Designing and Implementing Software for Financial Instrument Pricing / Edition 1

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Overview

One of the best languages for the development of financial engineering and instrument pricing applications is C++. This book has several features that allow developers to write robust, flexible and extensible software systems. The book is an ANSI/ISO standard, fully object-oriented and interfaces with many third-party applications. It has support for templates and generic programming, massive reusability using templates (?write once?) and support for legacy C applications.

In this book, author Daniel J. Duffy brings C++ to the next level by applying it to the design and implementation of classes, libraries and applications for option and derivative pricing models. He employs modern software engineering techniques to produce industrial-strength applications:

  • Using the Standard Template Library (STL) in finance
  • Creating your own template classes and functions
  • Reusable data structures for vectors, matrices and tensors
  • Classes for numerical analysis (numerical linear algebra ?)
  • Solving the Black Scholes equations, exact and approximate solutions
  • Implementing the Finite Difference Method in C++
  • Integration with the ?Gang of Four? Design Patterns
  • Interfacing with Excel (output and Add-Ins)
  • Financial engineering and XML
  • Cash flow and yield curves

Included with the book is a CD containing the source code in the Datasim Financial Toolkit. You can use this to get up to speed with your C++ applications by reusing existing classes and libraries.

'Unique... Let's all give a warm welcome to modern pricing tools.'
-- Paul Wilmott, mathematician, author and fund manager

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Product Details

  • ISBN-13: 9780470855096
  • Publisher: Wiley
  • Publication date: 5/14/2004
  • Series: Wiley Finance Series , #321
  • Edition description: BOOK & CD
  • Edition number: 1
  • Pages: 432
  • Product dimensions: 6.90 (w) x 9.80 (h) x 1.30 (d)

Meet the Author

Daniel Duffy works for Datasim, an Amsterdam-based trainer and software developer (www.datasim-component.com, www.datasim.nl). He has been working in IT since 1979 and with object-oriented technology since 1987. He received his MSc and PhD theses (in numerical analysis) from Trinity College, Dublin. His current interests are in the modelling of financial instruments using numerical methods (for example, finite difference method) and C++. He can be contacted at dduffy@datasim.nl
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Table of Contents

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

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First Chapter

Financial Instrument Pricing Using C++


By Daniel J. Duffy

John Wiley & Sons

ISBN: 0-470-85509-6


Chapter One

Executive Overview of this Book

1.1 WHAT IS THIS BOOK?

The goal of this book is to model financial instruments, such as options, bonds and interest-rate products by partial differential equations, finite differences and C++. It is intended for IT and quantitative finance professionals who know this material and wish to deepen their knowledge and for those readers who use techniques such as Monte Carlo, Fourier transform methods, stochastic differential equations and lattice methods (for example, the binomial method) for instrument pricing.

We integrate a number of well-known areas to create a traceable and maintainable path from when a financial engineer proposes a new model to when he or she codes the resulting equations in C++. When viewed as a black box, the core process in this book is to produce C++ classes and code for financial engineering applications. Furthermore, we give lots of examples of code that developers can use without much hassle. The accompanying CD contains all the source code in this book. We provide guidelines, algorithms and reusable code to help the reader to achieve these ends. The main activities that realise the core process are:

Activity 1: Map the financial model to a partial differential equation (PDE)

Activity 2: Approximate the PDE by the finite difference method (FDM)

Activity 3: Implement the FDM usingC++ and design patterns.

In this book we shall concentrate on Activities 2 and 3. Since this is a book on the application of C++ to financial engineering we concentrate on mapping the numerical algorithms from Activity 2 to robust and flexible C++ code and classes. However, we shall provide sufficient motivation and background information to help the reader to understand the complete 'instrument life cycle'. This life cycle describes the processes, activities, decisions and alternatives that describe how to program models for financial instruments in C++.

The topics in this book relate to finance, partial differential equations, numerical schemes and C++ code, and for this reason we use the term Computational Finance to sum up these related activities (see Seydel, 2003, where the same phrase is used). The foundations for partial differential equations and finite difference schemes for financial engineering applications are discussed in Duffy (2004b).

1.2 WHAT'S SPECIAL ABOUT THIS BOOK?

This book is part of a larger, ongoing project. It is the outcome of one part of this project and concentrates on showing how to program finite difference schemes in C++. Our approach is novel in a number of respects.

