Financial Instrument Pricing Using C++

1 Executive Overview of this Book 1

1.1 What is this book? 1

1.2 What’s special about this book? 1

1.3 Who is this book for? 2

1.4 Software requirements 3

1.5 The structure of this book 4

1.6 Pedagogical approach 5

1.7 What this book is not 6

1.8 Source code on the CD 6

PART I TEMPLATE PROGRAMMING IN C++

2 A Gentle Introduction to Templates in C++ 9

2.1 Introduction and objectives 9

2.2 Motivation and background 10

2.3 Defining a template 11

2.3.1 An example 13

2.4 Template instantiation 15

2.5 Function templates 16

2.5.1 An example 17

2.6 Default values and typedefs 18

2.7 Guidelines when implementing templates 18

2.8 Conclusions and summary 19

3 An Introduction to the Standard Template Library 20

3.1 Introduction and objectives 20

3.2 A Bird’s-eye view of STL 20

3.3 Sequence containers 23

3.4 Associative containers 27

3.5 Iterators in STL 30

3.6 Algorithms 33

3.7 Using STL for financial instruments 35

3.8 Conclusions and summary 35

4 STL for Financial Engineering Applications 36

4.1 Introduction and objectives 36

4.2 Clever data structures 36

4.3 Set theory and STL 40

4.4 Useful algorithms 43

4.5 STL adaptor containers 45

4.6 Conclusions and summary 46

5 The Property Pattern in Financial Engineering 47

5.1 Introduction and objectives 47

5.2 The Property pattern 47

5.3 An example 51

5.4 Extending the Property pattern: property sets and property lists 52

5.5 Properties and exotic options 57

5.6 Conclusions and summary 59

PART II BUILDING BLOCK CLASSES

6 Arrays, Vectors and Matrices 63

6.1 Introduction and objectives 63

6.2 Motivation and background 64

6.3 A layered approach 66

6.4 The Array and Matrix classes in detail 66

6.5 The Vector and NumericMatrix classes in detail 72

6.6 Associative arrays and matrices 74

6.7 Conclusions and summary 77

7 Arrays and Matrix Properties 78

7.1 Introduction and objectives 78

7.2 An overview of the functionality 78

7.3 Software requirements 79

7.4 The core processes 80

7.5 Other function categories 85

7.6 Using the functions 87

7.7 An introduction to exception handling 88

7.8 Conclusions and summary 90

8 Numerical Linear Algebra 91

8.1 Introduction and objectives 91

8.2 An introduction to numerical linear algebra 91

8.3 Tridiagonal systems 94

8.4 Block tridiagonal systems 99

8.5 What requirements should our matrix satisfy? 101

8.6 Conclusions and summary 102

9 Modelling Functions in C++ 103

9.1 Introduction and objectives 103

9.2 Function pointers in C++ 103

9.3 Function objects in STL 106

9.4 Some function types 109

9.5 Creating your own function classes 111

9.6 Arrays of functions 114

9.7 Vector functions 115

9.8 Real-valued functions 115

9.9 Vector-valued functions 116

9.10 Conclusions and summary 116

10 C++ Classes for Statistical Distributions 117

10.1 Introduction and objectives 117

10.2 Discrete and continuous probability distribution functions 117

10.3 Continuous distributions 119

10.4 Discrete distributions 124

10.5 Tests 127

10.6 Conclusions and summary 128

PART III ORDINARY AND STOCHASTIC DIFFERENTIAL EQUATIONS

11 Numerical Solution of Initial Value Problems: Fundamentals 131

11.1 Introduction and objectives 131

11.2 A model problem 132

11.3 Discretisation 133

11.4 Common schemes 134

11.5 Some theoretical issues 136

11.6 Fitting: Special schemes for difficult problems 137

11.7 Non-linear scalar problems and predictor–corrector methods 138

11.8 Extrapolation techniques 139

11.9 C++ design and implementation 140

11.10 Generalisations 143

11.11 Conclusions and summary 144

12 Stochastic Processes and Stochastic Differential Equations 145

12.1 Introduction and objectives 145

12.2 Random variables and random processes 145

12.3 An introduction to stochastic differential equations 151

12.4 Some finite difference schemes 152

12.5 Which scheme to use? 153

12.6 Systems of SDEs 154

12.7 Conclusions and summary 154

13 Two-Point Boundary Value Problems 155

13.1 Introduction and objectives 155

13.2 Description of problem 155

13.3 (Traditional) centred-difference schemes 157

13.4 Approximation of the boundary conditions 158

13.5 Exponentially fitted schemes and convection–diffusion 160

13.6 Approximating the derivatives 160

13.7 Design issues 161

13.8 Conclusions and summary 163

14 Matrix Iterative Methods 164

14.1 Introduction and objectives 164

14.2 Iterative methods 165

14.3 The Jacobi method 165

14.4 Gauss–Seidel method 166

14.5 Successive overrelaxation (SOR) 166

14.6 Other methods 166

14.7 The linear complementarity problem 168

14.8 Implementation 169

14.9 Conclusions and summary 171

PART IV PROGRAMMING THE BLACK–SCHOLES ENVIRONMENT

15 An Overview of Computational Finance 175

15.1 Introduction and objectives 175

15.2 The development life cycle 175

15.3 Partial differential equations 176

15.4 Numerical approximation of PDEs 177

15.5 The class of finite difference schemes 179

15.6 Special schemes for special problems 179

15.7 Implementation issues and the choice of programming language 180

15.8 Origins and application areas 180

15.9 Conclusions and summary 181

