Finite Difference Methods in Financial Engineering

The world of quantitative finance (QF) is one of the fastest
growing areas of research and its practical applications to
derivatives pricing problem. Since the discovery of the famous
Black-Scholes equation in the 1970's we have seen a surge in the
number of models for a wide range of products such as plain and
exotic options, interest rate derivatives, real options and many
others. Gone are the days when it was possible to price these
derivatives analytically. For most problems we must resort to some
kind of approximate method.

In this book we employ partial differential equations (PDE) to
describe a range of one-factor and multi-factor derivatives
products such as plain European and American options, multi-asset
options, Asian options, interest rate options and real options. PDE
techniques allow us to create a framework for modeling complex and
interesting derivatives products. Having defined the PDE problem we
then approximate it using the Finite Difference Method (FDM). This
method has been used for many application areas such as fluid
dynamics, heat transfer, semiconductor simulation and astrophysics,
to name just a few. In this book we apply the same techniques to
pricing real-life derivative products. We use both traditional (or
well-known) methods as well as a number of advanced schemes that
are making their way into the QF literature:

• Crank-Nicolson, exponentially fitted and higher-order schemes
  for one-factor and multi-factor options

• Early exercise features and approximation using front-fixing,
  penalty and variational methods

• Modelling stochastic volatility models using Splitting
  methods

• Critique of ADI and Crank-Nicolson schemes; when they work and
  when they don't work

• Modelling jumps using Partial Integro Differential Equations
  (PIDE)

• Free and moving boundary value problems in QF

Included with the book is a CD containing information on how to
set up FDM algorithms, how to map these algorithms to C++ as well
as several working programs for one-factor and two-factor models.
We also provide source code so that you can customize the
applications to suit your own needs.

Autorentext
Daniel Duffy is a numerical analyst who has been working in the IT business since 1979. He has been involved in the analysis, design and implementation of systems using object-oriented, component and (more recently) intelligent agent technologies to large industrial and financial applications. As early as 1993 he was involved in C++ projects for risk management and options applications with a large Dutch bank. His main interest is in finding robust and scalable numerical schemes that approximate the partial differential equations that model financial derivatives products. He has an M.Sc. in the Finite Element Method first-order hyperbolic systems and a Ph.D. in robust finite difference methods for convection-diffusion partial differential equations. Both degrees are from Trinity College, Dublin, Ireland.
Daniel Duffy is founder of Datasim Education and Datasim Component Technology, two companies involved in training, consultancy and software development.

Klappentext
The world of quantitative finance (QF) is one of the fastest growing areas of research and its practical applications to derivatives pricing problem. Since the discovery of the famous Black-Scholes equation in the 1970's we have seen a surge in the number of models for a wide range of products such as plain and exotic options, interest rate derivatives, real options and many others. Gone are the days when it was possible to price these derivatives analytically. For most problems we must resort to some kind of approximate method. In this book we employ partial differential equations (PDE) to describe a range of one-factor and multi-factor derivatives products such as plain European and American options, multi-asset options, Asian options, interest rate options and real options. PDE techniques allow us to create a framework for modeling complex and interesting derivatives products. Having defined the PDE problem we then approximate it using the Finite Difference Method (FDM). This method has been used for many application areas such as fluid dynamics, heat transfer, semiconductor simulation and astrophysics, to name just a few. In this book we apply the same techniques to pricing real-life derivative products. We use both traditional (or well-known) methods as well as a number of advanced schemes that are making their way into the QF literature:

• Crank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options
• Early exercise features and approximation using front-fixing, penalty and variational methods
• Modelling stochastic volatility models using Splitting methods
• Critique of ADI and Crank-Nicolson schemes; when they work and when they don't work
• Modelling jumps using Partial Integro Differential Equations (PIDE)
• Free and moving boundary value problems in QF
  Included with the book is a CD containing information on how to set up FDM algorithms, how to map these algorithms to C++ as well as several working programs for one-factor and two-factor models. We also provide source code so that you can customize the applications to suit your own needs.

Inhalt

0 Goals of this Book and Global Overview 1

0.1 What is this book? 1

0.2 Why has this book been written? 2

0.3 For whom is this book intended? 2

0.4 Why should I read this book? 2

0.5 The structure of this book 3

0.6 What this book does not cover 4

0.7 Contact, feedback and more information 4

Part I The Continuous Theory of Partial Differential Equations 5

1 An Introduction to Ordinary Differential Equations 7

1.1 Introduction and objectives 7

1.2 Two-point boundary value problem 8

1.3 Linear boundary value problems 9

1.4 Initial value problems 10

1.5 Some special cases 10

1.6 Summary and conclusions 11

2 An Introduction to Partial Differential Equations 13

2.1 Introduction and objectives 13

2.2 Partial differential equations 13

2.3 Specialisations 15

2.4 Parabolic partial differential equations 18

2.5 Hyperbolic equations 20

2.6 Systems of equations 22

2.7 Equations containing integrals 23

2.8 Summary and conclusions 24

3 Second-Order Parabolic Differential Equations 25

3.1 Introduction and objectives 25

3.2 Linear parabolic equations 25

3.3 The continuous problem 26

3.4 The maximum principle for parabolic equations 28

3.5 A special case: one-factor generalised BlackScholes models 29

3.6 Fundamental solution and the Green's function 30

3.7 Integral representation of the solution of parabolic PDEs 31

3.8 Parabolic equations in one space dimension 33

3.9 Summary and conclusions 35

4 An Introduction to the Heat Equation in One Dimension 37

4.1 Introduction and objectives 37

4.2 Motivation and background 38

4.3 The heat equation and financial engineering 39

4.4 The separation of variables technique 40

4.5 Transformation techniques for the heat equation 44

4.6 Summary and conclusions 46

5 An Introduction to the Method of Characteristics 47

5.1 Introduction and objectives 47

5.2 First-order hyperbolic equations 47

5.3 Second-order hyperbolic equations 50

5.4 Applications to financial engineering 53

5.5 Systems of equations 55

5.6 Propagation of discontinuities 57

5.7 Summary and conclusions 59

Part II Finite Difference Methods: the Fundamentals 61

6 An Introduction to the Finite Difference Method 63

6.1 Introduction and objectives 63

6.2 Fundamentals of numerical differentiation 63

6.3 Caveat: accuracy and round-off errors 65

6.4 Where are divided differences used in instrument pricing? 67

6.5 Initial value p…