Distribution (BVN) and its Applications 427

23.3.1 Computing BVN by Solving a Hyperbolic PDE 430

23.3.2 Analytical Solutions of Multi-Asset and Basket Options 433

23.4 PDE Models for Multi-Asset Option Problems: Requirements and Design 435

23.4.1 Domain Transformation 435

23.4.2 Numerical Boundary Conditions 435

23.5 An Overview of Finite Difference Schemes for Multi-Asset Option Problems 436

23.5.1 Common Design Principles 436

23.5.2 Detailed Design 438

23.5.3 Testing the Software 440

23.6 American Spread Options 440

23.7 Appendices 442

23.7.1 Traditional Approach to Numerical Boundary Conditions 442

23.7.2 Top-Down Design of Monte Carlo Applications 443

23.8 Summary and Conclusions 444

Chapter 24 : Asian (Average Value) Options 447

24.1 Introduction and Objectives 447

24.2 Background and Problem Statement 448

24.2.1 Challenges 449

24.3 Prototype PDE Model 450

24.3.1 Similarity Reduction 451

24.4 The Many Ways to Handle the Convective Term 452

24.4.1 Method of Lines (MOL) 452

24.4.2 Other Schemes 454

24.4.3 A Stable Monotone Upwind Scheme 455

24.5 ADE for Asian Options 455

24.6 ADI for Asian Options 456

24.6.1 Modern ADI Variations 458

24.7 Summary and Conclusions 459

Chapter 25 : Interest Rate Models 461

25.1 Introduction and Objectives 461

25.2 Main Use Cases 462

25.3 The CIR Model 462

25.3.1 Analytic Solutions 463

25.3.2 Initial Boundary Value Problem 466

25.4 Well-Posedness of the CIRPDE Model 466

25.4.1 Gronwall’s Inequalities 467

25.4.2 Energy Inequalities 468

25.5 Finite Difference Methods for the CIR Model 469

25.5.1 Numerical Boundary Conditions 470

25.6 Heston Model and the Feller Condition 471

25.7 Summary and Conclusion 475

Chapter 26 : Epilogue Models Follow-Up Chapters 1 to 25 477

26.1 Introduction and Objectives 477

26.2 Mixed Derivatives and Monotone Schemes 478

26.2.1 The Maximum Principle and Mixed Derivatives 478

26.2.2 Some Examples 480

26.2.3 Code Sample Method of Lines (MOL) for Two-Factor Hull-White Model 481

26.3 The Complex Step Method (CSM) Revisited 483

26.3.1 Black-Scholes Greeks Using CSM and the Faddeeva Function 483

26.3.2 CSM and Functions of Several Complex Variables 487

26.3.3 C++ Code for Extended CSM 488

26.3.4 CSM for Non-Linear Solvers 492

26.4 Extending the Hull-White: Possible Projects 493

26.5 Summary and Conclusions 495

Bibliography 497

Index 505

Authors

Daniel J. Duffy Datasim Education BV.