1. We use modern object-oriented and generic design patterns in C++ to solve a range of partial, stochastic and ordinary differential equations in financial engineering. Traditionally, engineers have used packages such as Matlab, Maple, the C language or other specialised libraries. Each alternative solution has its own benefits of course, but using C++ means that your code is portable, flexible and future-proof (C++ will still be used 20 years from now). Using C++ means that you are not tied into one vendor or operating system.

2. We give a thorough introduction to finite difference methods, how to apply them to Black-Scholes type equations and how to map them to C++ code. We avoid glib recipe-type schemes that work well for toy problems but do not always scale to real-life problems. In particular, we show how to program the famous Crank-Nicolson scheme and discuss when it breaks down, especially in applications with small volatility, discontinuous payoff functions or non-linearities. We propose new schemes that overcome these problems and produce uniformly good approximations to the delta of an option. The book discusses finite difference schemes for both one-factor and two-factor problems.

3. Successful software always needs to be adapted and extended, and to this end we design our classes and applications so that they can easily be modified. Our book is novel in the sense that we apply the powerful and proven design patterns (see Gamma et al., 1995; Buschmann et al., 1996) to help us to produce applications that can be extended or adapted to different organisational, hardware and software contexts.

4. Last, but not least, it is vital that our software artefacts are well documented. We document these artefacts at the design level and, in particular, we use the de-facto Unified Modeling Language (UML) to visually display the structural, functional and behavioural relationships between the classes and objects in our applications.

In short, this book describes in a step-by-step manner how to create 'good' software for financial engineering applications; it also integrates established techniques from fluid mechanics, numerical analysis and software design to produce a coherent and seamless approach to the design and implementation of financial models in C++.

1.3 WHO IS THIS BOOK FOR?

This book is meant for IT and quantitative finance professionals (risk managers, product development and derivatives research groups) who work in financial institutions and software companies and are involved in designing and implementing pricing models in C++. This book deals with fundamental issues such as C++ design and implementation, design patterns, finite difference methods and advanced software environments. Thus, it is of value to financial engineers ('Quants'), software developers and financial modellers.

We feel that the book is useful for universities and other educational institutes that deliver financial courses. This is not a book on instrument theory as such, and we assume that the reader has knowledge of option theory as presented in books by Hull (2000) and Wilmott (1998), for example. We also assume that the reader has had some exposure to differential equations, differential and integral calculus and matrix algebra. Finally, the reader should have a working knowledge of C++.

As we have already mentioned in this chapter, the book is suited not only to those readers from a partial differential equation (PDE) background but also to those who use techniques such as Monte Carlo, Fourier transform methods, stochastic differential equations (SDEs) and the binomial method for instrument pricing. We do our best to show that finite differences compare well with, and even outperform, these former methods, especially for complex and non-linear one-factor and two-factor Black-Scholes models.

Finally, the real option theory is emerging and many of the techniques in this book can be used in decision support systems in the oil, gas and energy industries. Thus, the book is also of interest to engineers, scientists and financial engineers in these fields.

1.4 SOFTWARE REQUIREMENTS

We have written this book from a number of viewpoints that have to do with what we call software quality. In general, we adopt the ISO 9126 quality characteristics (see Kitchenham and Pfleeger, 1996) as our working model. ISO 9126 describes how good a software product is. It consists of six top-level characteristics:

Functionality: The ability of the software to satisfy stated or implied customer needs.

Reliability: Does the software maintain its level of performance for a stated period of time?

Usability: Is the software easy to understand, learn or integrate with other applications?

Efficiency: Describes the response times in the application and the corresponding resources that are needed.

Maintainability: How easy is it to modify, adapt and test the application? How stable is the application under change?

Portability: The ease with which the application can be adapted to work in some new software or hardware environment.

Any one (or all) of the above requirements may be important for your new or existing software project. In general, the more requirements your applications must satisfy the more time it will take to satisfy them. In this book we classify applications into three broad categories, depending on the level of flexibility that they must have:

Low flexibility: These are either throwaway prototypes or simple programs in order to test a piece of code or check the validity of some new model

Medium flexibility: The code and classes in this category can be customised (by changing its source code if necessary) and used in your own applications

High flexibility: The code in this category can be used in your applications without any changes.