16 Finite Difference Schemes for Black–Scholes 182

16.1 Introduction and objectives 182

16.2 Model problem: The one-dimensional heat equation 182

16.3 The Black–Scholes equation 186

16.4 Initial conditions and exotic options payoffs 187

16.5 Implementation 190

16.6 Method of lines: A whirlwind introduction 190

16.7 Conclusions and summary 191

17 Implicit Finite Difference Schemes for Black–Scholes 192

17.1 Introduction and objectives 192

17.2 Fully implicit method 193

17.3 An introduction to the Crank–Nicolson method 194

17.4 A critique of Crank–Nicolson 195

17.5 Is there hope? the Keller scheme 199

17.6 Conclusions and summary 202

18 Special Schemes for Plain and Exotic Options 203

18.1 Introduction and objectives 203

18.2 Motivating exponentially fitted schemes 203

18.3 Exponentially fitted schemes for parabolic problems 205

18.4 What happens when the volatility goes to zero? 208

18.5 Exponential fitting with explicit time 209

18.6 Exponential fitting and exotic options 210

18.7 Some final remarks 211

19 My First Finite Difference Solver 212

19.1 Introduction and objectives 212

19.2 Modelling partial differential equations in C++ 214

19.3 Finite difference schemes as C++ classes, Part I 218

19.4 Finite difference schemes as C++ classes, Part II 219

19.5 Initialisation issues 220

19.6 Interfacing with Excel 224

19.7 Conclusions and summary 224

20 An Introduction to ADI and Splitting Schemes 225

20.1 Introduction and objectives 225

20.2 A model problem 226

20.3 Motivation and history 227

20.4 Basic ADI scheme for the heat equation 228

20.4.1 Three-dimensional heat equation 229

20.5 Basic splitting scheme for the heat equation 230

20.6 Approximating cross-derivatives 231

20.7 Handling boundary conditions 232

20.8 Algorithms and design issues 234

20.9 Conclusions and summary 236

21 Numerical Approximation of Two-Factor Derivative Models 237

21.1 Introduction and objectives 237

21.2 Two-factor models in financial engineering 237

21.3 Finite difference approximations 241

21.4 ADI schemes for Asian options 242

21.5 Splitting schemes 243

21.6 Conclusions and summary 243

PART V DESIGN PATTERNS

22 A C++ Application for Displaying Numeric Data 247

22.1 Introduction and objectives 247

22.2 Input mechanisms 248

22.3 Conversion and processing mechanisms 249

22.4 Output and display mechanisms 250

22.5 Putting it all together 252

22.6 Output 252

22.7 Other functionality 252

22.8 Using Excel and property sets 258

22.9 Extensions and the road to design patterns 259

22.10 Conclusions and summary 260

23 Object Creational Patterns 261

23.1 Introduction and objectives 261

23.2 The Singleton pattern 263

23.3 The Prototype pattern 270

23.4 Factory Method pattern (virtual constructor) 272

23.5 Abstract Factory pattern 275

23.6 Applications to financial engineering 279

23.7 Conclusions and summary 279

24 Object Structural Patterns 281

24.1 Introduction and objectives 281

24.2 Kinds of structural relationships between classes 281

24.3 Whole–Part pattern 286

24.4 The Composite pattern 288

24.5 The Fa¸cade pattern 289

24.6 The Bridge pattern 290

24.7 Conclusions and summary 295

25 Object Behavioural Patterns 296

25.1 Introduction and objectives 296

25.2 Kinds of behavioural patterns 297

25.3 Iterator pattern 298

25.4 The Visitor pattern 301

25.5 Notification patterns 305

25.6 Conclusions and summary 307

PART VI DESIGN AND DEPLOYMENT ISSUES

26 An Introduction to the Extensible Markup Language 311

26.1 Introduction and objectives 311

26.2 A short history of XML 312

26.3 The XML structure 312

26.4 Document Type Definition 315

26.5 Extensible Stylesheet Language Transformation (XSLT) 320

26.6 An application of XML: Financial products Markup Language 324

26.7 Conclusions and summary 327

27 Advanced XML and Programming Interface 328

27.1 Introduction and objectives 328

27.2 XML Schema 328

27.3 Accessing XML data: The Document Object Model 334

27.4 DOM and C++: The essentials 335

27.5 DOM, entities and property sets 338

27.6 XML structures for plain and barrier options 342

27.7 Conclusions and summary 345

28 Interfacing C++ and Excel 346

28.1 Introduction and objectives 346

28.2 Object model in Excel: An overview 346

28.3 Under the bonnet: Technical details of C++ interfacing to Excel 348

28.4 Implementing the core process 351

28.5 Extensions 354

28.6 Application areas 355

28.7 Conclusions and summary 355

29 Advanced Excel Interfacing 356

29.1 Introduction and objectives 356

29.2 Status report and new requirements 356

29.3 A gentle introduction to Excel add-ins 357

29.4 Automation add-in in detail 359

29.5 Creating a COM add-in 367

29.6 Future trends 373

29.7 Conclusions and summary 373

30 An Extended Application: Option Strategies and Portfolios 374

30.1 Introduction and objectives 374

30.2 Spreads 374

30.3 Combinations: Straddles and strangles 375

30.4 Designing and implementing spreads 376

30.5 Delta hedging 378

30.6 An example 379

30.7 Tips and guidelines 381

Appendices

A1 My C++ refresher 383

A2 Dates and other temporal types 394

References 397

Index 401