It is important to know at the outset how flexible our solutions must be; on the one hand, we do not want to 'over-engineer' our application, but nor do we want to produce code that is difficult to maintain, understand or falls apart when we modify it. This book will provide you with guidelines to help you to produce good designs for financial engineering applications.

We layer the software in this book by examining it at four different levels:

Foundation classes and building blocks: Reusable components for vectors, lists, matrices and other containers. We make ample use of the Standard Template Library (STL).

Mechanisms: Tightly coupled groups of generic functions that are related to a specific piece of functionality. An example is a set of functions for Date manipulations (cash flows, interest rate curves).

Half-products: Ready-to-use libraries that you can use as part of your own applications. We can place these half-products in assemblies and DLLs.

Applications: Dedicated applications for the user (not the developer). These applications are usually executables.

There are many advantages associated with taking this layered approach to software development, as we shall see in this book.

1.5 THE STRUCTURE OF THIS BOOK

This book is partitioned into six major parts, each of which deals with a major topic and consists of a number of chapters. These chapters deal with techniques that help to achieve the goals of each part.

Part I This part is an introduction to C++ template classes. We define what templates are, how to create them and how to use them in financial engineering applications. We give an overview of the Standard Template Library (STL). This is a C++ library consisting of template classes for a wide range of data containers, algorithms and functionality for navigating in these containers. We develop a number of template classes based on STL that we can use in financial engineering applications.

Part II In this part we create classes and code that will be used when approximating partial differential equations by finite difference methods. First, we create template classes for arrays, vectors and matrices as well as the corresponding mathematical operations on them. Furthermore, we introduce several classes that solve linear systems of equations. These classes implement direct and iterative matrix solvers in numerical linear algebra. Second, we create a number of other foundation classes that we need in numerical differentiation and integration. Finally, some useful classes for statistics are introduced.

Part III This part represents the start of the book as far as the mathematical core is concerned. We motivate the finite difference method by applying it to a simple first-order ordinary differential equation in Chapter 11. This chapter discusses the most important ideas and schemes that will serve us well in later chapters. Continuing from Chapter 11, we introduce stochastic differential equations and the finite difference schemes needed in order to approximate them. We also propose several schemes to approximate two-point boundary value problems. Special attention is paid to the Crank-Nicolson scheme and why it fails to approximate the solution of the convection-diffusion equation in certain circumstances. It is in this part of the book that we introduce the class of exponentially fitted schemes and explain how they resolve the spurious oscillation problems associated with Crank-Nicolson.

Part IV In this part we introduce the one-factor and two-factor Black-Scholes equations and devise appropriate finite difference schemes for them. Before we reach this level of Nirvana, we begin with the one-dimensional heat equation and discuss explicit and implicit finite difference schemes to approximate its solution. The schemes are extensions of the time-independent schemes that we introduced in Part III. Slightly increasing the level of difficulty, we discuss the Crank-Nicolson and fully implicit schemes for the one-dimensional convection-diffusion equation (and its specialisation, the Black-Scholes equation). We analyse the schemes in some detail, discussing why they work, when they do not work and how to produce fitted schemes that approximate the solution and the delta of the Black-Scholes equation.

Proceeding to two-factor problems, we propose Alternating Direction Implicit (ADI) and splitting methods and compare their relative merits.

Part V In this part we give an introduction to design patterns. Design is about alternatives and we have many choices when designing a system as the choices are determined by the software requirements. We begin with an introduction to some general design principles. In particular, we focus on templates and inheritance and why they are competitors. We also introduce the important notion of delegation whose understanding is fundamental to design patterns.

The main objective in Part V is to show how the famous design patterns of GOF (see Gamma et al., 1995) are applied to financial engineering applications. We pay special attention to choosing appropriate examples and to a discussion of the advantages of design patterns in this particular context. Three chapters are devoted to the Creational, Structural and Behavioural patterns.

Part VI This part contains a number of chapters that are of particular interest to financial engineers and IT personnel who write financial engineering applications. First, we give an introduction to the Extensible Markup Language (XML), a W3C standard for interoperable data representation. We also describe how it is used in option pricing applications in this book. XML will become more important in the financial world in the coming years as evidenced by the work seen with FpML and FIX. We also discuss classes and code that allow C++ code to communicate with Excel. Finally, we introduce a number of design patterns that are very useful for the current work.

1.6

Continues...


Excerpted from Financial Instrument Pricing Using C++ by Daniel J. Duffy Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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